3.20.70 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=354 \[ \frac {5 (2 c d-b e)^3 (-b e g-6 c d g+8 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{3/2} e^2}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}+\frac {5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-6 c d g+8 c e f)}{24 e^2}+\frac {(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-6 c d g+8 c e f)}{4 e^2 (2 c d-b e)}+\frac {5 (b+2 c x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-6 c d g+8 c e f)}{64 c e} \]

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Rubi [A]  time = 0.75, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {792, 654, 670, 640, 612, 621, 204} \begin {gather*} \frac {5 (2 c d-b e)^3 (-b e g-6 c d g+8 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 )}{128 c^{3/2} e^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 (d+e x)^3 (2 c d-b e)}+\frac {5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-6 c d g+8 c e f)}{24 e^2}+\frac {(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-6 c d g+8 c e f)}{4 e^2 (2 c d-b e)}+\frac {5 (b+2 c x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-6 c d g+8 c e f)}{64 c e} \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)^3,x]

[Out]

(5*(2*c*d - b*e)*(8*c*e*f - 6*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])/(64*c*e) +
(5*(8*c*e*f - 6*c*d*g - b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(24*e^2) + ((8*c*e*f - 6*c*d*g -
b*e*g)*(c*d - b*e - c*e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(4*e^2*(2*c*d - b*e)) + (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)^3) + (5*(2*c*d - b*e)^3*(8*c*e*f -
6*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])])/(128*c^(3/2)*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 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 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 654

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(a + b*x + c*x^2)^(m +
p)/(a/d + (c*x)/e)^m, 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]
&&  !IntegerQ[p] && IntegerQ[m] && RationalQ[p] && (LtQ[0, -m, p] || LtQ[p, -m, 0]) && NeQ[m, 2] && NeQ[m, -1
]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && 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)^3} \, 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)^3}+\frac {(8 c e f-6 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)^2} \, dx}{e (2 c d-b e)}\\ &=\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)^3}+\frac {(8 c e f-6 c d g-b e g) \int \left (\frac {c d^2-b d e}{d}-c e x\right )^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{e (2 c d-b e)}\\ &=\frac {(8 c e f-6 c d g-b e g) (c d-b e-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e)}+\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)^3}+\frac {(5 (8 c e f-6 c d g-b e g)) \int \left (\frac {c d^2-b d e}{d}-c e x\right ) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{8 e}\\ &=\frac {5 (8 c e f-6 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2}+\frac {(8 c e f-6 c d g-b e g) (c d-b e-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e)}+\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)^3}+\frac {(5 (2 c d-b e) (8 c e f-6 c d g-b e g)) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{16 e}\\ &=\frac {5 (2 c d-b e) (8 c e f-6 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}}{64 c e}+\frac {5 (8 c e f-6 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2}+\frac {(8 c e f-6 c d g-b e g) (c d-b e-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e)}+\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)^3}+\frac {\left (5 (2 c d-b e)^3 (8 c e f-6 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}{128 c e}\\ &=\frac {5 (2 c d-b e) (8 c e f-6 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}}{64 c e}+\frac {5 (8 c e f-6 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2}+\frac {(8 c e f-6 c d g-b e g) (c d-b e-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e)}+\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)^3}+\frac {\left (5 (2 c d-b e)^3 (8 c e f-6 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 )}{64 c e}\\ &=\frac {5 (2 c d-b e) (8 c e f-6 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}}{64 c e}+\frac {5 (8 c e f-6 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2}+\frac {(8 c e f-6 c d g-b e g) (c d-b e-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e)}+\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)^3}+\frac {5 (2 c d-b e)^3 (8 c e f-6 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 )}{128 c^{3/2} e^2}\\ \end {align*}

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Mathematica [A]  time = 1.37, size = 294, normalized size = 0.83 \begin {gather*} \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left (\sqrt {c} \sqrt {e} \left (15 b^3 e^3 g+2 b^2 c e^2 (-118 d g+132 e f+59 e g x)+4 b c^2 e \left (173 d^2 g-2 d e (106 f+51 g x)+2 e^2 x (26 f+17 g x)\right )-8 c^3 \left (72 d^3 g-d^2 e (88 f+45 g x)+12 d e^2 x (3 f+2 g x)-2 e^3 x^2 (4 f+3 g x)\right )\right )+\frac {15 \sqrt {e (2 c d-b e)} (b e-2 c d)^2 (-b e g-6 c d g+8 c e f) \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )}{\sqrt {d+e x} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}\right )}{192 c^{3/2} e^{5/2}} \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)^3,x]

