3.20.71 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=342 \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (b e g-8 c d g+6 c e f)}{e^2 (d+e x)^2 (2 c d-b e)}-\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-8 c d g+6 c e f)}{3 e^2 (2 c d-b e)}-\frac {5 (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-8 c d g+6 c e f)}{8 e}-\frac {5 (2 c d-b e)^2 (b e g-8 c d g+6 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 \sqrt {c} e^2} \]
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Rubi [A] time = 0.60, antiderivative size = 342, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 6, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used =
{792, 662, 664, 612, 621, 204} \begin {gather*} -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (b e g-8 c d g+6 c e f)}{e^2 (d+e x)^2 (2 c d-b e)}-\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-8 c d g+6 c e f)}{3 e^2 (2 c d-b e)}-\frac {5 (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-8 c d g+6 c e f)}{8 e}-\frac {5 (2 c d-b e)^2 (b e g-8 c d g+6 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 \sqrt {c} e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^4,x]
[Out]
(-5*(6*c*e*f - 8*c*d*g + b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*e) - (5*c*(6*c*e*f -
8*c*d*g + b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)) - (2*(6*c*e*f - 8*c*d*g +
b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^2) - (2*(e*f - d*g)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^4) - (5*(2*c*d - b*e)^2*(6*c*e*f - 8*c*d*g +
b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(16*Sqrt[c]*e^2)
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 612
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]
Rule 621
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 662
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]
Rule 664
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]
Rule 792
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac {(6 c e f-8 c d g+b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx}{e (2 c d-b e)}\\ &=-\frac {2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac {(5 c (6 c e f-8 c d g+b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{d+e x} \, dx}{e (2 c d-b e)}\\ &=-\frac {5 c (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}-\frac {2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac {(5 c (6 c e f-8 c d g+b e g)) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{2 e}\\ &=-\frac {5 (6 c e f-8 c d g+b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e}-\frac {5 c (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}-\frac {2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac {\left (5 (2 c d-b e)^2 (6 c e f-8 c d g+b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 e}\\ &=-\frac {5 (6 c e f-8 c d g+b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e}-\frac {5 c (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}-\frac {2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac {\left (5 (2 c d-b e)^2 (6 c e f-8 c d g+b e g)\right ) \operatorname {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{8 e}\\ &=-\frac {5 (6 c e f-8 c d g+b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e}-\frac {5 c (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}-\frac {2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac {5 (2 c d-b e)^2 (6 c e f-8 c d g+b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 \sqrt {c} e^2}\\ \end {align*}
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Mathematica [A] time = 1.89, size = 309, normalized size = 0.90 \begin {gather*} \frac {2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (e^3 (e f-d g) (b e-c d+c e x)^3-\frac {e^{5/2} \sqrt {d+e x} (b e g-8 c d g+6 c e f) \left (\sqrt {c} \sqrt {e} \sqrt {d+e x} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} \left (33 b^2 e^2+2 b c e (13 e x-53 d)+4 c^2 \left (22 d^2-9 d e x+2 e^2 x^2\right )\right )+15 \sqrt {e (2 c d-b e)} (b e-2 c d)^2 \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )\right )}{48 \sqrt {c} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}\right )}{e^5 (d+e x)^3 (2 c d-b e) (b e-c d+c e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^4,x]
[Out]
(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(e^3*(e*f - d*g)*(-(c*d) + b*e + c*e*x)^3 - (e^(5/2)*(6*c*e*f - 8*
c*d*g + b*e*g)*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]*(33*b^
2*e^2 + 2*b*c*e*(-53*d + 13*e*x) + 4*c^2*(22*d^2 - 9*d*e*x + 2*e^2*x^2)) + 15*Sqrt[e*(2*c*d - b*e)]*(-2*c*d +
