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

Optimal. Leaf size=46 \[ a^2 A \log (x)+\frac{2}{3} a A b x^3+\frac{B \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} A b^2 x^6 \]

[Out]

(2*a*A*b*x^3)/3 + (A*b^2*x^6)/6 + (B*(a + b*x^3)^3)/(9*b) + a^2*A*Log[x]

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Rubi [A]  time = 0.0921061, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ a^2 A \log (x)+\frac{2}{3} a A b x^3+\frac{B \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} A b^2 x^6 \]

Antiderivative was successfully verified.

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

[Out]

(2*a*A*b*x^3)/3 + (A*b^2*x^6)/6 + (B*(a + b*x^3)^3)/(9*b) + a^2*A*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{A a^{2} \log{\left (x^{3} \right )}}{3} + \frac{2 A a b x^{3}}{3} + \frac{A b^{2} \int ^{x^{3}} x\, dx}{3} + \frac{B \left (a + b x^{3}\right )^{3}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*log(x**3)/3 + 2*A*a*b*x**3/3 + A*b**2*Integral(x, (x, x**3))/3 + B*(a + b
*x**3)**3/(9*b)

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Mathematica [A]  time = 0.0332811, size = 51, normalized size = 1.11 \[ a^2 A \log (x)+\frac{1}{6} b x^6 (2 a B+A b)+\frac{1}{3} a x^3 (a B+2 A b)+\frac{1}{9} b^2 B x^9 \]

Antiderivative was successfully verified.

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

[Out]

(a*(2*A*b + a*B)*x^3)/3 + (b*(A*b + 2*a*B)*x^6)/6 + (b^2*B*x^9)/9 + a^2*A*Log[x]

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Maple [A]  time = 0.003, size = 52, normalized size = 1.1 \[{\frac{B{x}^{9}{b}^{2}}{9}}+{\frac{A{b}^{2}{x}^{6}}{6}}+{\frac{B{x}^{6}ab}{3}}+{\frac{2\,aAb{x}^{3}}{3}}+{\frac{B{x}^{3}{a}^{2}}{3}}+{a}^{2}A\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2*(B*x^3+A)/x,x)

[Out]

1/9*B*x^9*b^2+1/6*A*b^2*x^6+1/3*B*x^6*a*b+2/3*a*A*b*x^3+1/3*B*x^3*a^2+a^2*A*ln(x
)

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Maxima [A]  time = 1.37785, size = 70, normalized size = 1.52 \[ \frac{1}{9} \, B b^{2} x^{9} + \frac{1}{6} \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac{1}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} + \frac{1}{3} \, A a^{2} \log \left (x^{3}\right ) \]

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]

1/9*B*b^2*x^9 + 1/6*(2*B*a*b + A*b^2)*x^6 + 1/3*(B*a^2 + 2*A*a*b)*x^3 + 1/3*A*a^
2*log(x^3)

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Fricas [A]  time = 0.222815, size = 66, normalized size = 1.43 \[ \frac{1}{9} \, B b^{2} x^{9} + \frac{1}{6} \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac{1}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{3} + A a^{2} \log \left (x\right ) \]

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]

1/9*B*b^2*x^9 + 1/6*(2*B*a*b + A*b^2)*x^6 + 1/3*(B*a^2 + 2*A*a*b)*x^3 + A*a^2*lo
g(x)

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Sympy [A]  time = 1.16031, size = 53, normalized size = 1.15 \[ A a^{2} \log{\left (x \right )} + \frac{B b^{2} x^{9}}{9} + x^{6} \left (\frac{A b^{2}}{6} + \frac{B a b}{3}\right ) + x^{3} \left (\frac{2 A a b}{3} + \frac{B a^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**2*log(x) + B*b**2*x**9/9 + x**6*(A*b**2/6 + B*a*b/3) + x**3*(2*A*a*b/3 + B*
a**2/3)

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GIAC/XCAS [A]  time = 0.222278, size = 70, normalized size = 1.52 \[ \frac{1}{9} \, B b^{2} x^{9} + \frac{1}{3} \, B a b x^{6} + \frac{1}{6} \, A b^{2} x^{6} + \frac{1}{3} \, B a^{2} x^{3} + \frac{2}{3} \, A a b x^{3} + A a^{2}{\rm ln}\left ({\left | x \right |}\right ) \]

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]

1/9*B*b^2*x^9 + 1/3*B*a*b*x^6 + 1/6*A*b^2*x^6 + 1/3*B*a^2*x^3 + 2/3*A*a*b*x^3 +
A*a^2*ln(abs(x))