Optimal. Leaf size=87 \[ \frac{b^2 \log \left (a+b x^4\right )}{4 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^4\right )}{4 c^2 (b c-a d)}-\frac{1}{4 a c x^4} \]
[Out]
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Rubi [A] time = 0.228493, 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 \[ \frac{b^2 \log \left (a+b x^4\right )}{4 a^2 (b c-a d)}-\frac{\log (x) (a d+b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^4\right )}{4 c^2 (b c-a d)}-\frac{1}{4 a c x^4} \]
Antiderivative was successfully verified.
[In] Int[1/(x^5*(a + b*x^4)*(c + d*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 30.3838, size = 76, normalized size = 0.87 \[ \frac{d^{2} \log{\left (c + d x^{4} \right )}}{4 c^{2} \left (a d - b c\right )} - \frac{1}{4 a c x^{4}} - \frac{b^{2} \log{\left (a + b x^{4} \right )}}{4 a^{2} \left (a d - b c\right )} - \frac{\left (a d + b c\right ) \log{\left (x^{4} \right )}}{4 a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**5/(b*x**4+a)/(d*x**4+c),x)
[Out]
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Mathematica [A] time = 0.0705019, size = 88, normalized size = 1.01 \[ -\frac{b^2 \log \left (a+b x^4\right )}{4 a^2 (a d-b c)}+\frac{\log (x) (-a d-b c)}{a^2 c^2}-\frac{d^2 \log \left (c+d x^4\right )}{4 c^2 (b c-a d)}-\frac{1}{4 a c x^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^5*(a + b*x^4)*(c + d*x^4)),x]
[Out]
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Maple [A] time = 0.016, size = 87, normalized size = 1. \[{\frac{{d}^{2}\ln \left ( d{x}^{4}+c \right ) }{4\,{c}^{2} \left ( ad-bc \right ) }}-{\frac{1}{4\,ac{x}^{4}}}-{\frac{\ln \left ( x \right ) d}{a{c}^{2}}}-{\frac{\ln \left ( x \right ) b}{{a}^{2}c}}-{\frac{{b}^{2}\ln \left ( b{x}^{4}+a \right ) }{4\,{a}^{2} \left ( ad-bc \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^5/(b*x^4+a)/(d*x^4+c),x)
[Out]
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Maxima [A] time = 1.39795, size = 117, normalized size = 1.34 \[ \frac{b^{2} \log \left (b x^{4} + a\right )}{4 \,{\left (a^{2} b c - a^{3} d\right )}} - \frac{d^{2} \log \left (d x^{4} + c\right )}{4 \,{\left (b c^{3} - a c^{2} d\right )}} - \frac{{\left (b c + a d\right )} \log \left (x^{4}\right )}{4 \, a^{2} c^{2}} - \frac{1}{4 \, a c x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)*(d*x^4 + c)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 7.1214, size = 134, normalized size = 1.54 \[ \frac{b^{2} c^{2} x^{4} \log \left (b x^{4} + a\right ) - a^{2} d^{2} x^{4} \log \left (d x^{4} + c\right ) - 4 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{4} \log \left (x\right ) - a b c^{2} + a^{2} c d}{4 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)*(d*x^4 + c)*x^5),x, algorithm="fricas")
[Out]
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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**5/(b*x**4+a)/(d*x**4+c),x)
[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^4 + a)*(d*x^4 + c)*x^5),x, algorithm="giac")
[Out]