3.1843 \(\int (A+B x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=308 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^5 (a+b x)} \]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e
^5*(a + b*x)) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(11/2)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (6*b*(b*d - a*e)*(2*b*B*d - A*
b*e - a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x))
- (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(15*e^5*(a + b*x)) + (2*b^3*B*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(17*e^5*(a + b*x))

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Rubi [A]  time = 0.512707, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e
^5*(a + b*x)) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(11/2)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (6*b*(b*d - a*e)*(2*b*B*d - A*
b*e - a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x))
- (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(15*e^5*(a + b*x)) + (2*b^3*B*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(17*e^5*(a + b*x))

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Rubi in Sympy [A]  time = 46.6718, size = 311, normalized size = 1.01 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{\frac{9}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{17 b e} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (17 A b e - 9 B a e - 8 B b d\right )}{255 b e^{2}} + \frac{4 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (17 A b e - 9 B a e - 8 B b d\right )}{3315 b e^{3}} + \frac{16 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (17 A b e - 9 B a e - 8 B b d\right )}{12155 b e^{4}} + \frac{32 \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (17 A b e - 9 B a e - 8 B b d\right )}{109395 b e^{5} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**(9/2)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(17*b*e) +
2*(d + e*x)**(9/2)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(17*A*b*e - 9*B*a*e - 8*B
*b*d)/(255*b*e**2) + 4*(3*a + 3*b*x)*(d + e*x)**(9/2)*(a*e - b*d)*sqrt(a**2 + 2*
a*b*x + b**2*x**2)*(17*A*b*e - 9*B*a*e - 8*B*b*d)/(3315*b*e**3) + 16*(d + e*x)**
(9/2)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(17*A*b*e - 9*B*a*e - 8*B*
b*d)/(12155*b*e**4) + 32*(d + e*x)**(9/2)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b
**2*x**2)*(17*A*b*e - 9*B*a*e - 8*B*b*d)/(109395*b*e**5*(a + b*x))

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Mathematica [A]  time = 0.423703, size = 245, normalized size = 0.8 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{9/2} \left (1105 a^3 e^3 (11 A e-2 B d+9 B e x)+255 a^2 b e^2 \left (13 A e (9 e x-2 d)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )-51 a b^2 e \left (B \left (16 d^3-72 d^2 e x+198 d e^2 x^2-429 e^3 x^3\right )-5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+b^3 \left (17 A e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+B \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )\right )}{109395 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(9/2)*(1105*a^3*e^3*(-2*B*d + 11*A*e + 9*B*e*x) +
255*a^2*b*e^2*(13*A*e*(-2*d + 9*e*x) + B*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) - 51*
a*b^2*e*(-5*A*e*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + B*(16*d^3 - 72*d^2*e*x + 198*d
*e^2*x^2 - 429*e^3*x^3)) + b^3*(17*A*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 4
29*e^3*x^3) + B*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 643
5*e^4*x^4))))/(109395*e^5*(a + b*x))

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Maple [A]  time = 0.014, size = 317, normalized size = 1. \[{\frac{12870\,B{x}^{4}{b}^{3}{e}^{4}+14586\,A{x}^{3}{b}^{3}{e}^{4}+43758\,B{x}^{3}a{b}^{2}{e}^{4}-6864\,B{x}^{3}{b}^{3}d{e}^{3}+50490\,A{x}^{2}a{b}^{2}{e}^{4}-6732\,A{x}^{2}{b}^{3}d{e}^{3}+50490\,B{x}^{2}{a}^{2}b{e}^{4}-20196\,B{x}^{2}a{b}^{2}d{e}^{3}+3168\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+59670\,Ax{a}^{2}b{e}^{4}-18360\,Axa{b}^{2}d{e}^{3}+2448\,Ax{b}^{3}{d}^{2}{e}^{2}+19890\,Bx{a}^{3}{e}^{4}-18360\,Bx{a}^{2}bd{e}^{3}+7344\,Bxa{b}^{2}{d}^{2}{e}^{2}-1152\,Bx{b}^{3}{d}^{3}e+24310\,A{a}^{3}{e}^{4}-13260\,Ad{e}^{3}{a}^{2}b+4080\,Aa{b}^{2}{d}^{2}{e}^{2}-544\,A{b}^{3}{d}^{3}e-4420\,Bd{e}^{3}{a}^{3}+4080\,B{a}^{2}b{d}^{2}{e}^{2}-1632\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{109395\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{9}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

