3.2100 \(\int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=264 \[ \frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^5 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^5 (a+b x)} \]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)
) - (8*b*(b*d - a*e)^3*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a
+ b*x)) + (12*b^2*(b*d - a*e)^2*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(11*e^5*(a + b*x)) - (8*b^3*(b*d - a*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^4*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(15*e^5*(a + b*x))

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Rubi [A]  time = 0.335017, 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 \[ \frac{12 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^5 (a+b x)}-\frac{8 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^5 (a+b x)}+\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(b*d - a*e)^4*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)
) - (8*b*(b*d - a*e)^3*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a
+ b*x)) + (12*b^2*(b*d - a*e)^2*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(11*e^5*(a + b*x)) - (8*b^3*(b*d - a*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^4*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(15*e^5*(a + b*x))

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Rubi in Sympy [A]  time = 38.9624, size = 226, normalized size = 0.86 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{7}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{15 e} + \frac{16 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{195 e^{2}} + \frac{32 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2145 e^{3}} + \frac{128 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6435 e^{4}} + \frac{256 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{45045 e^{5} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

2*(a + b*x)*(d + e*x)**(7/2)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(15*e) + 16*(d
+ e*x)**(7/2)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(195*e**2) + 32*(3
*a + 3*b*x)*(d + e*x)**(7/2)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(21
45*e**3) + 128*(d + e*x)**(7/2)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/
(6435*e**4) + 256*(d + e*x)**(7/2)*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**
2)/(45045*e**5*(a + b*x))

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Mathematica [A]  time = 0.219837, size = 172, normalized size = 0.65 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (6435 a^4 e^4+2860 a^3 b e^3 (7 e x-2 d)+390 a^2 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+60 a b^3 e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^4 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(6435*a^4*e^4 + 2860*a^3*b*e^3*(-2*d + 7*e*
x) + 390*a^2*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 60*a*b^3*e*(-16*d^3 + 56*
d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^4*(128*d^4 - 448*d^3*e*x + 1008*d^2*e
^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5*(a + b*x))

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Maple [A]  time = 0.012, size = 202, normalized size = 0.8 \[{\frac{6006\,{x}^{4}{b}^{4}{e}^{4}+27720\,{x}^{3}a{b}^{3}{e}^{4}-3696\,{x}^{3}{b}^{4}d{e}^{3}+49140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-15120\,{x}^{2}a{b}^{3}d{e}^{3}+2016\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+40040\,x{a}^{3}b{e}^{4}-21840\,x{a}^{2}{b}^{2}d{e}^{3}+6720\,xa{b}^{3}{d}^{2}{e}^{2}-896\,x{b}^{4}{d}^{3}e+12870\,{a}^{4}{e}^{4}-11440\,{a}^{3}bd{e}^{3}+6240\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-1920\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{45045\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

2/45045*(e*x+d)^(7/2)*(3003*b^4*e^4*x^4+13860*a*b^3*e^4*x^3-1848*b^4*d*e^3*x^3+2
4570*a^2*b^2*e^4*x^2-7560*a*b^3*d*e^3*x^2+1008*b^4*d^2*e^2*x^2+20020*a^3*b*e^4*x
-10920*a^2*b^2*d*e^3*x+3360*a*b^3*d^2*e^2*x-448*b^4*d^3*e*x+6435*a^4*e^4-5720*a^
3*b*d*e^3+3120*a^2*b^2*d^2*e^2-960*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 = 0.730179, size = 799, normalized size = 3.03 \[ \frac{2 \,{\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \,{\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \,{\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} +{\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d} a}{3003 \, e^{4}} + \frac{2 \,{\left (3003 \, b^{3} e^{7} x^{7} + 128 \, b^{3} d^{7} - 720 \, a b^{2} d^{6} e + 1560 \, a^{2} b d^{5} e^{2} - 1430 \, a^{3} d^{4} e^{3} + 231 \,{\left (31 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{3} d^{2} e^{5} + 405 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 35 \,{\left (b^{3} d^{3} e^{4} + 477 \, a b^{2} d^{2} e^{5} + 897 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{4} e^{3} - 45 \, a b^{2} d^{3} e^{4} - 4407 \, a^{2} b d^{2} e^{5} - 2717 \, a^{3} d e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{5} e^{2} - 90 \, a b^{2} d^{4} e^{3} + 195 \, a^{2} b d^{3} e^{4} + 3575 \, a^{3} d^{2} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{6} e - 360 \, a b^{2} d^{5} e^{2} + 780 \, a^{2} b d^{4} e^{3} - 715 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d} b}{45045 \, e^{5}} \]

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)^(5/2),x, algorithm="maxima")

[Out]

