3.1799 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=214 \[ -\frac{2 b^3 (d+e x)^{3/2} (-4 a B e-A b e+5 b B d)}{3 e^6}+\frac{4 b^2 \sqrt{d+e x} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{4 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 \sqrt{d+e x}}-\frac{2 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6 (d+e x)^{3/2}}+\frac{2 (b d-a e)^4 (B d-A e)}{5 e^6 (d+e x)^{5/2}}+\frac{2 b^4 B (d+e x)^{5/2}}{5 e^6} \]

[Out]

(2*(b*d - a*e)^4*(B*d - A*e))/(5*e^6*(d + e*x)^(5/2)) - (2*(b*d - a*e)^3*(5*b*B*
d - 4*A*b*e - a*B*e))/(3*e^6*(d + e*x)^(3/2)) + (4*b*(b*d - a*e)^2*(5*b*B*d - 3*
A*b*e - 2*a*B*e))/(e^6*Sqrt[d + e*x]) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e -
3*a*B*e)*Sqrt[d + e*x])/e^6 - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(3/2)
)/(3*e^6) + (2*b^4*B*(d + e*x)^(5/2))/(5*e^6)

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Rubi [A]  time = 0.289292, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{2 b^3 (d+e x)^{3/2} (-4 a B e-A b e+5 b B d)}{3 e^6}+\frac{4 b^2 \sqrt{d+e x} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{4 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 \sqrt{d+e x}}-\frac{2 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6 (d+e x)^{3/2}}+\frac{2 (b d-a e)^4 (B d-A e)}{5 e^6 (d+e x)^{5/2}}+\frac{2 b^4 B (d+e x)^{5/2}}{5 e^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^(7/2),x]

[Out]

(2*(b*d - a*e)^4*(B*d - A*e))/(5*e^6*(d + e*x)^(5/2)) - (2*(b*d - a*e)^3*(5*b*B*
d - 4*A*b*e - a*B*e))/(3*e^6*(d + e*x)^(3/2)) + (4*b*(b*d - a*e)^2*(5*b*B*d - 3*
A*b*e - 2*a*B*e))/(e^6*Sqrt[d + e*x]) + (4*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e -
3*a*B*e)*Sqrt[d + e*x])/e^6 - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(3/2)
)/(3*e^6) + (2*b^4*B*(d + e*x)^(5/2))/(5*e^6)

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Rubi in Sympy [A]  time = 100.63, size = 218, normalized size = 1.02 \[ \frac{2 B b^{4} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{6}} + \frac{2 b^{3} \left (d + e x\right )^{\frac{3}{2}} \left (A b e + 4 B a e - 5 B b d\right )}{3 e^{6}} + \frac{4 b^{2} \sqrt{d + e x} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{e^{6}} - \frac{4 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{e^{6} \sqrt{d + e x}} - \frac{2 \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (A e - B d\right ) \left (a e - b d\right )^{4}}{5 e^{6} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(7/2),x)

[Out]

2*B*b**4*(d + e*x)**(5/2)/(5*e**6) + 2*b**3*(d + e*x)**(3/2)*(A*b*e + 4*B*a*e -
5*B*b*d)/(3*e**6) + 4*b**2*sqrt(d + e*x)*(a*e - b*d)*(2*A*b*e + 3*B*a*e - 5*B*b*
d)/e**6 - 4*b*(a*e - b*d)**2*(3*A*b*e + 2*B*a*e - 5*B*b*d)/(e**6*sqrt(d + e*x))
- 2*(a*e - b*d)**3*(4*A*b*e + B*a*e - 5*B*b*d)/(3*e**6*(d + e*x)**(3/2)) - 2*(A*
e - B*d)*(a*e - b*d)**4/(5*e**6*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.438177, size = 337, normalized size = 1.57 \[ -\frac{2 \left (a^4 e^4 (3 A e+2 B d+5 B e x)+4 a^3 b e^3 \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )-6 a^2 b^2 e^2 \left (3 B \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )-A e \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+4 a b^3 e \left (B \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-3 A e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )+b^4 \left (A e \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-B \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )\right )}{15 e^6 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(a^4*e^4*(2*B*d + 3*A*e + 5*B*e*x) + 4*a^3*b*e^3*(A*e*(2*d + 5*e*x) + B*(8*d
^2 + 20*d*e*x + 15*e^2*x^2)) - 6*a^2*b^2*e^2*(-(A*e*(8*d^2 + 20*d*e*x + 15*e^2*x
^2)) + 3*B*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3)) + 4*a*b^3*e*(-3*A*e
*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + B*(128*d^4 + 320*d^3*e*x + 2
40*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4)) + b^4*(A*e*(128*d^4 + 320*d^3*e*x +
240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) - B*(256*d^5 + 640*d^4*e*x + 480*d^3
*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5))))/(15*e^6*(d + e*x)^(5/2)
)