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(Sqrt[c]*Sqrt[e]*(15*b^3*e^3*g + 2*b^2*c*e^2*(132*e*f - 118*d*g + 59*e
*g*x) - 8*c^3*(72*d^3*g + 12*d*e^2*x*(3*f + 2*g*x) - 2*e^3*x^2*(4*f + 3*g*x) - d^2*e*(88*f + 45*g*x)) + 4*b*c^
2*e*(173*d^2*g + 2*e^2*x*(26*f + 17*g*x) - 2*d*e*(106*f + 51*g*x))) + (15*Sqrt[e*(2*c*d - b*e)]*(-2*c*d + b*e)
^2*(8*c*e*f - 6*c*d*g - b*e*g)*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/(Sqrt[d + e*x]*S
qrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])))/(192*c^(3/2)*e^(5/2))

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IntegrateAlgebraic [F]  time = 180.14, 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)^3,x]

[Out]

$Aborted

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fricas [A]  time = 0.97, size = 813, normalized size = 2.30 \begin {gather*} \left [-\frac {15 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f - {\left (48 \, c^{4} d^{4} - 64 \, b c^{3} d^{3} e + 24 \, b^{2} c^{2} d^{2} e^{2} - b^{4} e^{4}\right )} g\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 (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f - {\left (24 \, c^{4} d e^{2} - 17 \, b c^{3} e^{3}\right )} g\right )} x^{2} + 8 \, {\left (88 \, c^{4} d^{2} e - 106 \, b c^{3} d e^{2} + 33 \, b^{2} c^{2} e^{3}\right )} f - {\left (576 \, c^{4} d^{3} - 692 \, b c^{3} d^{2} e + 236 \, b^{2} c^{2} d e^{2} - 15 \, b^{3} c e^{3}\right )} g - 2 \, {\left (8 \, {\left (18 \, c^{4} d e^{2} - 13 \, b c^{3} e^{3}\right )} f - {\left (180 \, c^{4} d^{2} e - 204 \, b c^{3} d e^{2} + 59 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{768 \, c^{2} e^{2}}, -\frac {15 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f - {\left (48 \, c^{4} d^{4} - 64 \, b c^{3} d^{3} e + 24 \, b^{2} c^{2} d^{2} e^{2} - b^{4} e^{4}\right )} g\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 (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f - {\left (24 \, c^{4} d e^{2} - 17 \, b c^{3} e^{3}\right )} g\right )} x^{2} + 8 \, {\left (88 \, c^{4} d^{2} e - 106 \, b c^{3} d e^{2} + 33 \, b^{2} c^{2} e^{3}\right )} f - {\left (576 \, c^{4} d^{3} - 692 \, b c^{3} d^{2} e + 236 \, b^{2} c^{2} d e^{2} - 15 \, b^{3} c e^{3}\right )} g - 2 \, {\left (8 \, {\left (18 \, c^{4} d e^{2} - 13 \, b c^{3} e^{3}\right )} f - {\left (180 \, c^{4} d^{2} e - 204 \, b c^{3} d e^{2} + 59 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{384 \, c^{2} e^{2}}\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)^3,x, algorithm="fricas")

[Out]

