Optimal. Leaf size=155 \[ \frac{b^4 \log \left (a+b x^2\right )}{2 a^4 (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}-\frac{\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}-\frac{a^2 d^2+a b c d+b^2 c^2}{2 a^3 c^3 x^2}-\frac{d^4 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)}-\frac{1}{6 a c x^6} \]
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Rubi [A] time = 0.386154, antiderivative size = 155, 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 \[ \frac{b^4 \log \left (a+b x^2\right )}{2 a^4 (b c-a d)}+\frac{a d+b c}{4 a^2 c^2 x^4}-\frac{\log (x) (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{a^4 c^4}-\frac{a^2 d^2+a b c d+b^2 c^2}{2 a^3 c^3 x^2}-\frac{d^4 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)}-\frac{1}{6 a c x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(a + b*x^2)*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 53.3115, size = 141, normalized size = 0.91 \[ \frac{d^{4} \log{\left (c + d x^{2} \right )}}{2 c^{4} \left (a d - b c\right )} - \frac{1}{6 a c x^{6}} + \frac{a d + b c}{4 a^{2} c^{2} x^{4}} - \frac{a^{2} d^{2} + a b c d + b^{2} c^{2}}{2 a^{3} c^{3} x^{2}} - \frac{b^{4} \log{\left (a + b x^{2} \right )}}{2 a^{4} \left (a d - b c\right )} - \frac{\left (a d + b c\right ) \left (a^{2} d^{2} + b^{2} c^{2}\right ) \log{\left (x^{2} \right )}}{2 a^{4} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(b*x**2+a)/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.106573, size = 147, normalized size = 0.95 \[ \frac{12 x^6 \log (x) \left (b^4 c^4-a^4 d^4\right )+a \left (a^3 c d \left (-2 c^2+3 c d x^2-6 d^2 x^4\right )+6 a^3 d^4 x^6 \log \left (c+d x^2\right )+2 a^2 b c^4-3 a b^2 c^4 x^2+6 b^3 c^4 x^4\right )-6 b^4 c^4 x^6 \log \left (a+b x^2\right )}{12 a^4 c^4 x^6 (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*(a + b*x^2)*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.022, size = 184, normalized size = 1.2 \[ -{\frac{1}{6\,ac{x}^{6}}}+{\frac{d}{4\,{x}^{4}a{c}^{2}}}+{\frac{b}{4\,{a}^{2}c{x}^{4}}}-{\frac{{d}^{2}}{2\,a{c}^{3}{x}^{2}}}-{\frac{bd}{2\,{a}^{2}{c}^{2}{x}^{2}}}-{\frac{{b}^{2}}{2\,{a}^{3}c{x}^{2}}}-{\frac{\ln \left ( x \right ){d}^{3}}{a{c}^{4}}}-{\frac{b\ln \left ( x \right ){d}^{2}}{{a}^{2}{c}^{3}}}-{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}{c}^{2}}}-{\frac{\ln \left ( x \right ){b}^{3}}{{a}^{4}c}}+{\frac{{d}^{4}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{4} \left ( ad-bc \right ) }}-{\frac{{b}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4} \left ( ad-bc \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(b*x^2+a)/(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.36096, size = 223, normalized size = 1.44 \[ \frac{b^{4} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{4} b c - a^{5} d\right )}} - \frac{d^{4} \log \left (d x^{2} + c\right )}{2 \,{\left (b c^{5} - a c^{4} d\right )}} - \frac{{\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{4} c^{4}} - \frac{6 \,{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} - 3 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}}{12 \, a^{3} c^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^7),x, algorithm="maxima")
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Fricas [A] time = 1.69139, size = 209, normalized size = 1.35 \[ \frac{6 \, b^{4} c^{4} x^{6} \log \left (b x^{2} + a\right ) - 6 \, a^{4} d^{4} x^{6} \log \left (d x^{2} + c\right ) - 2 \, a^{3} b c^{4} + 2 \, a^{4} c^{3} d - 12 \,{\left (b^{4} c^{4} - a^{4} d^{4}\right )} x^{6} \log \left (x\right ) - 6 \,{\left (a b^{3} c^{4} - a^{4} c d^{3}\right )} x^{4} + 3 \,{\left (a^{2} b^{2} c^{4} - a^{4} c^{2} d^{2}\right )} x^{2}}{12 \,{\left (a^{4} b c^{5} - a^{5} c^{4} d\right )} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)*x^7),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**7/(b*x**2+a)/(d*x**2+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^2 + a)*(d*x^2 + c)*x^7),x, algorithm="giac")
[Out]