3.2101 \(\int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=264 \[ \frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{5 e^5 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^5 (a+b x)} \]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)
) - (8*b*(b*d - a*e)^3*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a
+ b*x)) + (4*b^2*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3
*e^5*(a + b*x)) - (8*b^3*(b*d - a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(11*e^5*(a + b*x)) + (2*b^4*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(13*e^5*(a + b*x))

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Rubi [A]  time = 0.305685, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{3 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3}{7 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}{5 e^5 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)
) - (8*b*(b*d - a*e)^3*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a
+ b*x)) + (4*b^2*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3
*e^5*(a + b*x)) - (8*b^3*(b*d - a*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(11*e^5*(a + b*x)) + (2*b^4*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(13*e^5*(a + b*x))

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Rubi in Sympy [A]  time = 39.3244, size = 226, normalized size = 0.86 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{13 e} + \frac{16 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{143 e^{2}} + \frac{32 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{1287 e^{3}} + \frac{128 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3003 e^{4}} + \frac{256 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15015 e^{5} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a + b*x)*(d + e*x)**(5/2)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(13*e) + 16*(d
+ e*x)**(5/2)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(143*e**2) + 32*(3
*a + 3*b*x)*(d + e*x)**(5/2)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(12
87*e**3) + 128*(d + e*x)**(5/2)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/
(3003*e**4) + 256*(d + e*x)**(5/2)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**
2)/(15015*e**5*(a + b*x))

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Mathematica [A]  time = 0.213807, size = 172, normalized size = 0.65 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{5/2} \left (3003 a^4 e^4+1716 a^3 b e^3 (5 e x-2 d)+286 a^2 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+52 a b^3 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+b^4 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 e^5 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(3003*a^4*e^4 + 1716*a^3*b*e^3*(-2*d + 5*e*
x) + 286*a^2*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 52*a*b^3*e*(-16*d^3 + 40*
d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + b^4*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2
*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(15015*e^5*(a + b*x))

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Maple [A]  time = 0.013, size = 202, normalized size = 0.8 \[{\frac{2310\,{x}^{4}{b}^{4}{e}^{4}+10920\,{x}^{3}a{b}^{3}{e}^{4}-1680\,{x}^{3}{b}^{4}d{e}^{3}+20020\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-7280\,{x}^{2}a{b}^{3}d{e}^{3}+1120\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+17160\,x{a}^{3}b{e}^{4}-11440\,x{a}^{2}{b}^{2}d{e}^{3}+4160\,xa{b}^{3}{d}^{2}{e}^{2}-640\,x{b}^{4}{d}^{3}e+6006\,{a}^{4}{e}^{4}-6864\,{a}^{3}bd{e}^{3}+4576\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-1664\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{15015\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(e*x+d)^(5/2)*(1155*b^4*e^4*x^4+5460*a*b^3*e^4*x^3-840*b^4*d*e^3*x^3+100
10*a^2*b^2*e^4*x^2-3640*a*b^3*d*e^3*x^2+560*b^4*d^2*e^2*x^2+8580*a^3*b*e^4*x-572
0*a^2*b^2*d*e^3*x+2080*a*b^3*d^2*e^2*x-320*b^4*d^3*e*x+3003*a^4*e^4-3432*a^3*b*d
*e^3+2288*a^2*b^2*d^2*e^2-832*a*b^3*d^3*e+128*b^4*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*
x+a)^3