b*e)^2*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]]))/(48*Sqrt[c]*Sqrt[(-(c*d) + b*e + c*e*x)
/(-2*c*d + b*e)])))/(e^5*(2*c*d - b*e)*(d + e*x)^3*(-(c*d) + b*e + c*e*x)^2)
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IntegrateAlgebraic [F] time = 180.10, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^4,x]
[Out]
$Aborted
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fricas [A] time = 2.17, size = 931, normalized size = 2.72 \begin {gather*} \left [-\frac {15 \, {\left (6 \, {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (32 \, c^{3} d^{4} - 36 \, b c^{2} d^{3} e + 12 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} g + {\left (6 \, {\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (32 \, c^{3} d^{3} e - 36 \, b c^{2} d^{2} e^{2} + 12 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} g\right )} x\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (8 \, c^{3} e^{3} g x^{3} + 2 \, {\left (6 \, c^{3} e^{3} f - {\left (20 \, c^{3} d e^{2} - 13 \, b c^{2} e^{3}\right )} g\right )} x^{2} - 6 \, {\left (48 \, c^{3} d^{2} e - 41 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + {\left (376 \, c^{3} d^{3} - 352 \, b c^{2} d^{2} e + 81 \, b^{2} c d e^{2}\right )} g - {\left (6 \, {\left (14 \, c^{3} d e^{2} - 9 \, b c^{2} e^{3}\right )} f - {\left (136 \, c^{3} d^{2} e - 134 \, b c^{2} d e^{2} + 33 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{96 \, {\left (c e^{3} x + c d e^{2}\right )}}, \frac {15 \, {\left (6 \, {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (32 \, c^{3} d^{4} - 36 \, b c^{2} d^{3} e + 12 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} g + {\left (6 \, {\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (32 \, c^{3} d^{3} e - 36 \, b c^{2} d^{2} e^{2} + 12 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, {\left (8 \, c^{3} e^{3} g x^{3} + 2 \, {\left (6 \, c^{3} e^{3} f - {\left (20 \, c^{3} d e^{2} - 13 \, b c^{2} e^{3}\right )} g\right )} x^{2} - 6 \, {\left (48 \, c^{3} d^{2} e - 41 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + {\left (376 \, c^{3} d^{3} - 352 \, b c^{2} d^{2} e + 81 \, b^{2} c d e^{2}\right )} g - {\left (6 \, {\left (14 \, c^{3} d e^{2} - 9 \, b c^{2} e^{3}\right )} f - {\left (136 \, c^{3} d^{2} e - 134 \, b c^{2} d e^{2} + 33 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, {\left (c e^{3} x + c d e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="fricas")
[Out]
[-1/96*(15*(6*(4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (32*c^3*d^4 - 36*b*c^2*d^3*e + 12*b^2*c*d^2*e^
2 - b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (32*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 12*b^
2*c*d*e^3 - b^3*e^4)*g)*x)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 4*sqrt
(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(8*c^3*e^3*g*x^3 + 2*(6*c^3*e^3*f - (20*c
^3*d*e^2 - 13*b*c^2*e^3)*g)*x^2 - 6*(48*c^3*d^2*e - 41*b*c^2*d*e^2 + 8*b^2*c*e^3)*f + (376*c^3*d^3 - 352*b*c^2
*d^2*e + 81*b^2*c*d*e^2)*g - (6*(14*c^3*d*e^2 - 9*b*c^2*e^3)*f - (136*c^3*d^2*e - 134*b*c^2*d*e^2 + 33*b^2*c*e
^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2), 1/48*(15*(6*(4*c^3*d^3*e - 4*b*c^2*
d^2*e^2 + b^2*c*d*e^3)*f - (32*c^3*d^4 - 36*b*c^2*d^3*e + 12*b^2*c*d^2*e^2 - b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2
- 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (32*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 12*b^2*c*d*e^3 - b^3*e^4)*g)*x)*sqrt(c)*ar
ctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2
+ b*c*d*e)) + 2*(8*c^3*e^3*g*x^3 + 2*(6*c^3*e^3*f - (20*c^3*d*e^2 - 13*b*c^2*e^3)*g)*x^2 - 6*(48*c^3*d^2*e -
41*b*c^2*d*e^2 + 8*b^2*c*e^3)*f + (376*c^3*d^3 - 352*b*c^2*d^2*e + 81*b^2*c*d*e^2)*g - (6*(14*c^3*d*e^2 - 9*b*
c^2*e^3)*f - (136*c^3*d^2*e - 134*b*c^2*d*e^2 + 33*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)
)/(c*e^3*x + c*d*e^2)]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((16*exp(1)^4*c^4*g*1/96/exp(1)^4/c^2*
x-(-24*exp(1)^4*c^4*f-52*exp(1)^4*c^3*g*b+96*exp(1)^3*c^4*g*d)*1/96/exp(1)^4/c^2)*x-(-108*exp(1)^4*c^3*b*f-66*
exp(1)^4*c^2*g*b^2+192*exp(1)^3*c^4*d*f+320*exp(1)^3*c^3*g*b*d-368*exp(1)^2*c^4*g*d^2)*1/96/exp(1)^4/c^2)*sqrt
(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))+2*((-5*sqrt(-c*exp(2))*b^3*g*exp(1)^3+60*c*sqrt(-c*exp(2))*b^2*g*e
xp(1)^2*d-30*c*sqrt(-c*exp(2))*b^2*exp(1)^3*f-180*c^2*sqrt(-c*exp(2))*b*g*exp(1)*d^2+120*c^2*sqrt(-c*exp(2))*b
*exp(1)^2*d*f+160*c^3*sqrt(-c*exp(2))*g*d^3-120*c^3*sqrt(-c*exp(2))*exp(1)*d^2*f)/32/c/exp(2)/exp(1)*ln(abs(2*