[-1/768*(15*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f - (48*c^4*d^4 - 64*b*c^3*d^3*e
+ 24*b^2*c^2*d^2*e^2 - b^4*e^4)*g)*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*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^3
*f - (24*c^4*d*e^2 - 17*b*c^3*e^3)*g)*x^2 + 8*(88*c^4*d^2*e - 106*b*c^3*d*e^2 + 33*b^2*c^2*e^3)*f - (576*c^4*d
^3 - 692*b*c^3*d^2*e + 236*b^2*c^2*d*e^2 - 15*b^3*c*e^3)*g - 2*(8*(18*c^4*d*e^2 - 13*b*c^3*e^3)*f - (180*c^4*d
^2*e - 204*b*c^3*d*e^2 + 59*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^2*e^2), -1/384*(
15*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f - (48*c^4*d^4 - 64*b*c^3*d^3*e + 24*b^2
*c^2*d^2*e^2 - b^4*e^4)*g)*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*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^3*f - (24*c^4*d*e^2 - 17*
b*c^3*e^3)*g)*x^2 + 8*(88*c^4*d^2*e - 106*b*c^3*d*e^2 + 33*b^2*c^2*e^3)*f - (576*c^4*d^3 - 692*b*c^3*d^2*e + 2
36*b^2*c^2*d*e^2 - 15*b^3*c*e^3)*g - 2*(8*(18*c^4*d*e^2 - 13*b*c^3*e^3)*f - (180*c^4*d^2*e - 204*b*c^3*d*e^2 +
59*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^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)^(5/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(((96*exp(1)^5*c^5*g*1/768/exp(1)^4/c^
3*x+(128*exp(1)^5*c^5*f+272*exp(1)^5*c^4*g*b-384*exp(1)^4*c^5*g*d)*1/768/exp(1)^4/c^3)*x+(416*exp(1)^5*c^4*b*f
+236*exp(1)^5*c^3*g*b^2-576*exp(1)^4*c^5*d*f-816*exp(1)^4*c^4*g*b*d+720*exp(1)^3*c^5*g*d^2)*1/768/exp(1)^4/c^3
)*x+(528*exp(1)^5*c^3*b^2*f+30*exp(1)^5*c^2*g*b^3-1696*exp(1)^4*c^4*b*d*f-472*exp(1)^4*c^3*g*b^2*d+1408*exp(1)
^3*c^5*d^2*f+1384*exp(1)^3*c^4*g*b*d^2-1152*exp(1)^2*c^5*g*d^3)*1/768/exp(1)^4/c^3)*sqrt(-b*d*exp(1)-b*x*exp(2
)+c*d^2-c*x^2*exp(2))+2*(-(5*b^4*g*exp(1)^4-40*c*b^3*exp(1)^4*f-120*c^2*b^2*g*exp(1)^2*d^2+240*c^2*b^2*exp(1)^
3*f*d+320*c^3*b*g*exp(1)*d^3-480*c^3*b*exp(1)^2*f*d^2-240*c^4*g*d^4+320*c^4*exp(1)*f*d^3)/256/c/sqrt(-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))+(-
20*exp(2)*b^3*g*exp(1)^7*d^2+55*exp(2)^2*b^3*g*exp(1)^5*d^2-35*exp(2)^3*b^3*g*exp(1)^3*d^2-15*exp(2)^2*b^3*exp
(1)^6*f*d+15*exp(2)^3*b^3*exp(1)^4*f*d+100*c*exp(2)*b^2*g*exp(1)^6*d^3-275*c*exp(2)^2*b^2*g*exp(1)^4*d^3+175*c
*exp(2)^3*b^2*g*exp(1)^2*d^3-20*c*exp(2)*b^2*exp(1)^7*f*d^2+115*c*exp(2)^2*b^2*exp(1)^5*f*d^2-95*c*exp(2)^3*b^
2*exp(1)^3*f*d^2-140*c^2*exp(2)*b*g*exp(1)^5*d^4+400*c^2*exp(2)^2*b*g*exp(1)^3*d^4-260*c^2*exp(2)^3*b*g*exp(1)
*d^4+40*c^2*exp(2)*b*exp(1)^6*f*d^3-200*c^2*exp(2)^2*b*exp(1)^4*f*d^3+160*c^2*exp(2)^3*b*exp(1)^2*f*d^3+60*c^3
*exp(2)*g*exp(1)^4*d^5-180*c^3*exp(2)^2*g*exp(1)^2*d^5+120*c^3*exp(2)^3*g*d^5-20*c^3*exp(2)*exp(1)^5*f*d^4+100
*c^3*exp(2)^2*exp(1)^3*f*d^4-80*c^3*exp(2)^3*exp(1)*f*d^4)/4/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*e
xp(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)))-(-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)^3*b^3*g*exp(1)^8*d^2+17*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)^3*b^3*g*exp(1)^6*d^2-13*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)^4*d^2-9*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*ex
p(2))-sqrt(-c*exp(2))*x)^3*b^3*exp(1)^7*f*d+9*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)^5*f*d+20*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
)^3*b^2*g*exp(1)^7*d^3-85*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^2
*g*exp(1)^5*d^3+65*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^2*g*exp(
1)^3*d^3-4*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)^3*b^2*exp(1)^8*f*d^2+5
