3.20.59 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=276 \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^4 (2 c d-b e)}+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-8 c d g+2 c e f)}{3 e^2 (d+e x)^2 (2 c d-b e)}+\frac {c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-8 c d g+2 c e f)}{e^2 (2 c d-b e)}+\frac {\sqrt {c} (3 b e g-8 c d g+2 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 e^2} \]
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Rubi [A] time = 0.45, antiderivative size = 276, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used =
{792, 662, 664, 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 )^{5/2}}{3 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 )^{3/2} (3 b e g-8 c d g+2 c e f)}{3 e^2 (d+e x)^2 (2 c d-b e)}+\frac {c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-8 c d g+2 c e f)}{e^2 (2 c d-b e)}+\frac {\sqrt {c} (3 b e g-8 c d g+2 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 )}{2 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)^(3/2))/(d + e*x)^4,x]
[Out]
(c*(2*c*e*f - 8*c*d*g + 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(2*c*d - b*e)) + (2*(2*c*e*f
- 8*c*d*g + 3*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)*(d + e*x)^2) - (2*(e*f
- d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^4) + (Sqrt[c]*(2*c*e*f - 8*
c*d*g + 3*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])])/(2*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 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 )^{3/2}}{(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 )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^4}-\frac {(2 c e f-8 c d g+3 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)^3} \, dx}{3 e (2 c d-b e)}\\ &=\frac {2 (2 c e f-8 c d g+3 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) (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 )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^4}+\frac {(c (2 c e f-8 c d g+3 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx}{e (2 c d-b e)}\\ &=\frac {c (2 c e f-8 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e)}+\frac {2 (2 c e f-8 c d g+3 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) (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 )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^4}+\frac {(c (2 c e f-8 c d g+3 b e g)) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 e}\\ &=\frac {c (2 c e f-8 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e)}+\frac {2 (2 c e f-8 c d g+3 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) (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 )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^4}+\frac {(c (2 c e f-8 c d g+3 b e g)) \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 )}{e}\\ &=\frac {c (2 c e f-8 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e)}+\frac {2 (2 c e f-8 c d g+3 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) (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 )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^4}+\frac {\sqrt {c} (2 c e f-8 c d g+3 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 )}{2 e^2}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 150, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {(d+e x) (c (d-e x)-b e)} \left (\frac {e (d+e x) (3 b e g-8 c d g+2 c e f) \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {c (d+e x)}{2 c d-b e}\right )}{\sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}+\frac {e (e f-d g) (b e-c d+c e x)^2}{b e-2 c d}\right )}{3 e^3 (d+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)^(3/2))/(d + e*x)^4,x]
[Out]
(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((e*(e*f - d*g)*(-(c*d) + b*e + c*e*x)^2)/(-2*c*d + b*e) + (e*(2*c*e
*f - 8*c*d*g + 3*b*e*g)*(d + e*x)*Hypergeometric2F1[-3/2, -1/2, 1/2, (c*(d + e*x))/(2*c*d - b*e)])/Sqrt[(-(c*d
) + b*e + c*e*x)/(-2*c*d + b*e)]))/(3*e^3*(d + e*x)^2)
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IntegrateAlgebraic [A] time = 4.55, size = 322, normalized size = 1.17 \begin {gather*} \frac {\sqrt {-c e^2} (3 b e g-8 c d g+2 c e f) \log \left (b^2 e^2-8 c x \sqrt {-c e^2} \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}-4 b c d e-4 b c e^2 x+4 c^2 d^2-8 c^2 e^2 x^2\right )}{4 e^3}+\frac {\left (-3 b \sqrt {c} e g+8 c^{3/2} d g-2 c^{3/2} e f\right ) \tan ^{-1}\left (\frac {\sqrt {c} \left (2 \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}-2 x \sqrt {-c e^2}\right )}{b e}\right )}{2 e^2}+\frac {\sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2} \left (4 b d e g+2 b e^2 f+6 b e^2 g x-19 c d^2 g+4 c d e f-26 c d e g x+8 c e^2 f x-3 c e^2 g x^2\right )}{3 e^2 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^4,x]
[Out]
(Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]*(4*c*d*e*f + 2*b*e^2*f - 19*c*d^2*g + 4*b*d*e*g + 8*c*e^2*f*x - 26*
c*d*e*g*x + 6*b*e^2*g*x - 3*c*e^2*g*x^2))/(3*e^2*(d + e*x)^2) + ((-2*c^(3/2)*e*f + 8*c^(3/2)*d*g - 3*b*Sqrt[c]