2/109395*(e*x+d)^(9/2)*(6435*B*b^3*e^4*x^4+7293*A*b^3*e^4*x^3+21879*B*a*b^2*e^4*
x^3-3432*B*b^3*d*e^3*x^3+25245*A*a*b^2*e^4*x^2-3366*A*b^3*d*e^3*x^2+25245*B*a^2*
b*e^4*x^2-10098*B*a*b^2*d*e^3*x^2+1584*B*b^3*d^2*e^2*x^2+29835*A*a^2*b*e^4*x-918
0*A*a*b^2*d*e^3*x+1224*A*b^3*d^2*e^2*x+9945*B*a^3*e^4*x-9180*B*a^2*b*d*e^3*x+367
2*B*a*b^2*d^2*e^2*x-576*B*b^3*d^3*e*x+12155*A*a^3*e^4-6630*A*a^2*b*d*e^3+2040*A*
a*b^2*d^2*e^2-272*A*b^3*d^3*e-2210*B*a^3*d*e^3+2040*B*a^2*b*d^2*e^2-816*B*a*b^2*
d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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Maxima [A]  time = 0.737583, size = 941, normalized size = 3.06 \[ \frac{2 \,{\left (429 \, b^{3} e^{7} x^{7} - 16 \, b^{3} d^{7} + 120 \, a b^{2} d^{6} e - 390 \, a^{2} b d^{5} e^{2} + 715 \, a^{3} d^{4} e^{3} + 33 \,{\left (46 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 9 \,{\left (206 \, b^{3} d^{2} e^{5} + 600 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 5 \,{\left (160 \, b^{3} d^{3} e^{4} + 1374 \, a b^{2} d^{2} e^{5} + 1326 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} + 5 \,{\left (b^{3} d^{4} e^{3} + 636 \, a b^{2} d^{3} e^{4} + 1794 \, a^{2} b d^{2} e^{5} + 572 \, a^{3} d e^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{5} e^{2} - 15 \, a b^{2} d^{4} e^{3} - 1560 \, a^{2} b d^{3} e^{4} - 1430 \, a^{3} d^{2} e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{6} e - 60 \, a b^{2} d^{5} e^{2} + 195 \, a^{2} b d^{4} e^{3} + 2860 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d} A}{6435 \, e^{4}} + \frac{2 \,{\left (6435 \, b^{3} e^{8} x^{8} + 128 \, b^{3} d^{8} - 816 \, a b^{2} d^{7} e + 2040 \, a^{2} b d^{6} e^{2} - 2210 \, a^{3} d^{5} e^{3} + 429 \,{\left (52 \, b^{3} d e^{7} + 51 \, a b^{2} e^{8}\right )} x^{7} + 33 \,{\left (802 \, b^{3} d^{2} e^{6} + 2346 \, a b^{2} d e^{7} + 765 \, a^{2} b e^{8}\right )} x^{6} + 9 \,{\left (1212 \, b^{3} d^{3} e^{5} + 10506 \, a b^{2} d^{2} e^{6} + 10200 \, a^{2} b d e^{7} + 1105 \, a^{3} e^{8}\right )} x^{5} + 5 \,{\left (7 \, b^{3} d^{4} e^{4} + 8160 \, a b^{2} d^{3} e^{5} + 23358 \, a^{2} b d^{2} e^{6} + 7514 \, a^{3} d e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{5} e^{3} - 51 \, a b^{2} d^{4} e^{4} - 10812 \, a^{2} b d^{3} e^{5} - 10166 \, a^{3} d^{2} e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{6} e^{2} - 102 \, a b^{2} d^{5} e^{3} + 255 \, a^{2} b d^{4} e^{4} + 8840 \, a^{3} d^{3} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{7} e - 408 \, a b^{2} d^{6} e^{2} + 1020 \, a^{2} b d^{5} e^{3} - 1105 \, a^{3} d^{4} e^{4}\right )} x\right )} \sqrt{e x + d} B}{109395 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/6435*(429*b^3*e^7*x^7 - 16*b^3*d^7 + 120*a*b^2*d^6*e - 390*a^2*b*d^5*e^2 + 715
*a^3*d^4*e^3 + 33*(46*b^3*d*e^6 + 45*a*b^2*e^7)*x^6 + 9*(206*b^3*d^2*e^5 + 600*a
*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 5*(160*b^3*d^3*e^4 + 1374*a*b^2*d^2*e^5 + 1326
*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 + 5*(b^3*d^4*e^3 + 636*a*b^2*d^3*e^4 + 1794*a^2*
b*d^2*e^5 + 572*a^3*d*e^6)*x^3 - 3*(2*b^3*d^5*e^2 - 15*a*b^2*d^4*e^3 - 1560*a^2*
b*d^3*e^4 - 1430*a^3*d^2*e^5)*x^2 + (8*b^3*d^6*e - 60*a*b^2*d^5*e^2 + 195*a^2*b*
d^4*e^3 + 2860*a^3*d^3*e^4)*x)*sqrt(e*x + d)*A/e^4 + 2/109395*(6435*b^3*e^8*x^8
+ 128*b^3*d^8 - 816*a*b^2*d^7*e + 2040*a^2*b*d^6*e^2 - 2210*a^3*d^5*e^3 + 429*(5
2*b^3*d*e^7 + 51*a*b^2*e^8)*x^7 + 33*(802*b^3*d^2*e^6 + 2346*a*b^2*d*e^7 + 765*a
^2*b*e^8)*x^6 + 9*(1212*b^3*d^3*e^5 + 10506*a*b^2*d^2*e^6 + 10200*a^2*b*d*e^7 +
1105*a^3*e^8)*x^5 + 5*(7*b^3*d^4*e^4 + 8160*a*b^2*d^3*e^5 + 23358*a^2*b*d^2*e^6
+ 7514*a^3*d*e^7)*x^4 - 5*(8*b^3*d^5*e^3 - 51*a*b^2*d^4*e^4 - 10812*a^2*b*d^3*e^
5 - 10166*a^3*d^2*e^6)*x^3 + 3*(16*b^3*d^6*e^2 - 102*a*b^2*d^5*e^3 + 255*a^2*b*d
^4*e^4 + 8840*a^3*d^3*e^5)*x^2 - (64*b^3*d^7*e - 408*a*b^2*d^6*e^2 + 1020*a^2*b*
d^5*e^3 - 1105*a^3*d^4*e^4)*x)*sqrt(e*x + d)*B/e^5