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Maple [B]  time = 0.013, size = 469, normalized size = 2.2 \[ -{\frac{-6\,{b}^{4}B{x}^{5}{e}^{5}-10\,A{b}^{4}{e}^{5}{x}^{4}-40\,Ba{b}^{3}{e}^{5}{x}^{4}+20\,B{b}^{4}d{e}^{4}{x}^{4}-120\,Aa{b}^{3}{e}^{5}{x}^{3}+80\,A{b}^{4}d{e}^{4}{x}^{3}-180\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}+320\,Ba{b}^{3}d{e}^{4}{x}^{3}-160\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+180\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-720\,Aa{b}^{3}d{e}^{4}{x}^{2}+480\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+120\,B{a}^{3}b{e}^{5}{x}^{2}-1080\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+1920\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-960\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+40\,A{a}^{3}b{e}^{5}x+240\,A{a}^{2}{b}^{2}d{e}^{4}x-960\,Aa{b}^{3}{d}^{2}{e}^{3}x+640\,A{b}^{4}{d}^{3}{e}^{2}x+10\,B{a}^{4}{e}^{5}x+160\,B{a}^{3}bd{e}^{4}x-1440\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x+2560\,Ba{b}^{3}{d}^{3}{e}^{2}x-1280\,B{b}^{4}{d}^{4}ex+6\,A{a}^{4}{e}^{5}+16\,Ad{a}^{3}b{e}^{4}+96\,A{a}^{2}{b}^{2}{d}^{2}{e}^{3}-384\,Aa{b}^{3}{d}^{3}{e}^{2}+256\,A{d}^{4}{b}^{4}e+4\,B{a}^{4}d{e}^{4}+64\,B{a}^{3}b{d}^{2}{e}^{3}-576\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+1024\,B{d}^{4}a{b}^{3}e-512\,{b}^{4}B{d}^{5}}{15\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(7/2),x)

[Out]

-2/15*(-3*B*b^4*e^5*x^5-5*A*b^4*e^5*x^4-20*B*a*b^3*e^5*x^4+10*B*b^4*d*e^4*x^4-60
*A*a*b^3*e^5*x^3+40*A*b^4*d*e^4*x^3-90*B*a^2*b^2*e^5*x^3+160*B*a*b^3*d*e^4*x^3-8
0*B*b^4*d^2*e^3*x^3+90*A*a^2*b^2*e^5*x^2-360*A*a*b^3*d*e^4*x^2+240*A*b^4*d^2*e^3
*x^2+60*B*a^3*b*e^5*x^2-540*B*a^2*b^2*d*e^4*x^2+960*B*a*b^3*d^2*e^3*x^2-480*B*b^
4*d^3*e^2*x^2+20*A*a^3*b*e^5*x+120*A*a^2*b^2*d*e^4*x-480*A*a*b^3*d^2*e^3*x+320*A
*b^4*d^3*e^2*x+5*B*a^4*e^5*x+80*B*a^3*b*d*e^4*x-720*B*a^2*b^2*d^2*e^3*x+1280*B*a
*b^3*d^3*e^2*x-640*B*b^4*d^4*e*x+3*A*a^4*e^5+8*A*a^3*b*d*e^4+48*A*a^2*b^2*d^2*e^
3-192*A*a*b^3*d^3*e^2+128*A*b^4*d^4*e+2*B*a^4*d*e^4+32*B*a^3*b*d^2*e^3-288*B*a^2
*b^2*d^3*e^2+512*B*a*b^3*d^4*e-256*B*b^4*d^5)/(e*x+d)^(5/2)/e^6

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Maxima [A]  time = 0.729691, size = 562, normalized size = 2.63 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B b^{4} - 5 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 30 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} \sqrt{e x + d}}{e^{5}} + \frac{3 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 6 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 30 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{5}}\right )}}{15 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/15*((3*(e*x + d)^(5/2)*B*b^4 - 5*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x + d)
^(3/2) + 30*(5*B*b^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d*e + (3*B*a^2*b^2 + 2*A*a*b^3)
*e^2)*sqrt(e*x + d))/e^5 + (3*B*b^4*d^5 - 3*A*a^4*e^5 - 3*(4*B*a*b^3 + A*b^4)*d^
4*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 6*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3
+ 3*(B*a^4 + 4*A*a^3*b)*d*e^4 + 30*(5*B*b^4*d^3 - 3*(4*B*a*b^3 + A*b^4)*d^2*e +
3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a^3*b + 3*A*a^2*b^2)*e^3)*(e*x + d)^2 -
5*(5*B*b^4*d^4 - 4*(4*B*a*b^3 + A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*
e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*(e*x + d))/((
e*x + d)^(5/2)*e^5))/e