3*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^2*exp(1)^6*f*d^2-49*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^2*exp(1)^4*f*d^2-28*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)^3*b*g*exp(1)^6*d^4+128*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*g*exp(1)^4*d^4-100*c^2*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)^3*b*g*exp(1)^2*d^4+8*c^2*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)^3*b*exp(1)^7*f*d^3-88*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*exp(1)^5*f*d^3+80*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*exp(1)^3*f*d^3+12*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)^3*g*exp(1)^5*d^5-60*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*g*exp(1)^3*d^5+48*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*g*exp(1)*d^5-4*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)^3*exp(1)^6
*f*d^4+44*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*exp(1)^4*f*d^4-40
*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*exp(1)^2*f*d^4+8*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^3*g*exp(1)^9*d^3-36*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)^7*d^3+21*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)^5*d^3+7*
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)^3*
d^3+24*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)
^8*f*d^2-21*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)^6*f*d^2-3*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^3*exp(1)^4*f*d^2-24*c*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)^8*d^4+108*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*ex
p(2))*x)^2*b^2*g*exp(1)^6*d^4-9*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))-sq
rt(-c*exp(2))*x)^2*b^2*g*exp(1)^4*d^4-75*c*exp(2)^3*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*e
xp(2))-sqrt(-c*exp(2))*x)^2*b^2*g*exp(1)^2*d^4-84*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^2*exp(1)^7*f*d^3+33*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^2*exp(1)^5*f*d^3+51*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^2*exp(1)^3*f*d^3+24*c^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)^7*d^5-108*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*g*exp(1)^5*d^5-72*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*g*exp(1)^3*d^5+156*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*g*exp(1)*d^5+96*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*exp(1)^6*f*d^4+24*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*exp(1)^4*f*d
^4-120*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*ex
p(1)^2*f*d^4-8*c^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)^6*d^6+36*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*
g*exp(1)^4*d^6+60*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*g*exp(1)^2*d^6-88*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*g*d^6-36*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*exp(1)^5*f*d^5-36*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*exp(1)^3*f*d^5+72*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*exp(1)*f*d^5-4*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^4*g*exp(1)^9*d^3+19*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*e
xp(1)^7*d^3-26*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)^5*d^3
+11*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)^3*d^3-7*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)^8*f*d^2+14*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)^6*f*d^2-7*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)^4*f*d^2+32*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^3*g*exp(1)^8*d^4-174*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^3*g*exp(1)^6*d^4+225*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^3*g*exp(1)^4*d^4-83*c*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*e
xp(2))*x)*b^3*g*exp(1)^2*d^4+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)*b^
3*exp(1)^9*f*d^3+78*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^3*exp(1)^
