3.12 \(\int x \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} a^2 A x^2+\frac{1}{8} b x^8 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{11} b^2 B x^{11} \]

[Out]

(a^2*A*x^2)/2 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^8)/8 + (b^2*B*x^11)
/11

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Rubi [A]  time = 0.113843, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{1}{2} a^2 A x^2+\frac{1}{8} b x^8 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{11} b^2 B x^{11} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*A*x^2)/2 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^8)/8 + (b^2*B*x^11)
/11

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a^{2} \int x\, dx + \frac{B b^{2} x^{11}}{11} + \frac{a x^{5} \left (2 A b + B a\right )}{5} + \frac{b x^{8} \left (A b + 2 B a\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*Integral(x, x) + B*b**2*x**11/11 + a*x**5*(2*A*b + B*a)/5 + b*x**8*(A*b +
2*B*a)/8

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Mathematica [A]  time = 0.0146997, size = 55, normalized size = 1. \[ \frac{1}{2} a^2 A x^2+\frac{1}{8} b x^8 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{11} b^2 B x^{11} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*A*x^2)/2 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^8)/8 + (b^2*B*x^11)
/11

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \[{\frac{{b}^{2}B{x}^{11}}{11}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{8}}{8}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{5}}{5}}+{\frac{{a}^{2}A{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a)^2*(B*x^3+A),x)

[Out]

1/11*b^2*B*x^11+1/8*(A*b^2+2*B*a*b)*x^8+1/5*(2*A*a*b+B*a^2)*x^5+1/2*a^2*A*x^2

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Maxima [A]  time = 1.37513, size = 69, normalized size = 1.25 \[ \frac{1}{11} \, B b^{2} x^{11} + \frac{1}{8} \,{\left (2 \, B a b + A b^{2}\right )} x^{8} + \frac{1}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{5} + \frac{1}{2} \, A a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*B*b^2*x^11 + 1/8*(2*B*a*b + A*b^2)*x^8 + 1/5*(B*a^2 + 2*A*a*b)*x^5 + 1/2*A*
a^2*x^2

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Fricas [A]  time = 0.21389, size = 1, normalized size = 0.02 \[ \frac{1}{11} x^{11} b^{2} B + \frac{1}{4} x^{8} b a B + \frac{1}{8} x^{8} b^{2} A + \frac{1}{5} x^{5} a^{2} B + \frac{2}{5} x^{5} b a A + \frac{1}{2} x^{2} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*x^11*b^2*B + 1/4*x^8*b*a*B + 1/8*x^8*b^2*A + 1/5*x^5*a^2*B + 2/5*x^5*b*a*A
+ 1/2*x^2*a^2*A

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Sympy [A]  time = 0.111307, size = 54, normalized size = 0.98 \[ \frac{A a^{2} x^{2}}{2} + \frac{B b^{2} x^{11}}{11} + x^{8} \left (\frac{A b^{2}}{8} + \frac{B a b}{4}\right ) + x^{5} \left (\frac{2 A a b}{5} + \frac{B a^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*x**2/2 + B*b**2*x**11/11 + x**8*(A*b**2/8 + B*a*b/4) + x**5*(2*A*a*b/5 +
B*a**2/5)

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GIAC/XCAS [A]  time = 0.225562, size = 72, normalized size = 1.31 \[ \frac{1}{11} \, B b^{2} x^{11} + \frac{1}{4} \, B a b x^{8} + \frac{1}{8} \, A b^{2} x^{8} + \frac{1}{5} \, B a^{2} x^{5} + \frac{2}{5} \, A a b x^{5} + \frac{1}{2} \, A a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*B*b^2*x^11 + 1/4*B*a*b*x^8 + 1/8*A*b^2*x^8 + 1/5*B*a^2*x^5 + 2/5*A*a*b*x^5
+ 1/2*A*a^2*x^2