c*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)-sqrt(-c*exp(2))*b))-(30*exp(2)^2*b^3*g*e
xp(1)^5*d-35*exp(2)^3*b^3*g*exp(1)^3*d+5*exp(2)^3*b^3*exp(1)^4*f+40*c*exp(2)*b^2*g*exp(1)^6*d^2-290*c*exp(2)^2
*b^2*g*exp(1)^4*d^2+280*c*exp(2)^3*b^2*g*exp(1)^2*d^2+60*c*exp(2)^2*b^2*exp(1)^5*d*f-90*c*exp(2)^3*b^2*exp(1)^
3*d*f-80*c^2*exp(2)*b*g*exp(1)^5*d^3+580*c^2*exp(2)^2*b*g*exp(1)^3*d^3-560*c^2*exp(2)^3*b*g*exp(1)*d^3-180*c^2
*exp(2)^2*b*exp(1)^4*d^2*f+240*c^2*exp(2)^3*b*exp(1)^2*d^2*f+40*c^3*exp(2)*g*exp(1)^4*d^4-320*c^3*exp(2)^2*g*e
xp(1)^2*d^4+320*c^3*exp(2)^3*g*d^4+120*c^3*exp(2)^2*exp(1)^3*d^3*f-160*c^3*exp(2)^3*exp(1)*d^3*f)/8/exp(1)^7/2
/sqrt(b*d*exp(1)^3-c*d^2*exp(1)^2+c*d^2*exp(2)-b*d*exp(1)*exp(2))*atan((-d*sqrt(-c*exp(2))+(sqrt(-b*d*exp(1)-b
*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*exp(1))/sqrt(b*d*exp(1)^3-c*d^2*exp(1)^2+c*d^2*exp(2)-b*d*exp
(1)*exp(2)))-(-905969664*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^3*g*
exp(1)^7*d+1459617792*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^3*g*exp
(1)^5*d-553648128*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^3*exp(1)^6*
f-402653184*c*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^2*g*exp(1)^8*d^2+
7147094016*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^2*g*exp(1)^6*d^2
-10066329600*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^2*g*exp(1)^4*d
^2-1811939328*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^2*exp(1)^7*d*
f+5133828096*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b^2*exp(1)^5*d*f
+805306368*c^2*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b*g*exp(1)^7*d^3-1
4294188032*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b*g*exp(1)^5*d^3
+20132659200*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b*g*exp(1)^3*d
^3+5435817984*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b*exp(1)^6*d^
2*f-12079595520*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*b*exp(1)^4*
d^2*f-402653184*c^3*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*exp(1)^6*d^
4+8053063680*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*exp(1)^4*d^4
-12079595520*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*g*exp(1)^2*d^4
-3623878656*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*exp(1)^5*d^3*f+
8053063680*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^5*exp(1)^3*d^3*f+2
415919104*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^3*g*e
xp(1)^8*d^2-2717908992*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2
))*x)^4*b^3*g*exp(1)^6*d^2-2466250752*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2)
)-sqrt(-c*exp(2))*x)^4*b^3*g*exp(1)^4*d^2+2415919104*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d
^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^3*exp(1)^7*d*f+352321536*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*
x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^3*exp(1)^5*d*f-10066329600*c*exp(2)*sqrt(-c*exp(2))*(sqrt(
-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^2*g*exp(1)^7*d^3+503316480*c*exp(2)^2*sqrt(-
c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^2*g*exp(1)^5*d^3+26172456960
*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^2*g*exp(1)
^3*d^3+2415919104*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)
^4*b^2*exp(1)^8*d^2*f-7851737088*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-s
qrt(-c*exp(2))*x)^4*b^2*exp(1)^6*d^2*f-11173625856*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d
^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^2*exp(1)^4*d^2*f+12884901888*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp
(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b*g*exp(1)^6*d^4+18320719872*c^2*exp(2)^2*sqrt(-c*exp(
2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b*g*exp(1)^4*d^4-64424509440*c^2*exp
(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b*g*exp(1)^2*d^4-4
831838208*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b*e
xp(1)^7*d^3*f+1811939328*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c
*exp(2))*x)^4*b*exp(1)^5*d^3*f+36238786560*c^2*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x
^2*exp(2))-sqrt(-c*exp(2))*x)^4*b*exp(1)^3*d^3*f-5234491392*c^3*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*e
xp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*exp(1)^5*d^5-16106127360*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-
b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*exp(1)^3*d^5+43486543872*c^3*exp(2)^3*sqrt(-c