7*f*d^3-141*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^3*exp(1)^5*f*d^3+
59*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^3*exp(1)^3*f*d^3-72*c^2*ex
p(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)^7*d^5+439*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^2*g*exp(1)^5*d^5-527*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^2*g*exp(1)^3*d^5+160*c^2*exp(2)^4*(sqrt(-b*d*ex
p(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*g*exp(1)*d^5-12*c^2*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)*b^2*exp(1)^8*f*d^4-235*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^2*exp(1)^6*f*d^4+371*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^2*exp(1)^4*f*d^4-124*c^2*exp(2)^4*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*e
xp(2))-sqrt(-c*exp(2))*x)*b^2*exp(1)^2*f*d^4+64*c^3*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sq
rt(-c*exp(2))*x)*b*g*exp(1)^6*d^6-432*c^3*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)*b*g*exp(1)^4*d^6+456*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*g*exp(1)^2*d^6-88*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*g*d^6+
12*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*exp(1)^7*f*d^5+264*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*exp(1)^5*f*d^5-348*c^3*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*exp(1)^3*f*d^5+72*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*exp(1)*f*d^5-20*c^4*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)*g*exp(1)^5*d^7+148*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)*g*exp(1)^3*d^7-128*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)*g*exp(1)*d^7-4*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)*exp(1)^6*f*d^6-100*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)*ex
p(1)^4*f*d^6+104*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)*exp(1)^2*f*d
^6+8*sqrt(-c*exp(2))*b^4*g*exp(1)^10*d^4-44*exp(2)*sqrt(-c*exp(2))*b^4*g*exp(1)^8*d^4+77*exp(2)^2*sqrt(-c*exp(
2))*b^4*g*exp(1)^6*d^4-54*exp(2)^3*sqrt(-c*exp(2))*b^4*g*exp(1)^4*d^4+13*exp(2)^4*sqrt(-c*exp(2))*b^4*g*exp(1)
^2*d^4+16*exp(2)*sqrt(-c*exp(2))*b^4*exp(1)^9*f*d^3-41*exp(2)^2*sqrt(-c*exp(2))*b^4*exp(1)^7*f*d^3+34*exp(2)^3
*sqrt(-c*exp(2))*b^4*exp(1)^5*f*d^3-9*exp(2)^4*sqrt(-c*exp(2))*b^4*exp(1)^3*f*d^3-32*c*sqrt(-c*exp(2))*b^3*g*e
xp(1)^9*d^5+184*c*exp(2)*sqrt(-c*exp(2))*b^3*g*exp(1)^7*d^5-298*c*exp(2)^2*sqrt(-c*exp(2))*b^3*g*exp(1)^5*d^5+
181*c*exp(2)^3*sqrt(-c*exp(2))*b^3*g*exp(1)^3*d^5-35*c*exp(2)^4*sqrt(-c*exp(2))*b^3*g*exp(1)*d^5-84*c*exp(2)*s
qrt(-c*exp(2))*b^3*exp(1)^8*f*d^4+186*c*exp(2)^2*sqrt(-c*exp(2))*b^3*exp(1)^6*f*d^4-129*c*exp(2)^3*sqrt(-c*exp
(2))*b^3*exp(1)^4*f*d^4+27*c*exp(2)^4*sqrt(-c*exp(2))*b^3*exp(1)^2*f*d^4+48*c^2*sqrt(-c*exp(2))*b^2*g*exp(1)^8
*d^6-288*c^2*exp(2)*sqrt(-c*exp(2))*b^2*g*exp(1)^6*d^6+409*c^2*exp(2)^2*sqrt(-c*exp(2))*b^2*g*exp(1)^4*d^6-191
*c^2*exp(2)^3*sqrt(-c*exp(2))*b^2*g*exp(1)^2*d^6+22*c^2*exp(2)^4*sqrt(-c*exp(2))*b^2*g*d^6+156*c^2*exp(2)*sqrt
(-c*exp(2))*b^2*exp(1)^7*f*d^5-285*c^2*exp(2)^2*sqrt(-c*exp(2))*b^2*exp(1)^5*f*d^5+147*c^2*exp(2)^3*sqrt(-c*ex
p(2))*b^2*exp(1)^3*f*d^5-18*c^2*exp(2)^4*sqrt(-c*exp(2))*b^2*exp(1)*f*d^5-32*c^3*sqrt(-c*exp(2))*b*g*exp(1)^7*
d^7+200*c^3*exp(2)*sqrt(-c*exp(2))*b*g*exp(1)^5*d^7-232*c^3*exp(2)^2*sqrt(-c*exp(2))*b*g*exp(1)^3*d^7+64*c^3*e
xp(2)^3*sqrt(-c*exp(2))*b*g*exp(1)*d^7-124*c^3*exp(2)*sqrt(-c*exp(2))*b*exp(1)^6*f*d^6+176*c^3*exp(2)^2*sqrt(-
c*exp(2))*b*exp(1)^4*f*d^6-52*c^3*exp(2)^3*sqrt(-c*exp(2))*b*exp(1)^2*f*d^6+8*c^4*sqrt(-c*exp(2))*g*exp(1)^6*d
^8-52*c^4*exp(2)*sqrt(-c*exp(2))*g*exp(1)^4*d^8+44*c^4*exp(2)^2*sqrt(-c*exp(2))*g*exp(1)^2*d^8+36*c^4*exp(2)*s
qrt(-c*exp(2))*exp(1)^5*f*d^7-36*c^4*exp(2)^2*sqrt(-c*exp(2))*exp(1)^3*f*d^7)/8/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)^2)