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Fricas [A]  time = 0.287915, size = 581, normalized size = 2.71 \[ \frac{2 \,{\left (3 \, B b^{4} e^{5} x^{5} + 256 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} - 128 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 96 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 16 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 2 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 5 \,{\left (2 \, B b^{4} d e^{4} -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 10 \,{\left (8 \, B b^{4} d^{2} e^{3} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 30 \,{\left (16 \, B b^{4} d^{3} e^{2} - 8 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 5 \,{\left (128 \, B b^{4} d^{4} e - 64 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 48 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 8 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} -{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )}}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(3*B*b^4*e^5*x^5 + 256*B*b^4*d^5 - 3*A*a^4*e^5 - 128*(4*B*a*b^3 + A*b^4)*d^
4*e + 96*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 16*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^
3 - 2*(B*a^4 + 4*A*a^3*b)*d*e^4 - 5*(2*B*b^4*d*e^4 - (4*B*a*b^3 + A*b^4)*e^5)*x^
4 + 10*(8*B*b^4*d^2*e^3 - 4*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b
^3)*e^5)*x^3 + 30*(16*B*b^4*d^3*e^2 - 8*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 6*(3*B*a^2
*b^2 + 2*A*a*b^3)*d*e^4 - (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 5*(128*B*b^4*d^4*
e - 64*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 48*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - 8*(2
*B*a^3*b + 3*A*a^2*b^2)*d*e^4 - (B*a^4 + 4*A*a^3*b)*e^5)*x)/((e^8*x^2 + 2*d*e^7*
x + d^2*e^6)*sqrt(e*x + d))

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Sympy [A]  time = 15.6886, size = 2440, normalized size = 11.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a**4*e**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x
) + 15*e**8*x**2*sqrt(d + e*x)) - 16*A*a**3*b*d*e**4/(15*d**2*e**6*sqrt(d + e*x)
+ 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 40*A*a**3*b*e**5*x/
(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d +
e*x)) - 96*A*a**2*b**2*d**2*e**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(
d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 240*A*a**2*b**2*d*e**4*x/(15*d**2*e**6*
sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 180*A*
a**2*b**2*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15
*e**8*x**2*sqrt(d + e*x)) + 384*A*a*b**3*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) +
30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 960*A*a*b**3*d**2*e**
3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(
d + e*x)) + 720*A*a*b**3*d*e**4*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*s
qrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 120*A*a*b**3*e**5*x**3/(15*d**2*e**
6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 256*
A*b**4*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*
x**2*sqrt(d + e*x)) - 640*A*b**4*d**3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*
e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 480*A*b**4*d**2*e**3*x**2/(
15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e
*x)) - 80*A*b**4*d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d +
e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 10*A*b**4*e**5*x**4/(15*d**2*e**6*sqrt(d +
e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 4*B*a**4*d*e**4
/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d +
e*x)) - 10*B*a**4*e**5*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x
) + 15*e**8*x**2*sqrt(d + e*x)) - 64*B*a**3*b*d**2*e**3/(15*d**2*e**6*sqrt(d + e
*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 160*B*a**3*b*d*e
**4*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqr
t(d + e*x)) - 120*B*a**3*b*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*s
qrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 576*B*a**2*b**2*d**3*e**2/(15*d**2*
e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 1
440*B*a**2*b**2*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e
*x) + 15*e**8*x**2*sqrt(d + e*x)) + 1080*B*a**2*b**2*d*e**4*x**2/(15*d**2*e**6*s
qrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 180*B*a
**2*b**2*e**5*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*
e**8*x**2*sqrt(d + e*x)) - 1024*B*a*b**3*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30
*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 2560*B*a*b**3*d**3*e**2*
x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d
+ e*x)) - 1920*B*a*b**3*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x
*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 320*B*a*b**3*d*e**4*x**3/(15*d**2
*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) +
40*B*a*b**3*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) +
15*e**8*x**2*sqrt(d + e*x)) + 512*B*b**4*d**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d
*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 1280*B*b**4*d**4*e*x/(15*d
**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x))
+ 960*B*b**4*d**3*e**2*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d +
e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 160*B*b**4*d**2*e**3*x**3/(15*d**2*e**6*sqr
t(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 20*B*b**4
*d*e**4*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x
**2*sqrt(d + e*x)) + 6*B*b**4*e**5*x**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*
x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a**4*x + 2*A*a**3*
b*x**2 + 2*A*a**2*b**2*x**3 + A*a*b**3*x**4 + A*b**4*x**5/5 + B*a**4*x**2/2 + 4*
B*a**3*b*x**3/3 + 3*B*a**2*b**2*x**4/2 + 4*B*a*b**3*x**5/5 + B*b**4*x**6/6)/d**(
7/2), True))

