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

Optimal. Leaf size=85 \[ \frac{2}{5} a^3 A x^{5/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{23} b^2 x^{23/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{29} b^3 B x^{29/2} \]

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(
17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(23/2))/23 + (2*b^3*B*x^(29/2))/29

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Rubi [A]  time = 0.12639, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2}{5} a^3 A x^{5/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{23} b^2 x^{23/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{29} b^3 B x^{29/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(
17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(23/2))/23 + (2*b^3*B*x^(29/2))/29

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Rubi in Sympy [A]  time = 14.0824, size = 85, normalized size = 1. \[ \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{2 B b^{3} x^{\frac{29}{2}}}{29} + \frac{2 a^{2} x^{\frac{11}{2}} \left (3 A b + B a\right )}{11} + \frac{6 a b x^{\frac{17}{2}} \left (A b + B a\right )}{17} + \frac{2 b^{2} x^{\frac{23}{2}} \left (A b + 3 B a\right )}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(5/2)/5 + 2*B*b**3*x**(29/2)/29 + 2*a**2*x**(11/2)*(3*A*b + B*a)/11
+ 6*a*b*x**(17/2)*(A*b + B*a)/17 + 2*b**2*x**(23/2)*(A*b + 3*B*a)/23

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Mathematica [A]  time = 0.0469156, size = 85, normalized size = 1. \[ \frac{2}{5} a^3 A x^{5/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{23} b^2 x^{23/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{29} b^3 B x^{29/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(
17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(23/2))/23 + (2*b^3*B*x^(29/2))/29

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Maple [A]  time = 0.009, size = 80, normalized size = 0.9 \[{\frac{43010\,B{b}^{3}{x}^{12}+54230\,{x}^{9}A{b}^{3}+162690\,{x}^{9}a{b}^{2}B+220110\,{x}^{6}a{b}^{2}A+220110\,{x}^{6}B{a}^{2}b+340170\,{x}^{3}A{a}^{2}b+113390\,{x}^{3}B{a}^{3}+249458\,A{a}^{3}}{623645}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/623645*x^(5/2)*(21505*B*b^3*x^12+27115*A*b^3*x^9+81345*B*a*b^2*x^9+110055*A*a*
b^2*x^6+110055*B*a^2*b*x^6+170085*A*a^2*b*x^3+56695*B*a^3*x^3+124729*A*a^3)

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Maxima [A]  time = 1.43999, size = 99, normalized size = 1.16 \[ \frac{2}{29} \, B b^{3} x^{\frac{29}{2}} + \frac{2}{23} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{23}{2}} + \frac{6}{17} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{17}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} + \frac{2}{11} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/29*B*b^3*x^(29/2) + 2/23*(3*B*a*b^2 + A*b^3)*x^(23/2) + 6/17*(B*a^2*b + A*a*b^
2)*x^(17/2) + 2/5*A*a^3*x^(5/2) + 2/11*(B*a^3 + 3*A*a^2*b)*x^(11/2)

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Fricas [A]  time = 0.228435, size = 105, normalized size = 1.24 \[ \frac{2}{623645} \,{\left (21505 \, B b^{3} x^{14} + 27115 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{11} + 110055 \,{\left (B a^{2} b + A a b^{2}\right )} x^{8} + 124729 \, A a^{3} x^{2} + 56695 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/623645*(21505*B*b^3*x^14 + 27115*(3*B*a*b^2 + A*b^3)*x^11 + 110055*(B*a^2*b +
A*a*b^2)*x^8 + 124729*A*a^3*x^2 + 56695*(B*a^3 + 3*A*a^2*b)*x^5)*sqrt(x)

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Sympy [A]  time = 124.276, size = 114, normalized size = 1.34 \[ \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{6 A a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 A a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 A b^{3} x^{\frac{23}{2}}}{23} + \frac{2 B a^{3} x^{\frac{11}{2}}}{11} + \frac{6 B a^{2} b x^{\frac{17}{2}}}{17} + \frac{6 B a b^{2} x^{\frac{23}{2}}}{23} + \frac{2 B b^{3} x^{\frac{29}{2}}}{29} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(5/2)/5 + 6*A*a**2*b*x**(11/2)/11 + 6*A*a*b**2*x**(17/2)/17 + 2*A*b*
*3*x**(23/2)/23 + 2*B*a**3*x**(11/2)/11 + 6*B*a**2*b*x**(17/2)/17 + 6*B*a*b**2*x
**(23/2)/23 + 2*B*b**3*x**(29/2)/29

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GIAC/XCAS [A]  time = 0.211814, size = 104, normalized size = 1.22 \[ \frac{2}{29} \, B b^{3} x^{\frac{29}{2}} + \frac{6}{23} \, B a b^{2} x^{\frac{23}{2}} + \frac{2}{23} \, A b^{3} x^{\frac{23}{2}} + \frac{6}{17} \, B a^{2} b x^{\frac{17}{2}} + \frac{6}{17} \, A a b^{2} x^{\frac{17}{2}} + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}} + \frac{6}{11} \, A a^{2} b x^{\frac{11}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/29*B*b^3*x^(29/2) + 6/23*B*a*b^2*x^(23/2) + 2/23*A*b^3*x^(23/2) + 6/17*B*a^2*b
*x^(17/2) + 6/17*A*a*b^2*x^(17/2) + 2/11*B*a^3*x^(11/2) + 6/11*A*a^2*b*x^(11/2)
+ 2/5*A*a^3*x^(5/2)