*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*g*exp(1)*d^5+2415919104*c^3*exp
(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*exp(1)^6*d^4*f+36238
78656*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*exp(1
)^4*d^4*f-28185722880*c^3*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*ex
p(2))*x)^4*exp(1)^2*d^4*f-1610612736*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))
*x)^3*b^4*g*exp(1)^8*d^2+3892314112*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*
x)^3*b^4*g*exp(1)^6*d^2-2281701376*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x
)^3*b^4*g*exp(1)^4*d^2-671088640*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^
3*b^4*exp(1)^7*d*f+671088640*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^
4*exp(1)^5*d*f+20132659200*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^
3*g*exp(1)^7*d^3-45566918656*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*
b^3*g*exp(1)^5*d^3+19897778176*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^
3*b^3*g*exp(1)^3*d^3-805306368*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^
3*b^3*exp(1)^8*d^2*f+15971909632*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x
)^3*b^3*exp(1)^6*d^2*f-9630121984*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*
x)^3*b^3*exp(1)^4*d^2*f-68451041280*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(
2))*x)^3*b^2*g*exp(1)^6*d^4+133882183680*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c
*exp(2))*x)^3*b^2*g*exp(1)^4*d^4-32212254720*c^2*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqr
t(-c*exp(2))*x)^3*b^2*g*exp(1)^2*d^4+12884901888*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))
-sqrt(-c*exp(2))*x)^3*b^2*exp(1)^7*d^3*f-68048388096*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp
(2))-sqrt(-c*exp(2))*x)^3*b^2*exp(1)^5*d^3*f+21944598528*c^2*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2
*exp(2))-sqrt(-c*exp(2))*x)^3*b^2*exp(1)^3*d^3*f+82946555904*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c
*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b*g*exp(1)^5*d^5-129788542976*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^
2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b*g*exp(1)^3*d^5-19595788288*c^3*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*
d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b*g*exp(1)*d^5-23353884672*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*
d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b*exp(1)^6*d^4*f+83080773632*c^3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+
c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b*exp(1)^4*d^4*f+6710886400*c^3*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)
+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b*exp(1)^2*d^4*f-33017561088*c^4*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*g*exp(1)^4*d^6+37580963840*c^4*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*g*exp(1)^2*d^6+39728447488*c^4*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*g*d^6+11274289152*c^4*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-
c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*exp(1)^5*d^5*f-30333206528*c^4*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-
c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*exp(1)^3*d^5*f-25232932864*c^4*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-
c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*exp(1)*d^5*f+4026531840*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)
+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^4*g*exp(1)^9*d^3-9663676416*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*ex
p(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^4*g*exp(1)^7*d^3+5234491392*exp(2)^3*sqrt(-c*exp(2)
)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^4*g*exp(1)^5*d^3+402653184*exp(2)^4*
sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^4*g*exp(1)^3*d^3+24159
19104*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^4*exp(1
)^8*d^2*f-2818572288*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))
*x)^2*b^4*exp(1)^6*d^2*f+402653184*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-s
qrt(-c*exp(2))*x)^2*b^4*exp(1)^4*d^2*f-23353884672*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2
-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*g*exp(1)^8*d^4+38654705664*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1
)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*g*exp(1)^6*d^4+12280922112*c*exp(2)^3*sqrt(-c*exp(2)