*e*g)*ArcTan[(Sqrt[c]*(-2*Sqrt[-(c*e^2)]*x + 2*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]))/(b*e)])/(2*e^2) + (
Sqrt[-(c*e^2)]*(2*c*e*f - 8*c*d*g + 3*b*e*g)*Log[4*c^2*d^2 - 4*b*c*d*e + b^2*e^2 - 4*b*c*e^2*x - 8*c^2*e^2*x^2
- 8*c*Sqrt[-(c*e^2)]*x*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]])/(4*e^3)
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fricas [A] time = 2.15, size = 593, normalized size = 2.15 \begin {gather*} \left [\frac {3 \, {\left (2 \, c d^{2} e f + {\left (2 \, c e^{3} f - {\left (8 \, c d e^{2} - 3 \, b e^{3}\right )} g\right )} x^{2} - {\left (8 \, c d^{3} - 3 \, b d^{2} e\right )} g + 2 \, {\left (2 \, c d e^{2} f - {\left (8 \, c d^{2} e - 3 \, b d e^{2}\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 (3 \, c e^{2} g x^{2} - 2 \, {\left (2 \, c d e + b e^{2}\right )} f + {\left (19 \, c d^{2} - 4 \, b d e\right )} g - 2 \, {\left (4 \, c e^{2} f - {\left (13 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{12 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, -\frac {3 \, {\left (2 \, c d^{2} e f + {\left (2 \, c e^{3} f - {\left (8 \, c d e^{2} - 3 \, b e^{3}\right )} g\right )} x^{2} - {\left (8 \, c d^{3} - 3 \, b d^{2} e\right )} g + 2 \, {\left (2 \, c d e^{2} f - {\left (8 \, c d^{2} e - 3 \, b d e^{2}\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 (3 \, c e^{2} g x^{2} - 2 \, {\left (2 \, c d e + b e^{2}\right )} f + {\left (19 \, c d^{2} - 4 \, b d e\right )} g - 2 \, {\left (4 \, c e^{2} f - {\left (13 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{6 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} 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)^(3/2)/(e*x+d)^4,x, algorithm="fricas")
[Out]
[1/12*(3*(2*c*d^2*e*f + (2*c*e^3*f - (8*c*d*e^2 - 3*b*e^3)*g)*x^2 - (8*c*d^3 - 3*b*d^2*e)*g + 2*(2*c*d*e^2*f -
(8*c*d^2*e - 3*b*d*e^2)*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*(3*c*e^2*g*x^2 - 2*(2*c*d*e + b*e^2)*
f + (19*c*d^2 - 4*b*d*e)*g - 2*(4*c*e^2*f - (13*c*d*e - 3*b*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d
*e))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2), -1/6*(3*(2*c*d^2*e*f + (2*c*e^3*f - (8*c*d*e^2 - 3*b*e^3)*g)*x^2 - (8*c*
d^3 - 3*b*d^2*e)*g + 2*(2*c*d*e^2*f - (8*c*d^2*e - 3*b*d*e^2)*g)*x)*sqrt(c)*arctan(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*(3*c*e^2*g*x^2
- 2*(2*c*d*e + b*e^2)*f + (19*c*d^2 - 4*b*d*e)*g - 2*(4*c*e^2*f - (13*c*d*e - 3*b*e^2)*g)*x)*sqrt(-c*e^2*x^2 -
b*e^2*x + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3*x + d^2*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)^(3/2)/(e*x+d)^4,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: -4*exp(1)*c*g*1/4/exp(1)^3*sqrt(-b*d*exp
(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))+2*((3*c*sqrt(-c*exp(2))*b*g*exp(1)-8*c^2*sqrt(-c*exp(2))*g*d+2*c^2*sqrt(-c*
exp(2))*exp(1)*f)/4/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))+(12*exp(2)^2*b^3*g*exp(1)^5*d-10*exp(2)^3*b^3*g*exp(1)^3*d-2*exp(2)^3*b^3*exp(1)^4*f+
48*c*exp(2)*b^2*g*exp(1)^6*d^2-180*c*exp(2)^2*b^2*g*exp(1)^4*d^2+120*c*exp(2)^3*b^2*g*exp(1)^2*d^2+24*c*exp(2)
^2*b^2*exp(1)^5*d*f-12*c*exp(2)^3*b^2*exp(1)^3*d*f-96*c^2*exp(2)*b*g*exp(1)^5*d^3+360*c^2*exp(2)^2*b*g*exp(1)^
3*d^3-240*c^2*exp(2)^3*b*g*exp(1)*d^3-72*c^2*exp(2)^2*b*exp(1)^4*d^2*f+48*c^2*exp(2)^3*b*exp(1)^2*d^2*f+48*c^3
*exp(2)*g*exp(1)^4*d^4-192*c^3*exp(2)^2*g*exp(1)^2*d^4+128*c^3*exp(2)^3*g*d^4+48*c^3*exp(2)^2*exp(1)^3*d^3*f-3
2*c^3*exp(2)^3*exp(1)*d^3*f)/32/(b*exp(1)^8*d-exp(2)*b*exp(1)^6*d-c*exp(1)^7*d^2+c*exp(2)*exp(1)^5*d^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)))+(503316480*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-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*g*exp(1)^5*d+5
0331648*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+40265318
4*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^2-4328521728
*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+422785843
2*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+10066329
60*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-130862284
8*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+8657043456*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^3-8455716864*
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^3-3019898880
*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+362387865
6*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+40265318
4*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^4-4831838208*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+4831838208*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+2013265920*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-2415919104*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-1610612736*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*exp(1)^8*d^2+150
9949440*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*e
xp(1)^6*d^2+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*g*exp(1)^4*d^2-805306368*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+553648128*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+6039797760*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^4*b^2*g*exp(1)^7*d^3+1509949440*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^3-9059696640*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^3-1610612736*
