3.122 \(\int \frac{1}{x^7 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=119 \[ -\frac{b^3 \log \left (a+b x^3\right )}{3 a^3 (b c-a d)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{d^3 \log \left (c+d x^3\right )}{3 c^3 (b c-a d)}-\frac{1}{6 a c x^6} \]

[Out]

-1/(6*a*c*x^6) + (b*c + a*d)/(3*a^2*c^2*x^3) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*Lo
g[x])/(a^3*c^3) - (b^3*Log[a + b*x^3])/(3*a^3*(b*c - a*d)) + (d^3*Log[c + d*x^3]
)/(3*c^3*(b*c - a*d))

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Rubi [A]  time = 0.321633, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{b^3 \log \left (a+b x^3\right )}{3 a^3 (b c-a d)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{d^3 \log \left (c+d x^3\right )}{3 c^3 (b c-a d)}-\frac{1}{6 a c x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/(6*a*c*x^6) + (b*c + a*d)/(3*a^2*c^2*x^3) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*Lo
g[x])/(a^3*c^3) - (b^3*Log[a + b*x^3])/(3*a^3*(b*c - a*d)) + (d^3*Log[c + d*x^3]
)/(3*c^3*(b*c - a*d))

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Rubi in Sympy [A]  time = 37.4735, size = 109, normalized size = 0.92 \[ - \frac{d^{3} \log{\left (c + d x^{3} \right )}}{3 c^{3} \left (a d - b c\right )} - \frac{1}{6 a c x^{6}} + \frac{a d + b c}{3 a^{2} c^{2} x^{3}} + \frac{b^{3} \log{\left (a + b x^{3} \right )}}{3 a^{3} \left (a d - b c\right )} + \frac{\left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right ) \log{\left (x^{3} \right )}}{3 a^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(b*x**3+a)/(d*x**3+c),x)

[Out]

-d**3*log(c + d*x**3)/(3*c**3*(a*d - b*c)) - 1/(6*a*c*x**6) + (a*d + b*c)/(3*a**
2*c**2*x**3) + b**3*log(a + b*x**3)/(3*a**3*(a*d - b*c)) + (a**2*d**2 + a*b*c*d
+ b**2*c**2)*log(x**3)/(3*a**3*c**3)

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Mathematica [A]  time = 0.100781, size = 119, normalized size = 1. \[ \frac{b^3 \log \left (a+b x^3\right )}{3 a^3 (a d-b c)}+\frac{a d+b c}{3 a^2 c^2 x^3}+\frac{\log (x) \left (a^2 d^2+a b c d+b^2 c^2\right )}{a^3 c^3}+\frac{d^3 \log \left (c+d x^3\right )}{3 c^3 (b c-a d)}-\frac{1}{6 a c x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/(6*a*c*x^6) + (b*c + a*d)/(3*a^2*c^2*x^3) + ((b^2*c^2 + a*b*c*d + a^2*d^2)*Lo
g[x])/(a^3*c^3) + (b^3*Log[a + b*x^3])/(3*a^3*(-(b*c) + a*d)) + (d^3*Log[c + d*x
^3])/(3*c^3*(b*c - a*d))

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Maple [A]  time = 0.017, size = 124, normalized size = 1. \[ -{\frac{1}{6\,ac{x}^{6}}}+{\frac{d}{3\,a{c}^{2}{x}^{3}}}+{\frac{b}{3\,{a}^{2}c{x}^{3}}}+{\frac{\ln \left ( x \right ){d}^{2}}{a{c}^{3}}}+{\frac{\ln \left ( x \right ) bd}{{a}^{2}{c}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{3}c}}+{\frac{{b}^{3}\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3} \left ( ad-bc \right ) }}-{\frac{{d}^{3}\ln \left ( d{x}^{3}+c \right ) }{3\,{c}^{3} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^3+a)/(d*x^3+c),x)

[Out]

-1/6/a/c/x^6+1/3/a/c^2/x^3*d+1/3/a^2/c/x^3*b+1/a/c^3*ln(x)*d^2+1/a^2/c^2*ln(x)*b
*d+1/a^3/c*ln(x)*b^2+1/3*b^3/a^3/(a*d-b*c)*ln(b*x^3+a)-1/3*d^3/c^3/(a*d-b*c)*ln(
d*x^3+c)

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Maxima [A]  time = 1.48408, size = 158, normalized size = 1.33 \[ -\frac{b^{3} \log \left (b x^{3} + a\right )}{3 \,{\left (a^{3} b c - a^{4} d\right )}} + \frac{d^{3} \log \left (d x^{3} + c\right )}{3 \,{\left (b c^{4} - a c^{3} d\right )}} + \frac{{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} \log \left (x^{3}\right )}{3 \, a^{3} c^{3}} + \frac{2 \,{\left (b c + a d\right )} x^{3} - a c}{6 \, a^{2} c^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^7),x, algorithm="maxima")

[Out]

-1/3*b^3*log(b*x^3 + a)/(a^3*b*c - a^4*d) + 1/3*d^3*log(d*x^3 + c)/(b*c^4 - a*c^
3*d) + 1/3*(b^2*c^2 + a*b*c*d + a^2*d^2)*log(x^3)/(a^3*c^3) + 1/6*(2*(b*c + a*d)
*x^3 - a*c)/(a^2*c^2*x^6)

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Fricas [A]  time = 6.00004, size = 171, normalized size = 1.44 \[ -\frac{2 \, b^{3} c^{3} x^{6} \log \left (b x^{3} + a\right ) - 2 \, a^{3} d^{3} x^{6} \log \left (d x^{3} + c\right ) - 6 \,{\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{6} \log \left (x\right ) + a^{2} b c^{3} - a^{3} c^{2} d - 2 \,{\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{3}}{6 \,{\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^7),x, algorithm="fricas")

[Out]

-1/6*(2*b^3*c^3*x^6*log(b*x^3 + a) - 2*a^3*d^3*x^6*log(d*x^3 + c) - 6*(b^3*c^3 -
a^3*d^3)*x^6*log(x) + a^2*b*c^3 - a^3*c^2*d - 2*(a*b^2*c^3 - a^3*c*d^2)*x^3)/((
a^3*b*c^4 - a^4*c^3*d)*x^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^7),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError