3.10 \(\int \frac{\left (a+b x^3\right ) \left (A+B x^3\right )}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0717959, 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 \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{6 x^6}+b B \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)*(A + B*x^3))/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.35459, size = 31, normalized size = 1.07 \[ - \frac{A a}{6 x^{6}} + \frac{B b \log{\left (x^{3} \right )}}{3} - \frac{\frac{A b}{3} + \frac{B a}{3}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)*(B*x**3+A)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0319615, size = 31, normalized size = 1.07 \[ \frac{-a B-A b}{3 x^3}-\frac{a A}{6 x^6}+b B \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)*(A + B*x^3))/x^7,x]

[Out]

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

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Maple [A]  time = 0.007, size = 28, normalized size = 1. \[ bB\ln \left ( x \right ) -{\frac{Aa}{6\,{x}^{6}}}-{\frac{Ab}{3\,{x}^{3}}}-{\frac{Ba}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)*(B*x^3+A)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.37386, size = 41, normalized size = 1.41 \[ \frac{1}{3} \, B b \log \left (x^{3}\right ) - \frac{2 \,{\left (B a + A b\right )} x^{3} + A a}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.229256, size = 42, normalized size = 1.45 \[ \frac{6 \, B b x^{6} \log \left (x\right ) - 2 \,{\left (B a + A b\right )} x^{3} - A a}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.67505, size = 27, normalized size = 0.93 \[ B b \log{\left (x \right )} - \frac{A a + x^{3} \left (2 A b + 2 B a\right )}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)*(B*x**3+A)/x**7,x)

[Out]

B*b*log(x) - (A*a + x**3*(2*A*b + 2*B*a))/(6*x**6)

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GIAC/XCAS [A]  time = 0.224487, size = 50, normalized size = 1.72 \[ B b{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, B b x^{6} + 2 \, B a x^{3} + 2 \, A b x^{3} + A a}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)/x^7,x, algorithm="giac")

[Out]

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