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

Optimal. Leaf size=29 \[ \frac{1}{3} x^3 (a B+A b)+a A \log (x)+\frac{1}{6} b B x^6 \]

[Out]

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

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Rubi [A]  time = 0.0664195, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{3} x^3 (a B+A b)+a A \log (x)+\frac{1}{6} b B x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0161975, size = 29, normalized size = 1. \[ \frac{1}{3} x^3 (a B+A b)+a A \log (x)+\frac{1}{6} b B x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 28, normalized size = 1. \[{\frac{bB{x}^{6}}{6}}+{\frac{A{x}^{3}b}{3}}+{\frac{B{x}^{3}a}{3}}+aA\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(B*x^3+A)/x,x)

[Out]

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

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Maxima [A]  time = 1.36714, size = 38, normalized size = 1.31 \[ \frac{1}{6} \, B b x^{6} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} + \frac{1}{3} \, A a \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224996, size = 34, normalized size = 1.17 \[ \frac{1}{6} \, B b x^{6} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} + A a \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.04253, size = 27, normalized size = 0.93 \[ A a \log{\left (x \right )} + \frac{B b x^{6}}{6} + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222875, size = 38, normalized size = 1.31 \[ \frac{1}{6} \, B b x^{6} + \frac{1}{3} \, B a x^{3} + \frac{1}{3} \, A b x^{3} + A a{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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