3.7 \(\int \frac{\left (a+b x^3\right ) \left (A+B x^3\right )}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0786205, 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 \[ \log (x) (a B+A b)-\frac{a A}{3 x^3}+\frac{1}{3} b B x^3 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)*(A + B*x^3))/x^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)*(B*x**3+A)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0224484, size = 29, normalized size = 1. \[ \log (x) (a B+A b)-\frac{a A}{3 x^3}+\frac{1}{3} b B x^3 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)*(A + B*x^3))/x^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 26, normalized size = 0.9 \[{\frac{bB{x}^{3}}{3}}+A\ln \left ( x \right ) b+Ba\ln \left ( x \right ) -{\frac{Aa}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(B*x^3+A)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.37409, size = 38, normalized size = 1.31 \[ \frac{1}{3} \, B b x^{3} + \frac{1}{3} \,{\left (B a + A b\right )} \log \left (x^{3}\right ) - \frac{A a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224578, size = 41, normalized size = 1.41 \[ \frac{B b x^{6} + 3 \,{\left (B a + A b\right )} x^{3} \log \left (x\right ) - A a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.39514, size = 26, normalized size = 0.9 \[ - \frac{A a}{3 x^{3}} + \frac{B b x^{3}}{3} + \left (A b + B a\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)*(B*x**3+A)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226048, size = 54, normalized size = 1.86 \[ \frac{1}{3} \, B b x^{3} +{\left (B a + A b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{B a x^{3} + A b x^{3} + A a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^4,x, algorithm="giac")

[Out]

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