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maple [B]  time = 0.07, size = 4726, normalized size = 13.35 \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)^3,x)

[Out]

2/3*g/e^3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+2/3*g/e*c/(-b*e^2+2*c*d
*e)*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)-15*e^3*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*f-25*e^3*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*f-25*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^4*g+10/3*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*
d*g+25*e^4*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*f-25/8*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^2*g+25/2
*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^3*g+15/2*e^4*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*f+25/8*e^6*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*f+25*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^5*g-25/8*g*e*c^3/(-b*e^2+2*c*d*e)*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+5/4*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*d*g-15/2*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^2*g+25/8*g*e^2*c^2/(-b*e^2+2*c*d*e)*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+15/16*g*e^2*c/(-b*e^2+2*c*d*e)
*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d-25/16*g*e^3*c/(-b*e^2+2*c*d*e)*b^3/(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^2+1
5*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^3*g-25/2*e^5*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*f-15/8*g*e*c^2/(-b*e^2+2*c*d*e)*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1
/2)*x*d^2-16/3*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)*d*g+5/4*g*c^4/(-b*e^2+
2*c*d*e)*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))-5/24*g*e/(-b*e^2+2*c*d*e)*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+16/3*e
*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)*f-5/8*e^5/(-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)*f+2/e^3/(-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)*f+5/12*g*c/(-b*e^2+2*c*d*e)*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*b-5/64*g
*e^3/c/(-b*e^2+2*c*d*e)*b^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+5/6*g*c^2/(-b*e^2+2*c*d*e)*d*(-(
x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x+5/4*g*c^3/(-b*e^2+2*c*d*e)*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*
d*e)*(x+d/e))^(1/2)*x+5/8*g*c^2/(-b*e^2+2*c*d*e)*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^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)*d*f+5/16*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))*d*g-16/3/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)*
d*g+15/2*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^3*g+10*e^2*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))*f-10/3*e^3*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*f+5/3*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)*d*g-20/3*e*c^3/(
-b*e^2+2*c*d*e)^2*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*g+20/3*e^2*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*f-10/3*e*c^2/(-b*e^2+2*c*d*e)^2*d^2*(-(x+d/e)^2*c*e^2+(-b*
e^2+2*c*d*e)*(x+d/e))^(3/2)*b*g+10/3*e^2*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*f-10*e*c^4/(-b*e^2+2*c*d*e)^2*d^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*g+10*e^2*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*f-5*e*c^3/(-b*e^2+2*c*d*e)^2*d^4*(-(x+
d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*g+5*e^2*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*f-10*e*c^5/(-b*e^2+2*c*d*e)^2*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-5/4*e^5*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*f-15/4*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^2*g+25/64*g*e^4/(-b*e^2+2*c*d*e)*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-15/16*g*e*c/(-b*e^2+2*c*d*e)*b^2*(-(
x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d^2-5/12*g*e*c/(-b*e^2+2*c*d*e)*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c
*d*e)*(x+d/e))^(3/2)*x-5/128*g*e^5/c/(-b*e^2+2*c*d*e)*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))-15/2*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)*d^2*f+5/8*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)*d*g-5/32*g*e^3/(-b*e^2+2*c*d*e)*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+
15/32*g*e^2/(-b*e^2+2*c*d*e)*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d-2/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)*d*g-5/3*e^3*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)*f-5/16*e^7/(-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))*f+16/3/e*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)*f

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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)^3,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 )}^3} \,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)^3,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)^3, 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 )^{3}}\, 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)**3,x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**3, x)

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