)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*g*exp(1)^4*d^4-22045261824*c*exp(2
)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*g*exp(1)^2*d^4+3
221225472*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*e
xp(1)^9*d^3*f-21743271936*c*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*
exp(2))*x)^2*b^3*exp(1)^7*d^3*f+5234491392*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2
*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*exp(1)^5*d^3*f+7751073792*c*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*e
xp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^3*exp(1)^3*d^3*f+45902462976*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt
(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*g*exp(1)^7*d^5-28185722880*c^2*exp(2)^2*s
qrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*g*exp(1)^5*d^5-126634
426368*c^2*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*
g*exp(1)^3*d^5+75698798592*c^2*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(
-c*exp(2))*x)^2*b^2*g*exp(1)*d^5-9663676416*c^2*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^
2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*exp(1)^8*d^4*f+36238786560*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b
*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*exp(1)^6*d^4*f+47110422528*c^2*exp(2)^3*sqrt(-c*exp(2))
*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*exp(1)^4*d^4*f-40466644992*c^2*exp(
2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*exp(1)^2*d^4*f-
37849399296*c^3*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b
*g*exp(1)^6*d^6-20937965568*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt
(-c*exp(2))*x)^2*b*g*exp(1)^4*d^6+184817811456*c^3*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2
-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*g*exp(1)^2*d^6-59592671232*c^3*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1
)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*g*d^6+9663676416*c^3*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d
*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*exp(1)^7*d^5*f-16911433728*c^3*exp(2)^2*sqrt(-c*
exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*exp(1)^5*d^5*f-97039417344*c^3
*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*exp(1)^3*d^5
*f+37849399296*c^3*exp(2)^4*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x
)^2*b*exp(1)*d^5*f+11274289152*c^4*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqr
t(-c*exp(2))*x)^2*g*exp(1)^5*d^7+20132659200*c^4*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c
*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*exp(1)^3*d^7-75698798592*c^4*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*
x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*exp(1)*d^7-3221225472*c^4*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*
d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*exp(1)^6*d^6*f+47513075712*c^4*exp(2)^3*sqrt(-c*e
xp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*exp(1)^2*d^6*f-704643072*exp(2)^2
*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^5*g*exp(1)^9*d^3+2365587456*exp(2)^3*(s
qrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^5*g*exp(1)^7*d^3-2617245696*exp(2)^4*(sqrt
(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^5*g*exp(1)^5*d^3+956301312*exp(2)^5*(sqrt(-b*
d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^5*g*exp(1)^3*d^3-251658240*exp(2)^3*(sqrt(-b*d*ex
p(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^5*exp(1)^8*d^2*f+503316480*exp(2)^4*(sqrt(-b*d*exp(1)
-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^5*exp(1)^6*d^2*f-251658240*exp(2)^5*(sqrt(-b*d*exp(1)-b*x
*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^5*exp(1)^4*d^2*f+402653184*c*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp
(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*g*exp(1)^10*d^4+11978932224*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*g*exp(1)^8*d^4-40365981696*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*g*exp(1)^6*d^4+41171288064*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*g*exp(1)^4*d^4-13186891776*c*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*g*exp(1)^2*d^4-603979776*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(
2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*exp(1)^9*d^3*f+8657043456*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2
)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*exp(1)^7*d^3*f-13488881664*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2
