3.119 \(\int \frac{1}{x^4 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=87 \[ \frac{b^2 \log \left (a+b x^3\right )}{3 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^3\right )}{3 c^2 (b c-a d)}-\frac{1}{3 a c x^3} \]

[Out]

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

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Rubi [A]  time = 0.244589, antiderivative size = 87, 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 \[ \frac{b^2 \log \left (a+b x^3\right )}{3 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^3\right )}{3 c^2 (b c-a d)}-\frac{1}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 30.4183, size = 76, normalized size = 0.87 \[ \frac{d^{2} \log{\left (c + d x^{3} \right )}}{3 c^{2} \left (a d - b c\right )} - \frac{1}{3 a c x^{3}} - \frac{b^{2} \log{\left (a + b x^{3} \right )}}{3 a^{2} \left (a d - b c\right )} - \frac{\left (a d + b c\right ) \log{\left (x^{3} \right )}}{3 a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**3+a)/(d*x**3+c),x)

[Out]

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

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Mathematica [A]  time = 0.0797666, size = 88, normalized size = 1.01 \[ -\frac{b^2 \log \left (a+b x^3\right )}{3 a^2 (a d-b c)}+\frac{\log (x) (-a d-b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^3\right )}{3 c^2 (b c-a d)}-\frac{1}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 87, normalized size = 1. \[ -{\frac{1}{3\,ac{x}^{3}}}-{\frac{\ln \left ( x \right ) d}{a{c}^{2}}}-{\frac{\ln \left ( x \right ) b}{{a}^{2}c}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2} \left ( ad-bc \right ) }}+{\frac{{d}^{2}\ln \left ( d{x}^{3}+c \right ) }{3\,{c}^{2} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

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Maxima [A]  time = 1.40407, size = 117, normalized size = 1.34 \[ \frac{b^{2} \log \left (b x^{3} + a\right )}{3 \,{\left (a^{2} b c - a^{3} d\right )}} - \frac{d^{2} \log \left (d x^{3} + c\right )}{3 \,{\left (b c^{3} - a c^{2} d\right )}} - \frac{{\left (b c + a d\right )} \log \left (x^{3}\right )}{3 \, a^{2} c^{2}} - \frac{1}{3 \, a c x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.83515, size = 134, normalized size = 1.54 \[ \frac{b^{2} c^{2} x^{3} \log \left (b x^{3} + a\right ) - a^{2} d^{2} x^{3} \log \left (d x^{3} + c\right ) - 3 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3} \log \left (x\right ) - a b c^{2} + a^{2} c d}{3 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^4),x, algorithm="fricas")

[Out]

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

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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**4/(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^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError