3.9 \(\int \frac{\left (a+b x^3\right ) \left (A+B x^3\right )}{x^6} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{5 x^5}+b B x \]

[Out]

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

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Rubi [A]  time = 0.0548956, 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 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{5 x^5}+b B x \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)*(A + B*x^3))/x^6,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a}{5 x^{5}} + b \int B\, dx - \frac{\frac{A b}{2} + \frac{B a}{2}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)*(B*x**3+A)/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.0230193, size = 30, normalized size = 1.07 \[ \frac{-a B-A b}{2 x^2}-\frac{a A}{5 x^5}+b B x \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)*(A + B*x^3))/x^6,x]

[Out]

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

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Maple [A]  time = 0.008, size = 25, normalized size = 0.9 \[ bBx-{\frac{Ab+Ba}{2\,{x}^{2}}}-{\frac{Aa}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(B*x^3+A)/x^6,x)

[Out]

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

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Maxima [A]  time = 1.41876, size = 36, normalized size = 1.29 \[ B b x - \frac{5 \,{\left (B a + A b\right )} x^{3} + 2 \, A a}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.214746, size = 39, normalized size = 1.39 \[ \frac{10 \, B b x^{6} - 5 \,{\left (B a + A b\right )} x^{3} - 2 \, A a}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.6675, size = 27, normalized size = 0.96 \[ B b x - \frac{2 A a + x^{3} \left (5 A b + 5 B a\right )}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)*(B*x**3+A)/x**6,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223954, size = 39, normalized size = 1.39 \[ B b x - \frac{5 \, B a x^{3} + 5 \, A b x^{3} + 2 \, A a}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^6,x, algorithm="giac")

[Out]

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