)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*exp(1)^5*d^3*f+5435817984*c*exp(2)^5*(sqrt(-b*d*exp(1)-b*x*exp(2)
+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^4*exp(1)^3*d^3*f-1610612736*c^2*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+
c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*g*exp(1)^9*d^5-56673435648*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2
)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*g*exp(1)^7*d^5+173895843840*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*g*exp(1)^5*d^5-152102240256*c^2*exp(2)^4*(sqrt(-b*d*exp(1)-b*x
*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*g*exp(1)^3*d^5+39258685440*c^2*exp(2)^5*(sqrt(-b*d*exp(1)-b
*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*g*exp(1)*d^5+9462349824*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*
x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*exp(1)^8*d^4*f-61253615616*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-
b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*exp(1)^6*d^4*f+70363643904*c^2*exp(2)^4*(sqrt(-b*d*exp(1
)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*exp(1)^4*d^4*f-21340618752*c^2*exp(2)^5*(sqrt(-b*d*exp
(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*exp(1)^2*d^4*f+2415919104*c^3*exp(2)*(sqrt(-b*d*exp(
1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*g*exp(1)^8*d^6+105193144320*c^3*exp(2)^2*(sqrt(-b*d*e
xp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*g*exp(1)^6*d^6-283669168128*c^3*exp(2)^3*(sqrt(-b*
d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*g*exp(1)^4*d^6+189246996480*c^3*exp(2)^4*(sqrt(
-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*g*exp(1)^2*d^6-29796335616*c^3*exp(2)^5*(sqr
t(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*g*d^6-24763170816*c^3*exp(2)^2*(sqrt(-b*d*
exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*exp(1)^7*d^5*f+127339069440*c^3*exp(2)^3*(sqrt(-b
*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*exp(1)^5*d^5*f-104891154432*c^3*exp(2)^4*(sqrt
(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*exp(1)^3*d^5*f+18924699648*c^3*exp(2)^5*(sq
rt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*exp(1)*d^5*f-1610612736*c^4*exp(2)*(sqrt(
-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*g*exp(1)^7*d^7-84758495232*c^4*exp(2)^2*(sqrt(
-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*g*exp(1)^5*d^7+195286794240*c^4*exp(2)^3*(sqrt
(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*g*exp(1)^3*d^7-75698798592*c^4*exp(2)^4*(sqrt
(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*g*exp(1)*d^7+23555211264*c^4*exp(2)^2*(sqrt(-
b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*exp(1)^6*d^6*f-104287174656*c^4*exp(2)^3*(sqrt(
-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*exp(1)^4*d^6*f+47513075712*c^4*exp(2)^4*(sqrt(
-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*exp(1)^2*d^6*f+402653184*c^5*exp(2)*(sqrt(-b*d
*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*exp(1)^6*d^8+24964497408*c^5*exp(2)^2*(sqrt(-b*d*e
xp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*exp(1)^4*d^8-47513075712*c^5*exp(2)^3*(sqrt(-b*d*exp
(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*exp(1)^2*d^8-7650410496*c^5*exp(2)^2*(sqrt(-b*d*exp(1)
-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*exp(1)^5*d^7*f+29796335616*c^5*exp(2)^3*(sqrt(-b*d*exp(1)-b
*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*exp(1)^3*d^7*f+1610612736*exp(2)*sqrt(-c*exp(2))*b^5*g*exp(1)
^10*d^4-6543114240*exp(2)^2*sqrt(-c*exp(2))*b^5*g*exp(1)^8*d^4+9714008064*exp(2)^3*sqrt(-c*exp(2))*b^5*g*exp(1
)^6*d^4-6241124352*exp(2)^4*sqrt(-c*exp(2))*b^5*g*exp(1)^4*d^4+1459617792*exp(2)^5*sqrt(-c*exp(2))*b^5*g*exp(1
)^2*d^4+805306368*exp(2)^2*sqrt(-c*exp(2))*b^5*exp(1)^9*d^3*f-2164260864*exp(2)^3*sqrt(-c*exp(2))*b^5*exp(1)^7
*d^3*f+1912602624*exp(2)^4*sqrt(-c*exp(2))*b^5*exp(1)^5*d^3*f-553648128*exp(2)^5*sqrt(-c*exp(2))*b^5*exp(1)^3*
d^3*f-11945377792*c*exp(2)*sqrt(-c*exp(2))*b^4*g*exp(1)^9*d^5+43855642624*c*exp(2)^2*sqrt(-c*exp(2))*b^4*g*exp
(1)^7*d^5-57076088832*c*exp(2)^3*sqrt(-c*exp(2))*b^4*g*exp(1)^5*d^5+31037849600*c*exp(2)^4*sqrt(-c*exp(2))*b^4
*g*exp(1)^3*d^5-5872025600*c*exp(2)^5*sqrt(-c*exp(2))*b^4*g*exp(1)*d^5+1879048192*c*exp(2)*sqrt(-c*exp(2))*b^4
*exp(1)^10*d^4*f-12549357568*c*exp(2)^2*sqrt(-c*exp(2))*b^4*exp(1)^8*d^4*f+21944598528*c*exp(2)^3*sqrt(-c*exp(
2))*b^4*exp(1)^6*d^4*f-14428405760*c*exp(2)^4*sqrt(-c*exp(2))*b^4*exp(1)^4*d^4*f+3154116608*c*exp(2)^5*sqrt(-c