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Maxima [A]  time = 0.732059, size = 659, normalized size = 2.5 \[ \frac{2 \,{\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \,{\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \,{\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d} a}{1155 \, e^{4}} + \frac{2 \,{\left (1155 \, b^{3} e^{6} x^{6} + 128 \, b^{3} d^{6} - 624 \, a b^{2} d^{5} e + 1144 \, a^{2} b d^{4} e^{2} - 858 \, a^{3} d^{3} e^{3} + 105 \,{\left (14 \, b^{3} d e^{5} + 39 \, a b^{2} e^{6}\right )} x^{5} + 35 \,{\left (b^{3} d^{2} e^{4} + 156 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{3} e^{3} - 39 \, a b^{2} d^{2} e^{4} - 1430 \, a^{2} b d e^{5} - 429 \, a^{3} e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{4} e^{2} - 78 \, a b^{2} d^{3} e^{3} + 143 \, a^{2} b d^{2} e^{4} + 1144 \, a^{3} d e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{5} e - 312 \, a b^{2} d^{4} e^{2} + 572 \, a^{2} b d^{3} e^{3} - 429 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d} b}{15015 \, e^{5}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*
a^3*d^2*e^3 + 35*(4*b^3*d*e^4 + 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d
*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e^3 - 264*a^2*b*d*e^4
- 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 99*a^2*b*d^2*e^3 + 462*a^
3*d*e^4)*x)*sqrt(e*x + d)*a/e^4 + 2/15015*(1155*b^3*e^6*x^6 + 128*b^3*d^6 - 624*
a*b^2*d^5*e + 1144*a^2*b*d^4*e^2 - 858*a^3*d^3*e^3 + 105*(14*b^3*d*e^5 + 39*a*b^
2*e^6)*x^5 + 35*(b^3*d^2*e^4 + 156*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 - 5*(8*b^3*d
^3*e^3 - 39*a*b^2*d^2*e^4 - 1430*a^2*b*d*e^5 - 429*a^3*e^6)*x^3 + 3*(16*b^3*d^4*
e^2 - 78*a*b^2*d^3*e^3 + 143*a^2*b*d^2*e^4 + 1144*a^3*d*e^5)*x^2 - (64*b^3*d^5*e
- 312*a*b^2*d^4*e^2 + 572*a^2*b*d^3*e^3 - 429*a^3*d^2*e^4)*x)*sqrt(e*x + d)*b/e
^5

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Fricas [A]  time = 0.294163, size = 420, normalized size = 1.59 \[ \frac{2 \,{\left (1155 \, b^{4} e^{6} x^{6} + 128 \, b^{4} d^{6} - 832 \, a b^{3} d^{5} e + 2288 \, a^{2} b^{2} d^{4} e^{2} - 3432 \, a^{3} b d^{3} e^{3} + 3003 \, a^{4} d^{2} e^{4} + 210 \,{\left (7 \, b^{4} d e^{5} + 26 \, a b^{3} e^{6}\right )} x^{5} + 35 \,{\left (b^{4} d^{2} e^{4} + 208 \, a b^{3} d e^{5} + 286 \, a^{2} b^{2} e^{6}\right )} x^{4} - 20 \,{\left (2 \, b^{4} d^{3} e^{3} - 13 \, a b^{3} d^{2} e^{4} - 715 \, a^{2} b^{2} d e^{5} - 429 \, a^{3} b e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{4} d^{4} e^{2} - 104 \, a b^{3} d^{3} e^{3} + 286 \, a^{2} b^{2} d^{2} e^{4} + 4576 \, a^{3} b d e^{5} + 1001 \, a^{4} e^{6}\right )} x^{2} - 2 \,{\left (32 \, b^{4} d^{5} e - 208 \, a b^{3} d^{4} e^{2} + 572 \, a^{2} b^{2} d^{3} e^{3} - 858 \, a^{3} b d^{2} e^{4} - 3003 \, a^{4} d e^{5}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{5}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

2/15015*(1155*b^4*e^6*x^6 + 128*b^4*d^6 - 832*a*b^3*d^5*e + 2288*a^2*b^2*d^4*e^2
- 3432*a^3*b*d^3*e^3 + 3003*a^4*d^2*e^4 + 210*(7*b^4*d*e^5 + 26*a*b^3*e^6)*x^5
+ 35*(b^4*d^2*e^4 + 208*a*b^3*d*e^5 + 286*a^2*b^2*e^6)*x^4 - 20*(2*b^4*d^3*e^3 -
13*a*b^3*d^2*e^4 - 715*a^2*b^2*d*e^5 - 429*a^3*b*e^6)*x^3 + 3*(16*b^4*d^4*e^2 -
104*a*b^3*d^3*e^3 + 286*a^2*b^2*d^2*e^4 + 4576*a^3*b*d*e^5 + 1001*a^4*e^6)*x^2
- 2*(32*b^4*d^5*e - 208*a*b^3*d^4*e^2 + 572*a^2*b^2*d^3*e^3 - 858*a^3*b*d^2*e^4
- 3003*a^4*d*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)*(d + e*x)**(3/2)*((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.306728, 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)^(3/2),x, algorithm="giac")

[Out]

Done