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Fricas [A]  time = 0.288274, size = 855, normalized size = 2.78 \[ \frac{2 \,{\left (6435 \, B b^{3} e^{8} x^{8} + 128 \, B b^{3} d^{8} + 12155 \, A a^{3} d^{4} e^{4} - 272 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{7} e + 2040 \,{\left (B a^{2} b + A a b^{2}\right )} d^{6} e^{2} - 2210 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5} e^{3} + 429 \,{\left (52 \, B b^{3} d e^{7} + 17 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{8}\right )} x^{7} + 33 \,{\left (802 \, B b^{3} d^{2} e^{6} + 782 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{7} + 765 \,{\left (B a^{2} b + A a b^{2}\right )} e^{8}\right )} x^{6} + 9 \,{\left (1212 \, B b^{3} d^{3} e^{5} + 3502 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{6} + 10200 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{7} + 1105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{8}\right )} x^{5} + 5 \,{\left (7 \, B b^{3} d^{4} e^{4} + 2431 \, A a^{3} e^{8} + 2720 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{5} + 23358 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{6} + 7514 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{7}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{5} e^{3} - 9724 \, A a^{3} d e^{7} - 17 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{4} - 10812 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{5} - 10166 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{6} e^{2} + 24310 \, A a^{3} d^{2} e^{6} - 34 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{3} + 255 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{4} + 8840 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{7} e - 48620 \, A a^{3} d^{3} e^{5} - 136 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e^{2} + 1020 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{3} - 1105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/109395*(6435*B*b^3*e^8*x^8 + 128*B*b^3*d^8 + 12155*A*a^3*d^4*e^4 - 272*(3*B*a*
b^2 + A*b^3)*d^7*e + 2040*(B*a^2*b + A*a*b^2)*d^6*e^2 - 2210*(B*a^3 + 3*A*a^2*b)
*d^5*e^3 + 429*(52*B*b^3*d*e^7 + 17*(3*B*a*b^2 + A*b^3)*e^8)*x^7 + 33*(802*B*b^3
*d^2*e^6 + 782*(3*B*a*b^2 + A*b^3)*d*e^7 + 765*(B*a^2*b + A*a*b^2)*e^8)*x^6 + 9*
(1212*B*b^3*d^3*e^5 + 3502*(3*B*a*b^2 + A*b^3)*d^2*e^6 + 10200*(B*a^2*b + A*a*b^
2)*d*e^7 + 1105*(B*a^3 + 3*A*a^2*b)*e^8)*x^5 + 5*(7*B*b^3*d^4*e^4 + 2431*A*a^3*e
^8 + 2720*(3*B*a*b^2 + A*b^3)*d^3*e^5 + 23358*(B*a^2*b + A*a*b^2)*d^2*e^6 + 7514
*(B*a^3 + 3*A*a^2*b)*d*e^7)*x^4 - 5*(8*B*b^3*d^5*e^3 - 9724*A*a^3*d*e^7 - 17*(3*
B*a*b^2 + A*b^3)*d^4*e^4 - 10812*(B*a^2*b + A*a*b^2)*d^3*e^5 - 10166*(B*a^3 + 3*
A*a^2*b)*d^2*e^6)*x^3 + 3*(16*B*b^3*d^6*e^2 + 24310*A*a^3*d^2*e^6 - 34*(3*B*a*b^
2 + A*b^3)*d^5*e^3 + 255*(B*a^2*b + A*a*b^2)*d^4*e^4 + 8840*(B*a^3 + 3*A*a^2*b)*
d^3*e^5)*x^2 - (64*B*b^3*d^7*e - 48620*A*a^3*d^3*e^5 - 136*(3*B*a*b^2 + A*b^3)*d
^6*e^2 + 1020*(B*a^2*b + A*a*b^2)*d^5*e^3 - 1105*(B*a^3 + 3*A*a^2*b)*d^4*e^4)*x)
*sqrt(e*x + d)/e^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.363354, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done