*exp(2))*b^4*exp(1)^2*d^4*f+31675383808*c^2*exp(2)*sqrt(-c*exp(2))*b^3*g*exp(1)^8*d^6-102240354304*c^2*exp(2)^
2*sqrt(-c*exp(2))*b^3*g*exp(1)^6*d^6+108766691328*c^2*exp(2)^3*sqrt(-c*exp(2))*b^3*g*exp(1)^4*d^6-43721424896*
c^2*exp(2)^4*sqrt(-c*exp(2))*b^3*g*exp(1)^2*d^6+4966055936*c^2*exp(2)^5*sqrt(-c*exp(2))*b^3*g*d^6-7516192768*c
^2*exp(2)*sqrt(-c*exp(2))*b^3*exp(1)^9*d^5*f+39124467712*c^2*exp(2)^2*sqrt(-c*exp(2))*b^3*exp(1)^7*d^5*f-52294
582272*c^2*exp(2)^3*sqrt(-c*exp(2))*b^3*exp(1)^5*d^5*f+24394072064*c^2*exp(2)^4*sqrt(-c*exp(2))*b^3*exp(1)^3*d
^5*f-3154116608*c^2*exp(2)^5*sqrt(-c*exp(2))*b^3*exp(1)*d^5*f-39460012032*c^3*exp(2)*sqrt(-c*exp(2))*b^2*g*exp
(1)^7*d^7+109018349568*c^3*exp(2)^2*sqrt(-c*exp(2))*b^2*g*exp(1)^5*d^7-85161148416*c^3*exp(2)^3*sqrt(-c*exp(2)
)*b^2*g*exp(1)^3*d^7+18924699648*c^3*exp(2)^4*sqrt(-c*exp(2))*b^2*g*exp(1)*d^7+11274289152*c^3*exp(2)*sqrt(-c*
exp(2))*b^2*exp(1)^8*d^6*f-50130321408*c^3*exp(2)^2*sqrt(-c*exp(2))*b^2*exp(1)^6*d^6*f+47412412416*c^3*exp(2)^
3*sqrt(-c*exp(2))*b^2*exp(1)^4*d^6*f-11878268928*c^3*exp(2)^4*sqrt(-c*exp(2))*b^2*exp(1)^2*d^6*f+23622320128*c
^4*exp(2)*sqrt(-c*exp(2))*b*g*exp(1)^6*d^8-54022635520*c^4*exp(2)^2*sqrt(-c*exp(2))*b*g*exp(1)^4*d^8+237565378
56*c^4*exp(2)^3*sqrt(-c*exp(2))*b*g*exp(1)^2*d^8-7516192768*c^4*exp(2)*sqrt(-c*exp(2))*b*exp(1)^7*d^7*f+290581
38112*c^4*exp(2)^2*sqrt(-c*exp(2))*b*exp(1)^5*d^7*f-14898167808*c^4*exp(2)^3*sqrt(-c*exp(2))*b*exp(1)^3*d^7*f-
5502926848*c^5*exp(2)*sqrt(-c*exp(2))*g*exp(1)^5*d^9+9932111872*c^5*exp(2)^2*sqrt(-c*exp(2))*g*exp(1)^3*d^9+18
79048192*c^5*exp(2)*sqrt(-c*exp(2))*exp(1)^6*d^8*f-6308233216*c^5*exp(2)^2*sqrt(-c*exp(2))*exp(1)^4*d^8*f)/805
306368/exp(1)^7/((sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*exp(1)-2*sqrt(-c*exp(2)
)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*d+b*exp(1)^2*d-exp(2)*b*d-c*exp(1)*d^2)^
3)
________________________________________________________________________________________
maple [B] time = 0.07, size = 4987, normalized size = 14.58 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x)
[Out]
-15/8*e^7*c/(-b*e^2+2*c*d*e)^3*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+
d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d*g+45*e^4*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+
2*c*d*e)*(x+d/e))^(1/2)*x*d^2*g-15/2*e^5*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e)
)^(1/2)*x*d*g-75/4*e^7*c^2/(-b*e^2+2*c*d*e)^3*b^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*
e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d*f-20*e^3*c^3/(-b*e^2+2*c*d*e)^3*b*(-(x+d/e)^2*c
*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*d*g-75*e^5*c^3/(-b*e^2+2*c*d*e)^3*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2
)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^3*g+150*e^4*c^4/(-b*
e^2+2*c*d*e)^3*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b
*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4*g-150*e^5*c^4/(-b*e^2+2*c*d*e)^3*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d
/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^3*f-150*e^3*c^5/(-b*e^2+2*
c*d*e)^3*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c
*d*e)*(x+d/e))^(1/2))*d^5*g+150*e^4*c^5/(-b*e^2+2*c*d*e)^3*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b
*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4*f+25*g*e^3*c^3/(-b*e^2+2*c*d*e)^2*
b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(
x+d/e))^(1/2))*d^3-45*e^5*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d*f-9
0*e^3*c^4/(-b*e^2+2*c*d*e)^3*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^3*g+90*e^4*c^4/(-b*e^2+2*
c*d*e)^3*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^2*f+75/4*e^6*c^2/(-b*e^2+2*c*d*e)^3*b^4/(c*e^
2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(
1/2))*d^2*g+75*e^6*c^3/(-b*e^2+2*c*d*e)^3*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c
/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2*f-25*g*e^2*c^4/(-b*e^2+2*c*d*e)^2*b/(c*e^2)^(1/2)
*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^
4-25/2*g*e^4*c^2/(-b*e^2+2*c*d*e)^2*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/
(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2+15/2*g*e^3*c^2/(-b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2*c*e^2
+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d-15*g*e^2*c^3/(-b*e^2+2*c*d*e)^2*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d
/e))^(1/2)*x*d^2+25/8*g*e^5*c/(-b*e^2+2*c*d*e)^2*b^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c
