3.802 \(\int x^{3/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=320 \[ \frac{20 a^2 b^2 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac{2 b^4 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{15 (a+b x)}+\frac{10 a b^3 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{13 (a+b x)}+\frac{2 b^5 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)}+\frac{2 a^5 A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 a^4 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac{10 a^3 b x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 (a+b x)} \]

[Out]

(2*a^5*A*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*a^4*(5*A*b +
a*B)*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (10*a^3*b*(2*A*b + a
*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (20*a^2*b^2*(A*b + a*
B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (10*a*b^3*(A*b + 2*a
*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*b^4*(A*b + 5*a*B
)*x^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*(a + b*x)) + (2*b^5*B*x^(17/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x))

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Rubi [A]  time = 0.33741, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{20 a^2 b^2 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac{2 b^4 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{15 (a+b x)}+\frac{10 a b^3 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{13 (a+b x)}+\frac{2 b^5 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)}+\frac{2 a^5 A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 a^4 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac{10 a^3 b x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^5*A*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*a^4*(5*A*b +
a*B)*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (10*a^3*b*(2*A*b + a
*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (20*a^2*b^2*(A*b + a*
B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (10*a*b^3*(A*b + 2*a
*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*b^4*(A*b + 5*a*B
)*x^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*(a + b*x)) + (2*b^5*B*x^(17/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x))

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Rubi in Sympy [A]  time = 35.3739, size = 320, normalized size = 1. \[ \frac{B x^{\frac{5}{2}} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{17 b} + \frac{512 a^{5} x^{\frac{5}{2}} \left (17 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{765765 b \left (a + b x\right )} + \frac{256 a^{4} x^{\frac{5}{2}} \left (17 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{153153 b} + \frac{64 a^{3} x^{\frac{5}{2}} \left (3 a + 3 b x\right ) \left (17 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{65637 b} + \frac{32 a^{2} x^{\frac{5}{2}} \left (17 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7293 b} + \frac{4 a x^{\frac{5}{2}} \left (5 a + 5 b x\right ) \left (17 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3315 b} + \frac{2 x^{\frac{5}{2}} \left (17 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{255 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**(5/2)*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(17*b) + 512*a**5*x
**(5/2)*(17*A*b - 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(765765*b*(a + b*x)) +
256*a**4*x**(5/2)*(17*A*b - 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(153153*b)
+ 64*a**3*x**(5/2)*(3*a + 3*b*x)*(17*A*b - 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**
2)/(65637*b) + 32*a**2*x**(5/2)*(17*A*b - 5*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(
3/2)/(7293*b) + 4*a*x**(5/2)*(5*a + 5*b*x)*(17*A*b - 5*B*a)*(a**2 + 2*a*b*x + b*
*2*x**2)**(3/2)/(3315*b) + 2*x**(5/2)*(17*A*b - 5*B*a)*(a**2 + 2*a*b*x + b**2*x*
*2)**(5/2)/(255*b)

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Mathematica [A]  time = 0.0973107, size = 127, normalized size = 0.4 \[ \frac{2 x^{5/2} \sqrt{(a+b x)^2} \left (21879 a^5 (7 A+5 B x)+60775 a^4 b x (9 A+7 B x)+77350 a^3 b^2 x^2 (11 A+9 B x)+53550 a^2 b^3 x^3 (13 A+11 B x)+19635 a b^4 x^4 (15 A+13 B x)+3003 b^5 x^5 (17 A+15 B x)\right )}{765765 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(5/2)*Sqrt[(a + b*x)^2]*(21879*a^5*(7*A + 5*B*x) + 60775*a^4*b*x*(9*A + 7*B
*x) + 77350*a^3*b^2*x^2*(11*A + 9*B*x) + 53550*a^2*b^3*x^3*(13*A + 11*B*x) + 196
35*a*b^4*x^4*(15*A + 13*B*x) + 3003*b^5*x^5*(17*A + 15*B*x)))/(765765*(a + b*x))

