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

Optimal. Leaf size=28 \[ x (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{4} b B x^4 \]

[Out]

-(a*A)/(2*x^2) + (A*b + a*B)*x + (b*B*x^4)/4

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Rubi [A]  time = 0.0568581, antiderivative size = 28, 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 \[ x (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

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

[Out]

-(a*A)/(2*x^2) + (A*b + a*B)*x + (b*B*x^4)/4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a}{2 x^{2}} + \frac{B b x^{4}}{4} + \frac{\left (A b + B a\right ) \int A\, dx}{A} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/(2*x**2) + B*b*x**4/4 + (A*b + B*a)*Integral(A, x)/A

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Mathematica [A]  time = 0.0167454, size = 28, normalized size = 1. \[ x (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

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

[Out]

-(a*A)/(2*x^2) + (A*b + a*B)*x + (b*B*x^4)/4

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Maple [A]  time = 0.005, size = 24, normalized size = 0.9 \[{\frac{bB{x}^{4}}{4}}+Axb+Bxa-{\frac{Aa}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(B*x^3+A)/x^3,x)

[Out]

1/4*b*B*x^4+A*x*b+B*x*a-1/2*a*A/x^2

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Maxima [A]  time = 1.37077, size = 32, normalized size = 1.14 \[ \frac{1}{4} \, B b x^{4} +{\left (B a + A b\right )} x - \frac{A a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*b*x^4 + (B*a + A*b)*x - 1/2*A*a/x^2

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Fricas [A]  time = 0.214784, size = 38, normalized size = 1.36 \[ \frac{B b x^{6} + 4 \,{\left (B a + A b\right )} x^{3} - 2 \, A a}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(B*b*x^6 + 4*(B*a + A*b)*x^3 - 2*A*a)/x^2

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Sympy [A]  time = 1.05901, size = 24, normalized size = 0.86 \[ - \frac{A a}{2 x^{2}} + \frac{B b x^{4}}{4} + x \left (A b + B a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/(2*x**2) + B*b*x**4/4 + x*(A*b + B*a)

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GIAC/XCAS [A]  time = 0.222021, size = 31, normalized size = 1.11 \[ \frac{1}{4} \, B b x^{4} + B a x + A b x - \frac{A a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*b*x^4 + B*a*x + A*b*x - 1/2*A*a/x^2