*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d-32*e^2*c^3/(-b*e^2+2*c*d*e)^3*(-(x+d/e)^2*c*
e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*f-2/e^4/(-b*e^2+2*c*d*e)/(x+d/e)^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d
/e))^(7/2)*f+16/3*g*c^2/(-b*e^2+2*c*d*e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)-60*e^3*c^5/(-b*e^
2+2*c*d*e)^3*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+30*e^2*c^4/(-b*e^2+2*c*d*e)^3*d^4*(-(x+
d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*g-30*e^3*c^4/(-b*e^2+2*c*d*e)^3*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2
*c*d*e)*(x+d/e))^(1/2)*b*f+60*e^2*c^6/(-b*e^2+2*c*d*e)^3*d^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b
*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*g-60*e^3*c^6/(-b*e^2+2*c*d*e)^3*d^5/(c
*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e)
)^(1/2))*f+40*e^2*c^4/(-b*e^2+2*c*d*e)^3*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*g-40*e^3*c^4/
(-b*e^2+2*c*d*e)^3*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*f+20*e^2*c^3/(-b*e^2+2*c*d*e)^3*d^2*(
-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*b*g-20*e^3*c^3/(-b*e^2+2*c*d*e)^3*d*(-(x+d/e)^2*c*e^2+(-b*e^2
+2*c*d*e)*(x+d/e))^(3/2)*b*f-10/3*g*e^2*c^2/(-b*e^2+2*c*d*e)^2*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(
3/2)*x-15/2*g*e^2*c^2/(-b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d^2-5/4*g*e^4*c
/(-b*e^2+2*c*d*e)^2*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+15/4*g*e^3*c/(-b*e^2+2*c*d*e)^2*b^
3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d+10*g*e*c^4/(-b*e^2+2*c*d*e)^2*d^3*(-(x+d/e)^2*c*e^2+(-b*
e^2+2*c*d*e)*(x+d/e))^(1/2)*x+20*e^4*c^3/(-b*e^2+2*c*d*e)^3*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2
)*x*f-10*e^3*c^2/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*d*g+5*g*e*c^3/(-b*e^
2+2*c*d*e)^2*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b-15/4*e^5*c/(-b*e^2+2*c*d*e)^3*b^4*(-(x+d/
e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*g-45*e^3*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c
*d*e)*(x+d/e))^(1/2)*d^3*g+45*e^4*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)
*d^2*f+15/8*e^8*c/(-b*e^2+2*c*d*e)^3*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)
/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*f-32*c^2/(-b*e^2+2*c*d*e)^3/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-
b*e^2+2*c*d*e)*(x+d/e))^(7/2)*f+2*g/e^4/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))
^(7/2)-5/8*g*e^4/(-b*e^2+2*c*d*e)^2*b^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+32/e*c^2/(-b*e^2+2*c
*d*e)^3/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*d*g+12/e^3*c/(-b*e^2+2*c*d*e)^2/(x+d/e)^3*
(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*d*g+15/2*e^6*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-(x+d/e)^2*c*e^2+(-
b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+45/2*e^4*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/
e))^(1/2)*d^2*g-45/2*e^5*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*f+60*e
^2*c^5/(-b*e^2+2*c*d*e)^3*d^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*g+16/3*g/e^2*c/(-b*e^2+2*c*d
*e)^2/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-5/3*g*e^2*c/(-b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)
^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)-5/16*g*e^6/(-b*e^2+2*c*d*e)^2*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*
(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))+32*e*c^3/(-b*e^2+2*c*d*e
)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*d*g+10*e^4*c^2/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+
(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*f+15/4*e^6*c/(-b*e^2+2*c*d*e)^3*b^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e)
)^(1/2)*f-12/e^2*c/(-b*e^2+2*c*d*e)^2/(x+d/e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*f+2/e^5/(-b*
e^2+2*c*d*e)/(x+d/e)^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*d*g+20/3*g*e*c^3/(-b*e^2+2*c*d*e)^2*d
*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x+10/3*g*e*c^2/(-b*e^2+2*c*d*e)^2*d*(-(x+d/e)^2*c*e^2+(-b*e
^2+2*c*d*e)*(x+d/e))^(3/2)*b+10*g*e*c^5/(-b*e^2+2*c*d*e)^2*d^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(
-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^4,x)
[Out]
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^4, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**4,x)
[Out]
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**4, x)
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