3.2277 \(\int \frac{(d+e x)^{11/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=371 \[ \frac{2 (d+e x)^{7/2} (-8 b e g+11 c d g+5 c e f)}{15 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{4 (d+e x)^{5/2} (-8 b e g+11 c d g+5 c e f)}{15 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (d+e x)^{3/2} (2 c d-b e) (-8 b e g+11 c d g+5 c e f)}{15 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{32 \sqrt{d+e x} (2 c d-b e)^2 (-8 b e g+11 c d g+5 c e f)}{15 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(11/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
)) - (32*(2*c*d - b*e)^2*(5*c*e*f + 11*c*d*g - 8*b*e*g)*Sqrt[d + e*x])/(15*c^5*e^2*Sqrt[d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2]) + (16*(2*c*d - b*e)*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(3/2))/(15*c^4*e^2*Sqrt[d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2]) + (4*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(5/2))/(15*c^3*e^2*Sqrt[d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2]) + (2*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(7/2))/(15*c^2*e^2*(2*c*d - b*e)*Sqrt
[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

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Rubi [A]  time = 0.546034, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {788, 656, 648} \[ \frac{2 (d+e x)^{7/2} (-8 b e g+11 c d g+5 c e f)}{15 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{4 (d+e x)^{5/2} (-8 b e g+11 c d g+5 c e f)}{15 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (d+e x)^{3/2} (2 c d-b e) (-8 b e g+11 c d g+5 c e f)}{15 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{32 \sqrt{d+e x} (2 c d-b e)^2 (-8 b e g+11 c d g+5 c e f)}{15 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(11/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
)) - (32*(2*c*d - b*e)^2*(5*c*e*f + 11*c*d*g - 8*b*e*g)*Sqrt[d + e*x])/(15*c^5*e^2*Sqrt[d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2]) + (16*(2*c*d - b*e)*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(3/2))/(15*c^4*e^2*Sqrt[d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2]) + (4*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(5/2))/(15*c^3*e^2*Sqrt[d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2]) + (2*(5*c*e*f + 11*c*d*g - 8*b*e*g)*(d + e*x)^(7/2))/(15*c^2*e^2*(2*c*d - b*e)*Sqrt
[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

Rule 788

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rule 656

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[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

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*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \frac{(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{(5 c e f+11 c d g-8 b e g) \int \frac{(d+e x)^{9/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(2 (5 c e f+11 c d g-8 b e g)) \int \frac{(d+e x)^{7/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{5 c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(8 (2 c d-b e) (5 c e f+11 c d g-8 b e g)) \int \frac{(d+e x)^{5/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{15 c^3 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{16 (2 c d-b e) (5 c e f+11 c d g-8 b e g) (d+e x)^{3/2}}{15 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{\left (16 (2 c d-b e)^2 (5 c e f+11 c d g-8 b e g)\right ) \int \frac{(d+e x)^{3/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{15 c^4 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{11/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{32 (2 c d-b e)^2 (5 c e f+11 c d g-8 b e g) \sqrt{d+e x}}{15 c^5 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (2 c d-b e) (5 c e f+11 c d g-8 b e g) (d+e x)^{3/2}}{15 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{4 (5 c e f+11 c d g-8 b e g) (d+e x)^{5/2}}{15 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (5 c e f+11 c d g-8 b e g) (d+e x)^{7/2}}{15 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.23666, size = 263, normalized size = 0.71 \[ \frac{2 \sqrt{d+e x} \left (24 b^2 c^2 e^2 \left (67 d^2 g+3 d e (5 f-13 g x)+e^2 x (2 g x-5 f)\right )-16 b^3 c e^3 (47 d g+5 e f-12 e g x)+128 b^4 e^4 g-2 b c^3 e \left (3 d^2 e (85 f-246 g x)+741 d^3 g+3 d e^2 x (31 g x-70 f)+e^3 x^2 (15 f+4 g x)\right )+c^4 \left (3 d^2 e^2 x (61 g x-115 f)+9 d^3 e (25 f-83 g x)+498 d^4 g+d e^3 x^2 (75 f+23 g x)+e^4 x^3 (5 f+3 g x)\right )\right )}{15 c^5 e^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(128*b^4*e^4*g - 16*b^3*c*e^3*(5*e*f + 47*d*g - 12*e*g*x) + 24*b^2*c^2*e^2*(67*d^2*g + 3*d*e*
(5*f - 13*g*x) + e^2*x*(-5*f + 2*g*x)) - 2*b*c^3*e*(741*d^3*g + 3*d^2*e*(85*f - 246*g*x) + e^3*x^2*(15*f + 4*g
*x) + 3*d*e^2*x*(-70*f + 31*g*x)) + c^4*(498*d^4*g + 9*d^3*e*(25*f - 83*g*x) + e^4*x^3*(5*f + 3*g*x) + d*e^3*x
^2*(75*f + 23*g*x) + 3*d^2*e^2*x*(-115*f + 61*g*x))))/(15*c^5*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e
) + c*(d - e*x))])

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Maple [A]  time = 0.009, size = 367, normalized size = 1. \begin{align*}{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3\,g{e}^{4}{x}^{4}{c}^{4}-8\,b{c}^{3}{e}^{4}g{x}^{3}+23\,{c}^{4}d{e}^{3}g{x}^{3}+5\,{c}^{4}{e}^{4}f{x}^{3}+48\,{b}^{2}{c}^{2}{e}^{4}g{x}^{2}-186\,b{c}^{3}d{e}^{3}g{x}^{2}-30\,b{c}^{3}{e}^{4}f{x}^{2}+183\,{c}^{4}{d}^{2}{e}^{2}g{x}^{2}+75\,{c}^{4}d{e}^{3}f{x}^{2}+192\,{b}^{3}c{e}^{4}gx-936\,{b}^{2}{c}^{2}d{e}^{3}gx-120\,{b}^{2}{c}^{2}{e}^{4}fx+1476\,b{c}^{3}{d}^{2}{e}^{2}gx+420\,b{c}^{3}d{e}^{3}fx-747\,{c}^{4}{d}^{3}egx-345\,{c}^{4}{d}^{2}{e}^{2}fx+128\,{b}^{4}{e}^{4}g-752\,{b}^{3}cd{e}^{3}g-80\,{b}^{3}c{e}^{4}f+1608\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}g+360\,{b}^{2}{c}^{2}d{e}^{3}f-1482\,b{c}^{3}{d}^{3}eg-510\,b{c}^{3}{d}^{2}{e}^{2}f+498\,{c}^{4}{d}^{4}g+225\,f{d}^{3}{c}^{4}e \right ) }{15\,{c}^{5}{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

2/15*(c*e*x+b*e-c*d)*(3*c^4*e^4*g*x^4-8*b*c^3*e^4*g*x^3+23*c^4*d*e^3*g*x^3+5*c^4*e^4*f*x^3+48*b^2*c^2*e^4*g*x^
2-186*b*c^3*d*e^3*g*x^2-30*b*c^3*e^4*f*x^2+183*c^4*d^2*e^2*g*x^2+75*c^4*d*e^3*f*x^2+192*b^3*c*e^4*g*x-936*b^2*
c^2*d*e^3*g*x-120*b^2*c^2*e^4*f*x+1476*b*c^3*d^2*e^2*g*x+420*b*c^3*d*e^3*f*x-747*c^4*d^3*e*g*x-345*c^4*d^2*e^2
*f*x+128*b^4*e^4*g-752*b^3*c*d*e^3*g-80*b^3*c*e^4*f+1608*b^2*c^2*d^2*e^2*g+360*b^2*c^2*d*e^3*f-1482*b*c^3*d^3*
e*g-510*b*c^3*d^2*e^2*f+498*c^4*d^4*g+225*c^4*d^3*e*f)*(e*x+d)^(5/2)/c^5/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(5/2)

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Maxima [A]  time = 1.43286, size = 490, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (c^{3} e^{3} x^{3} + 45 \, c^{3} d^{3} - 102 \, b c^{2} d^{2} e + 72 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 3 \,{\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} - 3 \,{\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{3 \,{\left (c^{5} e^{2} x - c^{5} d e + b c^{4} e^{2}\right )} \sqrt{-c e x + c d - b e}} + \frac{2 \,{\left (3 \, c^{4} e^{4} x^{4} + 498 \, c^{4} d^{4} - 1482 \, b c^{3} d^{3} e + 1608 \, b^{2} c^{2} d^{2} e^{2} - 752 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} +{\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} + 3 \,{\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} - 3 \,{\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{15 \,{\left (c^{6} e^{3} x - c^{6} d e^{2} + b c^{5} e^{3}\right )} \sqrt{-c e x + c d - b e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

2/3*(c^3*e^3*x^3 + 45*c^3*d^3 - 102*b*c^2*d^2*e + 72*b^2*c*d*e^2 - 16*b^3*e^3 + 3*(5*c^3*d*e^2 - 2*b*c^2*e^3)*
x^2 - 3*(23*c^3*d^2*e - 28*b*c^2*d*e^2 + 8*b^2*c*e^3)*x)*f/((c^5*e^2*x - c^5*d*e + b*c^4*e^2)*sqrt(-c*e*x + c*
d - b*e)) + 2/15*(3*c^4*e^4*x^4 + 498*c^4*d^4 - 1482*b*c^3*d^3*e + 1608*b^2*c^2*d^2*e^2 - 752*b^3*c*d*e^3 + 12
8*b^4*e^4 + (23*c^4*d*e^3 - 8*b*c^3*e^4)*x^3 + 3*(61*c^4*d^2*e^2 - 62*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*x^2 - 3*(2
49*c^4*d^3*e - 492*b*c^3*d^2*e^2 + 312*b^2*c^2*d*e^3 - 64*b^3*c*e^4)*x)*g/((c^6*e^3*x - c^6*d*e^2 + b*c^5*e^3)
*sqrt(-c*e*x + c*d - b*e))

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Fricas [A]  time = 1.49564, size = 892, normalized size = 2.4 \begin{align*} -\frac{2 \,{\left (3 \, c^{4} e^{4} g x^{4} +{\left (5 \, c^{4} e^{4} f +{\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} g\right )} x^{3} + 3 \,{\left (5 \,{\left (5 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} f +{\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} + 5 \,{\left (45 \, c^{4} d^{3} e - 102 \, b c^{3} d^{2} e^{2} + 72 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} f + 2 \,{\left (249 \, c^{4} d^{4} - 741 \, b c^{3} d^{3} e + 804 \, b^{2} c^{2} d^{2} e^{2} - 376 \, b^{3} c d e^{3} + 64 \, b^{4} e^{4}\right )} g - 3 \,{\left (5 \,{\left (23 \, c^{4} d^{2} e^{2} - 28 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} f +{\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d}}{15 \,{\left (c^{7} e^{5} x^{3} + c^{7} d^{3} e^{2} - 2 \, b c^{6} d^{2} e^{3} + b^{2} c^{5} d e^{4} -{\left (c^{7} d e^{4} - 2 \, b c^{6} e^{5}\right )} x^{2} -{\left (c^{7} d^{2} e^{3} - b^{2} c^{5} e^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-2/15*(3*c^4*e^4*g*x^4 + (5*c^4*e^4*f + (23*c^4*d*e^3 - 8*b*c^3*e^4)*g)*x^3 + 3*(5*(5*c^4*d*e^3 - 2*b*c^3*e^4)
*f + (61*c^4*d^2*e^2 - 62*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*g)*x^2 + 5*(45*c^4*d^3*e - 102*b*c^3*d^2*e^2 + 72*b^2*
c^2*d*e^3 - 16*b^3*c*e^4)*f + 2*(249*c^4*d^4 - 741*b*c^3*d^3*e + 804*b^2*c^2*d^2*e^2 - 376*b^3*c*d*e^3 + 64*b^
4*e^4)*g - 3*(5*(23*c^4*d^2*e^2 - 28*b*c^3*d*e^3 + 8*b^2*c^2*e^4)*f + (249*c^4*d^3*e - 492*b*c^3*d^2*e^2 + 312
*b^2*c^2*d*e^3 - 64*b^3*c*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^7*e^5*x^3 + c
^7*d^3*e^2 - 2*b*c^6*d^2*e^3 + b^2*c^5*d*e^4 - (c^7*d*e^4 - 2*b*c^6*e^5)*x^2 - (c^7*d^2*e^3 - b^2*c^5*e^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(11/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x