3.1224 \(\int (A+B x) \sqrt{d+e x} (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=267 \[ -\frac{2 (d+e x)^{9/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{9 e^6}+\frac{2 (d+e x)^{7/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{7 e^6}-\frac{2 d^2 (d+e x)^{3/2} (B d-A e) (c d-b e)^2}{3 e^6}-\frac{2 c (d+e x)^{11/2} (-A c e-2 b B e+5 B c d)}{11 e^6}+\frac{2 d (d+e x)^{5/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6}+\frac{2 B c^2 (d+e x)^{13/2}}{13 e^6} \]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(3/2))/(3*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(5/2))/(5*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))*(d + e*x)^(7/2))/(7*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(
d + e*x)^(9/2))/(9*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(11/2))/(11*e^6) + (2*B*c^2*(d + e*x)^(13
/2))/(13*e^6)

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Rubi [A]  time = 0.152945, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ -\frac{2 (d+e x)^{9/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{9 e^6}+\frac{2 (d+e x)^{7/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{7 e^6}-\frac{2 d^2 (d+e x)^{3/2} (B d-A e) (c d-b e)^2}{3 e^6}-\frac{2 c (d+e x)^{11/2} (-A c e-2 b B e+5 B c d)}{11 e^6}+\frac{2 d (d+e x)^{5/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6}+\frac{2 B c^2 (d+e x)^{13/2}}{13 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*Sqrt[d + e*x]*(b*x + c*x^2)^2,x]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(3/2))/(3*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(5/2))/(5*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))*(d + e*x)^(7/2))/(7*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(
d + e*x)^(9/2))/(9*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(11/2))/(11*e^6) + (2*B*c^2*(d + e*x)^(13
/2))/(13*e^6)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (A+B x) \sqrt{d+e x} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2 \sqrt{d+e x}}{e^5}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{3/2}}{e^5}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{5/2}}{e^5}+\frac{\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{7/2}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{9/2}}{e^5}+\frac{B c^2 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac{2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{3/2}}{3 e^6}+\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{5/2}}{5 e^6}+\frac{2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{7/2}}{7 e^6}-\frac{2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^6}-\frac{2 c (5 B c d-2 b B e-A c e) (d+e x)^{11/2}}{11 e^6}+\frac{2 B c^2 (d+e x)^{13/2}}{13 e^6}\\ \end{align*}

Mathematica [A]  time = 0.200708, size = 273, normalized size = 1.02 \[ \frac{2 (d+e x)^{3/2} \left (13 A e \left (33 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 b c e \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )+c^2 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )+B \left (143 b^2 e^2 \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )+26 b c e \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )-5 c^2 \left (480 d^3 e^2 x^2-560 d^2 e^3 x^3-384 d^4 e x+256 d^5+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*Sqrt[d + e*x]*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(3/2)*(13*A*e*(33*b^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 22*b*c*e*(-16*d^3 + 24*d^2*e*x - 30*d
*e^2*x^2 + 35*e^3*x^3) + c^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) + B*(143
*b^2*e^2*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 26*b*c*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^
2 - 280*d*e^3*x^3 + 315*e^4*x^4) - 5*c^2*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^
4*x^4 - 693*e^5*x^5))))/(45045*e^6)

