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

Optimal. Leaf size=45 \[ \frac{\log \left (a+b x^4\right )}{4 (b c-a d)}-\frac{\log \left (c+d x^4\right )}{4 (b c-a d)} \]

[Out]

Log[a + b*x^4]/(4*(b*c - a*d)) - Log[c + d*x^4]/(4*(b*c - a*d))

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Rubi [A]  time = 0.093747, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\log \left (a+b x^4\right )}{4 (b c-a d)}-\frac{\log \left (c+d x^4\right )}{4 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x^4]/(4*(b*c - a*d)) - Log[c + d*x^4]/(4*(b*c - a*d))

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Rubi in Sympy [A]  time = 12.7466, size = 36, normalized size = 0.8 \[ - \frac{\log{\left (a + b x^{4} \right )}}{4 \left (a d - b c\right )} + \frac{\log{\left (c + d x^{4} \right )}}{4 \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(a + b*x**4)/(4*(a*d - b*c)) + log(c + d*x**4)/(4*(a*d - b*c))

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Mathematica [A]  time = 0.0316985, size = 31, normalized size = 0.69 \[ \frac{\log \left (a+b x^4\right )-\log \left (c+d x^4\right )}{4 b c-4 a d} \]

Antiderivative was successfully verified.

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

[Out]

(Log[a + b*x^4] - Log[c + d*x^4])/(4*b*c - 4*a*d)

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Maple [A]  time = 0.01, size = 42, normalized size = 0.9 \[{\frac{\ln \left ( d{x}^{4}+c \right ) }{4\,ad-4\,bc}}-{\frac{\ln \left ( b{x}^{4}+a \right ) }{4\,ad-4\,bc}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^4+a)/(d*x^4+c),x)

[Out]

1/4/(a*d-b*c)*ln(d*x^4+c)-1/4/(a*d-b*c)*ln(b*x^4+a)

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Maxima [A]  time = 1.36802, size = 55, normalized size = 1.22 \[ \frac{\log \left (b x^{4} + a\right )}{4 \,{\left (b c - a d\right )}} - \frac{\log \left (d x^{4} + c\right )}{4 \,{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.213667, size = 42, normalized size = 0.93 \[ \frac{\log \left (b x^{4} + a\right ) - \log \left (d x^{4} + c\right )}{4 \,{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.01317, size = 138, normalized size = 3.07 \[ \frac{\log{\left (x^{4} + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{4 \left (a d - b c\right )} - \frac{\log{\left (x^{4} + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{4 \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**4 + (-a**2*d**2/(a*d - b*c) + 2*a*b*c*d/(a*d - b*c) + a*d - b**2*c**2/(a*
d - b*c) + b*c)/(2*b*d))/(4*(a*d - b*c)) - log(x**4 + (a**2*d**2/(a*d - b*c) - 2
*a*b*c*d/(a*d - b*c) + a*d + b**2*c**2/(a*d - b*c) + b*c)/(2*b*d))/(4*(a*d - b*c
))

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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(x^3/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError