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

Optimal. Leaf size=85 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{25} b^2 x^{25/2} (3 a B+A b)+\frac{6}{19} a b x^{19/2} (a B+A b)+\frac{2}{31} b^3 B x^{31/2} \]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(
19/2))/19 + (2*b^2*(A*b + 3*a*B)*x^(25/2))/25 + (2*b^3*B*x^(31/2))/31

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Rubi [A]  time = 0.128127, 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 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{25} b^2 x^{25/2} (3 a B+A b)+\frac{6}{19} a b x^{19/2} (a B+A b)+\frac{2}{31} b^3 B x^{31/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(
19/2))/19 + (2*b^2*(A*b + 3*a*B)*x^(25/2))/25 + (2*b^3*B*x^(31/2))/31

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Rubi in Sympy [A]  time = 14.1236, size = 85, normalized size = 1. \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{31}{2}}}{31} + \frac{2 a^{2} x^{\frac{13}{2}} \left (3 A b + B a\right )}{13} + \frac{6 a b x^{\frac{19}{2}} \left (A b + B a\right )}{19} + \frac{2 b^{2} x^{\frac{25}{2}} \left (A b + 3 B a\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(7/2)/7 + 2*B*b**3*x**(31/2)/31 + 2*a**2*x**(13/2)*(3*A*b + B*a)/13
+ 6*a*b*x**(19/2)*(A*b + B*a)/19 + 2*b**2*x**(25/2)*(A*b + 3*B*a)/25

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Mathematica [A]  time = 0.0461847, size = 85, normalized size = 1. \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{25} b^2 x^{25/2} (3 a B+A b)+\frac{6}{19} a b x^{19/2} (a B+A b)+\frac{2}{31} b^3 B x^{31/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(
19/2))/19 + (2*b^2*(A*b + 3*a*B)*x^(25/2))/25 + (2*b^3*B*x^(31/2))/31

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Maple [A]  time = 0.009, size = 80, normalized size = 0.9 \[{\frac{86450\,B{b}^{3}{x}^{12}+107198\,{x}^{9}A{b}^{3}+321594\,{x}^{9}a{b}^{2}B+423150\,{x}^{6}a{b}^{2}A+423150\,{x}^{6}B{a}^{2}b+618450\,{x}^{3}A{a}^{2}b+206150\,{x}^{3}B{a}^{3}+382850\,A{a}^{3}}{1339975}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1339975*x^(7/2)*(43225*B*b^3*x^12+53599*A*b^3*x^9+160797*B*a*b^2*x^9+211575*A*
a*b^2*x^6+211575*B*a^2*b*x^6+309225*A*a^2*b*x^3+103075*B*a^3*x^3+191425*A*a^3)

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Maxima [A]  time = 1.40659, size = 99, normalized size = 1.16 \[ \frac{2}{31} \, B b^{3} x^{\frac{31}{2}} + \frac{2}{25} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{25}{2}} + \frac{6}{19} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{19}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} + \frac{2}{13} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/31*B*b^3*x^(31/2) + 2/25*(3*B*a*b^2 + A*b^3)*x^(25/2) + 6/19*(B*a^2*b + A*a*b^
2)*x^(19/2) + 2/7*A*a^3*x^(7/2) + 2/13*(B*a^3 + 3*A*a^2*b)*x^(13/2)

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Fricas [A]  time = 0.225451, size = 105, normalized size = 1.24 \[ \frac{2}{1339975} \,{\left (43225 \, B b^{3} x^{15} + 53599 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{12} + 211575 \,{\left (B a^{2} b + A a b^{2}\right )} x^{9} + 191425 \, A a^{3} x^{3} + 103075 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1339975*(43225*B*b^3*x^15 + 53599*(3*B*a*b^2 + A*b^3)*x^12 + 211575*(B*a^2*b +
A*a*b^2)*x^9 + 191425*A*a^3*x^3 + 103075*(B*a^3 + 3*A*a^2*b)*x^6)*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**(5/2)*(b*x**3+a)**3*(B*x**3+A),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21307, size = 104, normalized size = 1.22 \[ \frac{2}{31} \, B b^{3} x^{\frac{31}{2}} + \frac{6}{25} \, B a b^{2} x^{\frac{25}{2}} + \frac{2}{25} \, A b^{3} x^{\frac{25}{2}} + \frac{6}{19} \, B a^{2} b x^{\frac{19}{2}} + \frac{6}{19} \, A a b^{2} x^{\frac{19}{2}} + \frac{2}{13} \, B a^{3} x^{\frac{13}{2}} + \frac{6}{13} \, A a^{2} b x^{\frac{13}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/31*B*b^3*x^(31/2) + 6/25*B*a*b^2*x^(25/2) + 2/25*A*b^3*x^(25/2) + 6/19*B*a^2*b
*x^(19/2) + 6/19*A*a*b^2*x^(19/2) + 2/13*B*a^3*x^(13/2) + 6/13*A*a^2*b*x^(13/2)
+ 2/7*A*a^3*x^(7/2)