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Maple [A]  time = 0.007, size = 341, normalized size = 1.3 \begin{align*}{\frac{6930\,B{c}^{2}{x}^{5}{e}^{5}+8190\,A{c}^{2}{e}^{5}{x}^{4}+16380\,Bbc{e}^{5}{x}^{4}-6300\,B{c}^{2}d{e}^{4}{x}^{4}+20020\,Abc{e}^{5}{x}^{3}-7280\,A{c}^{2}d{e}^{4}{x}^{3}+10010\,B{b}^{2}{e}^{5}{x}^{3}-14560\,Bbcd{e}^{4}{x}^{3}+5600\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+12870\,A{b}^{2}{e}^{5}{x}^{2}-17160\,Abcd{e}^{4}{x}^{2}+6240\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-8580\,B{b}^{2}d{e}^{4}{x}^{2}+12480\,Bbc{d}^{2}{e}^{3}{x}^{2}-4800\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-10296\,A{b}^{2}d{e}^{4}x+13728\,Abc{d}^{2}{e}^{3}x-4992\,A{c}^{2}{d}^{3}{e}^{2}x+6864\,B{b}^{2}{d}^{2}{e}^{3}x-9984\,Bbc{d}^{3}{e}^{2}x+3840\,B{c}^{2}{d}^{4}ex+6864\,A{b}^{2}{d}^{2}{e}^{3}-9152\,Abc{d}^{3}{e}^{2}+3328\,A{c}^{2}{d}^{4}e-4576\,B{b}^{2}{d}^{3}{e}^{2}+6656\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(1/2)*(c*x^2+b*x)^2,x)

[Out]

2/45045*(e*x+d)^(3/2)*(3465*B*c^2*e^5*x^5+4095*A*c^2*e^5*x^4+8190*B*b*c*e^5*x^4-3150*B*c^2*d*e^4*x^4+10010*A*b
*c*e^5*x^3-3640*A*c^2*d*e^4*x^3+5005*B*b^2*e^5*x^3-7280*B*b*c*d*e^4*x^3+2800*B*c^2*d^2*e^3*x^3+6435*A*b^2*e^5*
x^2-8580*A*b*c*d*e^4*x^2+3120*A*c^2*d^2*e^3*x^2-4290*B*b^2*d*e^4*x^2+6240*B*b*c*d^2*e^3*x^2-2400*B*c^2*d^3*e^2
*x^2-5148*A*b^2*d*e^4*x+6864*A*b*c*d^2*e^3*x-2496*A*c^2*d^3*e^2*x+3432*B*b^2*d^2*e^3*x-4992*B*b*c*d^3*e^2*x+19
20*B*c^2*d^4*e*x+3432*A*b^2*d^2*e^3-4576*A*b*c*d^3*e^2+1664*A*c^2*d^4*e-2288*B*b^2*d^3*e^2+3328*B*b*c*d^4*e-12
80*B*c^2*d^5)/e^6

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Maxima [A]  time = 1.0942, size = 393, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} B c^{2} - 4095 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 5005 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 6435 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 15015 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

2/45045*(3465*(e*x + d)^(13/2)*B*c^2 - 4095*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(11/2) + 5005*(10*B*c^
2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(9/2) - 6435*(10*B*c^2*d^3 - A*b^2*e^3 - 6*
(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(7/2) + 9009*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(
2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(5/2) - 15015*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*
B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*(e*x + d)^(3/2))/e^6

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Fricas [A]  time = 1.79195, size = 824, normalized size = 3.09 \begin{align*} \frac{2 \,{\left (3465 \, B c^{2} e^{6} x^{6} - 1280 \, B c^{2} d^{6} + 3432 \, A b^{2} d^{3} e^{3} + 1664 \,{\left (2 \, B b c + A c^{2}\right )} d^{5} e - 2288 \,{\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{2} + 315 \,{\left (B c^{2} d e^{5} + 13 \,{\left (2 \, B b c + A c^{2}\right )} e^{6}\right )} x^{5} - 35 \,{\left (10 \, B c^{2} d^{2} e^{4} - 13 \,{\left (2 \, B b c + A c^{2}\right )} d e^{5} - 143 \,{\left (B b^{2} + 2 \, A b c\right )} e^{6}\right )} x^{4} + 5 \,{\left (80 \, B c^{2} d^{3} e^{3} + 1287 \, A b^{2} e^{6} - 104 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{4} + 143 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{5}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{4} e^{2} - 429 \, A b^{2} d e^{5} - 208 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{3} + 286 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{5} e - 429 \, A b^{2} d^{2} e^{4} - 208 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e^{2} + 286 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