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GIAC/XCAS [A]  time = 0.297136, size = 765, normalized size = 3.57 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} e^{24} - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d e^{24} + 150 \, \sqrt{x e + d} B b^{4} d^{2} e^{24} + 20 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} e^{25} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} e^{25} - 240 \, \sqrt{x e + d} B a b^{3} d e^{25} - 60 \, \sqrt{x e + d} A b^{4} d e^{25} + 90 \, \sqrt{x e + d} B a^{2} b^{2} e^{26} + 60 \, \sqrt{x e + d} A a b^{3} e^{26}\right )} e^{\left (-30\right )} + \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} B b^{4} d^{3} - 25 \,{\left (x e + d\right )} B b^{4} d^{4} + 3 \, B b^{4} d^{5} - 360 \,{\left (x e + d\right )}^{2} B a b^{3} d^{2} e - 90 \,{\left (x e + d\right )}^{2} A b^{4} d^{2} e + 80 \,{\left (x e + d\right )} B a b^{3} d^{3} e + 20 \,{\left (x e + d\right )} A b^{4} d^{3} e - 12 \, B a b^{3} d^{4} e - 3 \, A b^{4} d^{4} e + 270 \,{\left (x e + d\right )}^{2} B a^{2} b^{2} d e^{2} + 180 \,{\left (x e + d\right )}^{2} A a b^{3} d e^{2} - 90 \,{\left (x e + d\right )} B a^{2} b^{2} d^{2} e^{2} - 60 \,{\left (x e + d\right )} A a b^{3} d^{2} e^{2} + 18 \, B a^{2} b^{2} d^{3} e^{2} + 12 \, A a b^{3} d^{3} e^{2} - 60 \,{\left (x e + d\right )}^{2} B a^{3} b e^{3} - 90 \,{\left (x e + d\right )}^{2} A a^{2} b^{2} e^{3} + 40 \,{\left (x e + d\right )} B a^{3} b d e^{3} + 60 \,{\left (x e + d\right )} A a^{2} b^{2} d e^{3} - 12 \, B a^{3} b d^{2} e^{3} - 18 \, A a^{2} b^{2} d^{2} e^{3} - 5 \,{\left (x e + d\right )} B a^{4} e^{4} - 20 \,{\left (x e + d\right )} A a^{3} b e^{4} + 3 \, B a^{4} d e^{4} + 12 \, A a^{3} b d e^{4} - 3 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*b^4*e^24 - 25*(x*e + d)^(3/2)*B*b^4*d*e^24 + 150*sqrt(
x*e + d)*B*b^4*d^2*e^24 + 20*(x*e + d)^(3/2)*B*a*b^3*e^25 + 5*(x*e + d)^(3/2)*A*
b^4*e^25 - 240*sqrt(x*e + d)*B*a*b^3*d*e^25 - 60*sqrt(x*e + d)*A*b^4*d*e^25 + 90
*sqrt(x*e + d)*B*a^2*b^2*e^26 + 60*sqrt(x*e + d)*A*a*b^3*e^26)*e^(-30) + 2/15*(1
50*(x*e + d)^2*B*b^4*d^3 - 25*(x*e + d)*B*b^4*d^4 + 3*B*b^4*d^5 - 360*(x*e + d)^
2*B*a*b^3*d^2*e - 90*(x*e + d)^2*A*b^4*d^2*e + 80*(x*e + d)*B*a*b^3*d^3*e + 20*(
x*e + d)*A*b^4*d^3*e - 12*B*a*b^3*d^4*e - 3*A*b^4*d^4*e + 270*(x*e + d)^2*B*a^2*
b^2*d*e^2 + 180*(x*e + d)^2*A*a*b^3*d*e^2 - 90*(x*e + d)*B*a^2*b^2*d^2*e^2 - 60*
(x*e + d)*A*a*b^3*d^2*e^2 + 18*B*a^2*b^2*d^3*e^2 + 12*A*a*b^3*d^3*e^2 - 60*(x*e
+ d)^2*B*a^3*b*e^3 - 90*(x*e + d)^2*A*a^2*b^2*e^3 + 40*(x*e + d)*B*a^3*b*d*e^3 +
60*(x*e + d)*A*a^2*b^2*d*e^3 - 12*B*a^3*b*d^2*e^3 - 18*A*a^2*b^2*d^2*e^3 - 5*(x
*e + d)*B*a^4*e^4 - 20*(x*e + d)*A*a^3*b*e^4 + 3*B*a^4*d*e^4 + 12*A*a^3*b*d*e^4
- 3*A*a^4*e^5)*e^(-6)/(x*e + d)^(5/2)