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

Optimal. Leaf size=31 \[ \frac{1}{2} x^2 (a B+A b)-\frac{a A}{x}+\frac{1}{5} b B x^5 \]

[Out]

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

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Rubi [A]  time = 0.0554531, antiderivative size = 31, 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 \[ \frac{1}{2} x^2 (a B+A b)-\frac{a A}{x}+\frac{1}{5} b B x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/x + B*b*x**5/5 + (A*b + B*a)*Integral(x, x)

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Mathematica [A]  time = 0.0193497, size = 31, normalized size = 1. \[ \frac{1}{2} x^2 (a B+A b)-\frac{a A}{x}+\frac{1}{5} b B x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 30, normalized size = 1. \[{\frac{bB{x}^{5}}{5}}+{\frac{A{x}^{2}b}{2}}+{\frac{B{x}^{2}a}{2}}-{\frac{Aa}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(B*x^3+A)/x^2,x)

[Out]

1/5*b*B*x^5+1/2*A*x^2*b+1/2*B*x^2*a-a*A/x

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Maxima [A]  time = 1.37983, size = 36, normalized size = 1.16 \[ \frac{1}{5} \, B b x^{5} + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} - \frac{A a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21716, size = 39, normalized size = 1.26 \[ \frac{2 \, B b x^{6} + 5 \,{\left (B a + A b\right )} x^{3} - 10 \, A a}{10 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(2*B*b*x^6 + 5*(B*a + A*b)*x^3 - 10*A*a)/x

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Sympy [A]  time = 1.03608, size = 26, normalized size = 0.84 \[ - \frac{A a}{x} + \frac{B b x^{5}}{5} + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/x + B*b*x**5/5 + x**2*(A*b/2 + B*a/2)

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GIAC/XCAS [A]  time = 0.223506, size = 39, normalized size = 1.26 \[ \frac{1}{5} \, B b x^{5} + \frac{1}{2} \, B a x^{2} + \frac{1}{2} \, A b x^{2} - \frac{A a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*b*x^5 + 1/2*B*a*x^2 + 1/2*A*b*x^2 - A*a/x