2/3003*(231*b^3*e^6*x^6 - 16*b^3*d^6 + 104*a*b^2*d^5*e - 286*a^2*b*d^4*e^2 + 429
*a^3*d^3*e^3 + 63*(9*b^3*d*e^5 + 13*a*b^2*e^6)*x^5 + 7*(53*b^3*d^2*e^4 + 299*a*b
^2*d*e^5 + 143*a^2*b*e^6)*x^4 + (5*b^3*d^3*e^3 + 1469*a*b^2*d^2*e^4 + 2717*a^2*b
*d*e^5 + 429*a^3*e^6)*x^3 - 3*(2*b^3*d^4*e^2 - 13*a*b^2*d^3*e^3 - 715*a^2*b*d^2*
e^4 - 429*a^3*d*e^5)*x^2 + (8*b^3*d^5*e - 52*a*b^2*d^4*e^2 + 143*a^2*b*d^3*e^3 +
1287*a^3*d^2*e^4)*x)*sqrt(e*x + d)*a/e^4 + 2/45045*(3003*b^3*e^7*x^7 + 128*b^3*
d^7 - 720*a*b^2*d^6*e + 1560*a^2*b*d^5*e^2 - 1430*a^3*d^4*e^3 + 231*(31*b^3*d*e^
6 + 45*a*b^2*e^7)*x^6 + 63*(71*b^3*d^2*e^5 + 405*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^
5 + 35*(b^3*d^3*e^4 + 477*a*b^2*d^2*e^5 + 897*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 - 5
*(8*b^3*d^4*e^3 - 45*a*b^2*d^3*e^4 - 4407*a^2*b*d^2*e^5 - 2717*a^3*d*e^6)*x^3 +
3*(16*b^3*d^5*e^2 - 90*a*b^2*d^4*e^3 + 195*a^2*b*d^3*e^4 + 3575*a^3*d^2*e^5)*x^2
- (64*b^3*d^6*e - 360*a*b^2*d^5*e^2 + 780*a^2*b*d^4*e^3 - 715*a^3*d^3*e^4)*x)*s
qrt(e*x + d)*b/e^5

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Fricas [A]  time = 0.29314, size = 509, normalized size = 1.93 \[ \frac{2 \,{\left (3003 \, b^{4} e^{7} x^{7} + 128 \, b^{4} d^{7} - 960 \, a b^{3} d^{6} e + 3120 \, a^{2} b^{2} d^{5} e^{2} - 5720 \, a^{3} b d^{4} e^{3} + 6435 \, a^{4} d^{3} e^{4} + 231 \,{\left (31 \, b^{4} d e^{6} + 60 \, a b^{3} e^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{4} d^{2} e^{5} + 540 \, a b^{3} d e^{6} + 390 \, a^{2} b^{2} e^{7}\right )} x^{5} + 35 \,{\left (b^{4} d^{3} e^{4} + 636 \, a b^{3} d^{2} e^{5} + 1794 \, a^{2} b^{2} d e^{6} + 572 \, a^{3} b e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{4} d^{4} e^{3} - 60 \, a b^{3} d^{3} e^{4} - 8814 \, a^{2} b^{2} d^{2} e^{5} - 10868 \, a^{3} b d e^{6} - 1287 \, a^{4} e^{7}\right )} x^{3} + 3 \,{\left (16 \, b^{4} d^{5} e^{2} - 120 \, a b^{3} d^{4} e^{3} + 390 \, a^{2} b^{2} d^{3} e^{4} + 14300 \, a^{3} b d^{2} e^{5} + 6435 \, a^{4} d e^{6}\right )} x^{2} -{\left (64 \, b^{4} d^{6} e - 480 \, a b^{3} d^{5} e^{2} + 1560 \, a^{2} b^{2} d^{4} e^{3} - 2860 \, a^{3} b d^{3} e^{4} - 19305 \, a^{4} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^4*e^7*x^7 + 128*b^4*d^7 - 960*a*b^3*d^6*e + 3120*a^2*b^2*d^5*e^2
- 5720*a^3*b*d^4*e^3 + 6435*a^4*d^3*e^4 + 231*(31*b^4*d*e^6 + 60*a*b^3*e^7)*x^6
+ 63*(71*b^4*d^2*e^5 + 540*a*b^3*d*e^6 + 390*a^2*b^2*e^7)*x^5 + 35*(b^4*d^3*e^4
+ 636*a*b^3*d^2*e^5 + 1794*a^2*b^2*d*e^6 + 572*a^3*b*e^7)*x^4 - 5*(8*b^4*d^4*e^
3 - 60*a*b^3*d^3*e^4 - 8814*a^2*b^2*d^2*e^5 - 10868*a^3*b*d*e^6 - 1287*a^4*e^7)*
x^3 + 3*(16*b^4*d^5*e^2 - 120*a*b^3*d^4*e^3 + 390*a^2*b^2*d^3*e^4 + 14300*a^3*b*
d^2*e^5 + 6435*a^4*d*e^6)*x^2 - (64*b^4*d^6*e - 480*a*b^3*d^5*e^2 + 1560*a^2*b^2
*d^4*e^3 - 2860*a^3*b*d^3*e^4 - 19305*a^4*d^2*e^5)*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)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.316112, 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)^(5/2),x, algorithm="giac")

[Out]

Done