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Maple [A]  time = 0.011, size = 140, normalized size = 0.4 \[{\frac{90090\,B{b}^{5}{x}^{6}+102102\,A{x}^{5}{b}^{5}+510510\,B{x}^{5}a{b}^{4}+589050\,A{x}^{4}a{b}^{4}+1178100\,B{x}^{4}{a}^{2}{b}^{3}+1392300\,A{x}^{3}{a}^{2}{b}^{3}+1392300\,B{x}^{3}{a}^{3}{b}^{2}+1701700\,A{x}^{2}{a}^{3}{b}^{2}+850850\,B{x}^{2}{a}^{4}b+1093950\,Ax{a}^{4}b+218790\,Bx{a}^{5}+306306\,A{a}^{5}}{765765\, \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/765765*x^(5/2)*(45045*B*b^5*x^6+51051*A*b^5*x^5+255255*B*a*b^4*x^5+294525*A*a*
b^4*x^4+589050*B*a^2*b^3*x^4+696150*A*a^2*b^3*x^3+696150*B*a^3*b^2*x^3+850850*A*
a^3*b^2*x^2+425425*B*a^4*b*x^2+546975*A*a^4*b*x+109395*B*a^5*x+153153*A*a^5)*((b
*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [A]  time = 0.710018, size = 325, normalized size = 1.02 \[ \frac{2}{45045} \,{\left (231 \,{\left (13 \, b^{5} x^{2} + 15 \, a b^{4} x\right )} x^{\frac{11}{2}} + 1260 \,{\left (11 \, a b^{4} x^{2} + 13 \, a^{2} b^{3} x\right )} x^{\frac{9}{2}} + 2730 \,{\left (9 \, a^{2} b^{3} x^{2} + 11 \, a^{3} b^{2} x\right )} x^{\frac{7}{2}} + 2860 \,{\left (7 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x\right )} x^{\frac{5}{2}} + 1287 \,{\left (5 \, a^{4} b x^{2} + 7 \, a^{5} x\right )} x^{\frac{3}{2}}\right )} A + \frac{2}{765765} \,{\left (3003 \,{\left (15 \, b^{5} x^{2} + 17 \, a b^{4} x\right )} x^{\frac{13}{2}} + 15708 \,{\left (13 \, a b^{4} x^{2} + 15 \, a^{2} b^{3} x\right )} x^{\frac{11}{2}} + 32130 \,{\left (11 \, a^{2} b^{3} x^{2} + 13 \, a^{3} b^{2} x\right )} x^{\frac{9}{2}} + 30940 \,{\left (9 \, a^{3} b^{2} x^{2} + 11 \, a^{4} b x\right )} x^{\frac{7}{2}} + 12155 \,{\left (7 \, a^{4} b x^{2} + 9 \, a^{5} x\right )} x^{\frac{5}{2}}\right )} B \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^(3/2),x, algorithm="maxima")

[Out]

2/45045*(231*(13*b^5*x^2 + 15*a*b^4*x)*x^(11/2) + 1260*(11*a*b^4*x^2 + 13*a^2*b^
3*x)*x^(9/2) + 2730*(9*a^2*b^3*x^2 + 11*a^3*b^2*x)*x^(7/2) + 2860*(7*a^3*b^2*x^2
+ 9*a^4*b*x)*x^(5/2) + 1287*(5*a^4*b*x^2 + 7*a^5*x)*x^(3/2))*A + 2/765765*(3003
*(15*b^5*x^2 + 17*a*b^4*x)*x^(13/2) + 15708*(13*a*b^4*x^2 + 15*a^2*b^3*x)*x^(11/
2) + 32130*(11*a^2*b^3*x^2 + 13*a^3*b^2*x)*x^(9/2) + 30940*(9*a^3*b^2*x^2 + 11*a
^4*b*x)*x^(7/2) + 12155*(7*a^4*b*x^2 + 9*a^5*x)*x^(5/2))*B

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Fricas [A]  time = 0.276386, size = 167, normalized size = 0.52 \[ \frac{2}{765765} \,{\left (45045 \, B b^{5} x^{8} + 153153 \, A a^{5} x^{2} + 51051 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + 294525 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 696150 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + 425425 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 109395 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^(3/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^5*x^8 + 153153*A*a^5*x^2 + 51051*(5*B*a*b^4 + A*b^5)*x^7 + 2
94525*(2*B*a^2*b^3 + A*a*b^4)*x^6 + 696150*(B*a^3*b^2 + A*a^2*b^3)*x^5 + 425425*
(B*a^4*b + 2*A*a^3*b^2)*x^4 + 109395*(B*a^5 + 5*A*a^4*b)*x^3)*sqrt(x)

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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(x**(3/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275925, size = 266, normalized size = 0.83 \[ \frac{2}{17} \, B b^{5} x^{\frac{17}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, B a b^{4} x^{\frac{15}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{15} \, A b^{5} x^{\frac{15}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{13} \, B a^{2} b^{3} x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{13} \, A a b^{4} x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{11} \, B a^{3} b^{2} x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{11} \, A a^{2} b^{3} x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{9} \, B a^{4} b x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{9} \, A a^{3} b^{2} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, B a^{5} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{7} \, A a^{4} b x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{5} \, A a^{5} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)*x^(3/2),x, algorithm="giac")

[Out]

2/17*B*b^5*x^(17/2)*sign(b*x + a) + 2/3*B*a*b^4*x^(15/2)*sign(b*x + a) + 2/15*A*
b^5*x^(15/2)*sign(b*x + a) + 20/13*B*a^2*b^3*x^(13/2)*sign(b*x + a) + 10/13*A*a*
b^4*x^(13/2)*sign(b*x + a) + 20/11*B*a^3*b^2*x^(11/2)*sign(b*x + a) + 20/11*A*a^
2*b^3*x^(11/2)*sign(b*x + a) + 10/9*B*a^4*b*x^(9/2)*sign(b*x + a) + 20/9*A*a^3*b
^2*x^(9/2)*sign(b*x + a) + 2/7*B*a^5*x^(7/2)*sign(b*x + a) + 10/7*A*a^4*b*x^(7/2
)*sign(b*x + a) + 2/5*A*a^5*x^(5/2)*sign(b*x + a)