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+2214592512*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*exp(1)^6*d^2*f+905969664*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))-sqr
t(-c*exp(2))*x)^4*b^2*exp(1)^4*d^2*f-7247757312*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^4-14294188032*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+24561844224*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+3221225472*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*exp(1)^7*d^3*f+2214592
512*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-8455716864*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+2818572288*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*g*exp(1)^5*d^5+11274289152*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^5-16106127360*c^3*exp(2)^3*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)^4*g*exp(1)*d^5-1610612736*c^3*exp(2)*sqrt(-c*exp(2))*(s
qrt(-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-3623878656*c^3*exp(2)^2*sqr
t(-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+7247757312*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*exp(1)^2*d^4
*f+805306368*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
-1476395008*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+
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*g*exp(1)^4*d^2-13
4217728*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+134217
728*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-1127428915
2*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+19797114
880*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^3-801950
9248*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^3-22817
01376*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+1778
384896*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+386
54705664*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
-55767465984*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+14092861440*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*g*exp(
1)^2*d^4-6442450944*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+16508780544*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-7046430720*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-46707769344*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+46036680704*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+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*g*exp(1)*d^5+12884901888*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-18656264192*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-268435456*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+18522046464*c^4*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*ex
p(2))*x)^3*g*exp(1)^4*d^6-8589934592*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^6-13958643712*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^6-6442450944*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+4563402752*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*e
xp(1)^3*d^5*f+5905580032*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*ex
p(1)*d^5*f-2415919104*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+4831838208*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)^2*b^4*g*exp(1)^7*d^3-2818572288*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*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+402653184*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*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*b^4*exp(1)^4*d^2*f+13690208256*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^
4-16911433728*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^4-4227858432*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))-sqr
t(-c*exp(2))*x)^2*b^3*g*exp(1)^4*d^4+6945767424*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^4-1610612736*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*exp(1)^9*d^3*f+4831838208*c*exp(2)^2*sqrt(-c*exp(2))*(sq
rt(-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-2818572288*c*exp(2)^3*sq
rt(-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+1006632
96*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*exp(1)
^3*d^3*f-26575110144*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+4026531840*c^2*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*ex
p(2))-sqrt(-c*exp(2))*x)^2*b^2*g*exp(1)^5*d^5+53351546880*c^2*exp(2)^3*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^2*g*exp(1)^3*d^5-27783069696*c^2*exp(2)^4*sqrt(-c*exp(2))*(sq
rt(-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+4831838208*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-483183820
8*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-11676942336*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*ex
p(2))*x)^2*b^2*exp(1)^4*d^4*f+8657043456*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+21743271936*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+23353884672*c^3*exp(2)^2*sqrt(-c*exp(2))*(sqr
t(-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^6-72074919936*c^3*exp(2)^3*sq
rt(-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+209379655
68*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-
4831838208*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-4831838208*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+24561844224*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-8858370048*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-6442450944*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*g*exp(1)^5*d^7-15300820992*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+25769803776*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+16
10612736*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+4831838208*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*ex
p(2))*x)^2*exp(1)^4*d^6*f-10468982784*c^4*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*ex
p(2))-sqrt(-c*exp(2))*x)^2*exp(1)^2*d^6*f+301989888*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^3-855638016*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^5*g*exp(1)^7*d^3+805306368*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^3-251658240*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*g*exp(1)^3*d^3-50331648*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^5*exp(1)^8*d^2*f+100663296*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-50331648*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^4-5939134464*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*g*exp(1
)^8*d^4+17414750208*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*g*exp(1
)^6*d^4-15804137472*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*g*exp(1
)^4*d^4+4731174912*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*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-1409286144*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+1409286144*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-603979776*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+
29695672320*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^
5-76554436608*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*g*exp(1)^5*
d^5+59693334528*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^5-14696841216*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^5-6241124352*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+17465081856*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-15804137472*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+4831838208*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*ex
p(1)^8*d^6-56472109056*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*g*
exp(1)^6*d^6+120393302016*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^6-70464307200*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+10468982784*c^3*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^2*g*d^6+15099494400*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*ex
p(1)^7*d^5*f-36943429632*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+24763170816*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-4429185024*c^3*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
^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+46103789568*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^7-76504104960*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+25769803776*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^7-13891534848*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+27380416512*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-10468982784*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^8-13690208256*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)*g*exp(1)^4*d
^8+16106127360*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
+4429185024*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-64
42450944*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-80530
6368*exp(2)*sqrt(-c*exp(2))*b^5*g*exp(1)^10*d^4+2919235584*exp(2)^2*sqrt(-c*exp(2))*b^5*g*exp(1)^8*d^4-3976200
192*exp(2)^3*sqrt(-c*exp(2))*b^5*g*exp(1)^6*d^4+2415919104*exp(2)^4*sqrt(-c*exp(2))*b^5*g*exp(1)^4*d^4-5536481
28*exp(2)^5*sqrt(-c*exp(2))*b^5*g*exp(1)^2*d^4+50331648*exp(2)^3*sqrt(-c*exp(2))*b^5*exp(1)^7*d^3*f-100663296*
exp(2)^4*sqrt(-c*exp(2))*b^5*exp(1)^5*d^3*f+50331648*exp(2)^5*sqrt(-c*exp(2))*b^5*exp(1)^3*d^3*f+6308233216*c*
exp(2)*sqrt(-c*exp(2))*b^4*g*exp(1)^9*d^5-20099104768*c*exp(2)^2*sqrt(-c*exp(2))*b^4*g*exp(1)^7*d^5+2365587456
0*c*exp(2)^3*sqrt(-c*exp(2))*b^4*g*exp(1)^5*d^5-12113149952*c*exp(2)^4*sqrt(-c*exp(2))*b^4*g*exp(1)^3*d^5+2248
146944*c*exp(2)^5*sqrt(-c*exp(2))*b^4*g*exp(1)*d^5-1073741824*c*exp(2)*sqrt(-c*exp(2))*b^4*exp(1)^10*d^4*f+369
0987520*c*exp(2)^2*sqrt(-c*exp(2))*b^4*exp(1)^8*d^4*f-5033164800*c*exp(2)^3*sqrt(-c*exp(2))*b^4*exp(1)^6*d^4*f
+3154116608*c*exp(2)^4*sqrt(-c*exp(2))*b^4*exp(1)^4*d^4*f-738197504*c*exp(2)^5*sqrt(-c*exp(2))*b^4*exp(1)^2*d^
4*f-17179869184*c^2*exp(2)*sqrt(-c*exp(2))*b^3*g*exp(1)^8*d^6+46271561728*c^2*exp(2)^2*sqrt(-c*exp(2))*b^3*g*e
xp(1)^6*d^6-43436212224*c^2*exp(2)^3*sqrt(-c*exp(2))*b^3*g*exp(1)^4*d^6+16139681792*c^2*exp(2)^4*sqrt(-c*exp(2
))*b^3*g*exp(1)^2*d^6-1744830464*c^2*exp(2)^5*sqrt(-c*exp(2))*b^3*g*d^6+4294967296*c^2*exp(2)*sqrt(-c*exp(2))*
b^3*exp(1)^9*d^5*f-12549357568*c^2*exp(2)^2*sqrt(-c*exp(2))*b^3*exp(1)^7*d^5*f+13136560128*c^2*exp(2)^3*sqrt(-
c*exp(2))*b^3*exp(1)^5*d^5*f-5670699008*c^2*exp(2)^4*sqrt(-c*exp(2))*b^3*exp(1)^3*d^5*f+738197504*c^2*exp(2)^5
*sqrt(-c*exp(2))*b^3*exp(1)*d^5*f+21743271936*c^3*exp(2)*sqrt(-c*exp(2))*b^2*g*exp(1)^7*d^7-47412412416*c^3*ex
p(2)^2*sqrt(-c*exp(2))*b^2*g*exp(1)^5*d^7+31809601536*c^3*exp(2)^3*sqrt(-c*exp(2))*b^2*g*exp(1)^3*d^7-64424509
44*c^3*exp(2)^4*sqrt(-c*exp(2))*b^2*g*exp(1)*d^7-6442450944*c^3*exp(2)*sqrt(-c*exp(2))*b^2*exp(1)^8*d^6*f+1550
2147584*c^3*exp(2)^2*sqrt(-c*exp(2))*b^2*exp(1)^6*d^6*f-11374952448*c^3*exp(2)^3*sqrt(-c*exp(2))*b^2*exp(1)^4*
d^6*f+2617245696*c^3*exp(2)^4*sqrt(-c*exp(2))*b^2*exp(1)^2*d^6*f-13153337344*c^4*exp(2)*sqrt(-c*exp(2))*b*g*ex
p(1)^6*d^8+21810380800*c^4*exp(2)^2*sqrt(-c*exp(2))*b*g*exp(1)^4*d^8-8053063680*c^4*exp(2)^3*sqrt(-c*exp(2))*b
*g*exp(1)^2*d^8+4294967296*c^4*exp(2)*sqrt(-c*exp(2))*b*exp(1)^7*d^7*f-8120172544*c^4*exp(2)^2*sqrt(-c*exp(2))
*b*exp(1)^5*d^7*f+3221225472*c^4*exp(2)^3*sqrt(-c*exp(2))*b*exp(1)^3*d^7*f+3087007744*c^5*exp(2)*sqrt(-c*exp(2
))*g*exp(1)^5*d^9-3489660928*c^5*exp(2)^2*sqrt(-c*exp(2))*g*exp(1)^3*d^9-1073741824*c^5*exp(2)*sqrt(-c*exp(2))
*exp(1)^6*d^8*f+1476395008*c^5*exp(2)^2*sqrt(-c*exp(2))*exp(1)^4*d^8*f)/(-805306368*b*exp(1)^8*d+805306368*exp
(2)*b*exp(1)^6*d+805306368*c*exp(1)^7*d^2-805306368*c*exp(2)*exp(1)^5*d^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*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 = 2773, normalized size = 10.05 \begin {gather*} \text {Expression 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)^(3/2)/(e*x+d)^4,x)
[Out]
-12*e^4*c^4/(-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^2*f+4*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))^(1/2)*x*d*g+12*e^3*c^4/(-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^3*g+16/3*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))^(3/2)*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))^(5/2)-8*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))^(3/2)-2/3/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))^(5/2)*f+16/3*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))^(5/2)*f-9*g*e^3*c^2/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*ar
ctan((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+18*
g*e^2*c^3/(-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^2+6*e^5*c^3/(-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*f+e^5*c^2/(-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*g-6*e^4*c^3/(-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^2*g-8*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))^(5/2)+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))^(1/2)+2/3/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))
^(5/2)*d*g+4/3/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))^(5/2)*f-16/3*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))^(3/2)*d*g-2*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))^(1/2)*f+8*e^3*c^5/(-b*e^2+2*c*d*e)^3*d^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))*f-12*g*e*c^4/(-
b*e^2+2*c*d*e)^2*d^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))-12*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))^(1
/2)*x-6*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))^(1/2)*b+2*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))^(1/2)*d*g-e^6*c^2/(-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))*f-
8*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))^(1/2)*x*g+8*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))^(1/2)*x*f-4*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))^(1/2)*b*g+4*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))^(1/2)*b*f-8*e^2*c^5/(-b*e^2+2*c*d*e)^3*d^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))*g+3/2*g*e^4*c/(-b*e^2+2*c*d*e)^2*b^3/(c*e^2)^(1/2)*a
rctan((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))-4/3/
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))^(5/2)*d*g-16/3/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))^(5/2)*d*g-4*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))^(1/2)*x*f+6*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))^(1/2)*x
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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)^(3/2)/(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 )}^{3/2}}{{\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)^(3/2))/(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)^(3/2))/(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 {3}{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)**(3/2)/(e*x+d)**4,x)
[Out]
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**4, x)
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