2/45045*(3465*B*c^2*e^6*x^6 - 1280*B*c^2*d^6 + 3432*A*b^2*d^3*e^3 + 1664*(2*B*b*c + A*c^2)*d^5*e - 2288*(B*b^2
+ 2*A*b*c)*d^4*e^2 + 315*(B*c^2*d*e^5 + 13*(2*B*b*c + A*c^2)*e^6)*x^5 - 35*(10*B*c^2*d^2*e^4 - 13*(2*B*b*c +
A*c^2)*d*e^5 - 143*(B*b^2 + 2*A*b*c)*e^6)*x^4 + 5*(80*B*c^2*d^3*e^3 + 1287*A*b^2*e^6 - 104*(2*B*b*c + A*c^2)*d
^2*e^4 + 143*(B*b^2 + 2*A*b*c)*d*e^5)*x^3 - 3*(160*B*c^2*d^4*e^2 - 429*A*b^2*d*e^5 - 208*(2*B*b*c + A*c^2)*d^3
*e^3 + 286*(B*b^2 + 2*A*b*c)*d^2*e^4)*x^2 + 4*(160*B*c^2*d^5*e - 429*A*b^2*d^2*e^4 - 208*(2*B*b*c + A*c^2)*d^4
*e^2 + 286*(B*b^2 + 2*A*b*c)*d^3*e^3)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 9.92, size = 377, normalized size = 1.41 \begin{align*} \frac{2 \left (\frac{B c^{2} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{11 e^{5}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{9 e^{5}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 2 A b^{2} d e^{3} + 6 A b c d^{2} e^{2} - 4 A c^{2} d^{3} e + 3 B b^{2} d^{2} e^{2} - 8 B b c d^{3} e + 5 B c^{2} d^{4}\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A b^{2} d^{2} e^{3} - 2 A b c d^{3} e^{2} + A c^{2} d^{4} e - B b^{2} d^{3} e^{2} + 2 B b c d^{4} e - B c^{2} d^{5}\right )}{3 e^{5}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(1/2)*(c*x**2+b*x)**2,x)

[Out]

2*(B*c**2*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(A*c**2*e + 2*B*b*c*e - 5*B*c**2*d)/(11*e**5) + (d +
e*x)**(9/2)*(2*A*b*c*e**2 - 4*A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*d*e + 10*B*c**2*d**2)/(9*e**5) + (d + e*x)**
(7/2)*(A*b**2*e**3 - 6*A*b*c*d*e**2 + 6*A*c**2*d**2*e - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d**3)/(7
*e**5) + (d + e*x)**(5/2)*(-2*A*b**2*d*e**3 + 6*A*b*c*d**2*e**2 - 4*A*c**2*d**3*e + 3*B*b**2*d**2*e**2 - 8*B*b
*c*d**3*e + 5*B*c**2*d**4)/(5*e**5) + (d + e*x)**(3/2)*(A*b**2*d**2*e**3 - 2*A*b*c*d**3*e**2 + A*c**2*d**4*e -
B*b**2*d**3*e**2 + 2*B*b*c*d**4*e - B*c**2*d**5)/(3*e**5))/e

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Giac [A]  time = 1.37827, size = 510, normalized size = 1.91 \begin{align*} \frac{2}{45045} \,{\left (429 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} A b^{2} e^{\left (-2\right )} + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} B b^{2} e^{\left (-3\right )} + 286 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} A b c e^{\left (-3\right )} + 26 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} B b c e^{\left (-4\right )} + 13 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} A c^{2} e^{\left (-4\right )} + 5 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\right )} B c^{2} e^{\left (-5\right )}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2/45045*(429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*b^2*e^(-2) + 143*(35*(x*e
+ d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*b^2*e^(-3) + 286*(35
*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*b*c*e^(-3) + 2
6*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*
(x*e + d)^(3/2)*d^4)*B*b*c*e^(-4) + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d
^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*c^2*e^(-4) + 5*(693*(x*e + d)^(13/2) - 4095*(x*e +
d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e +
d)^(3/2)*d^5)*B*c^2*e^(-5))*e^(-1)