Optimal. Leaf size=156 \[ \frac{b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}-\frac{2 \log (x) (a d+b c)}{a^3 c^3}-\frac{b^3}{2 a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{1}{2 a^2 c^2 x^2}+\frac{d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}-\frac{d^3}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.448724, antiderivative size = 156, 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 \[ \frac{b^3 (b c-2 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^3}-\frac{2 \log (x) (a d+b c)}{a^3 c^3}-\frac{b^3}{2 a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{1}{2 a^2 c^2 x^2}+\frac{d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}-\frac{d^3}{2 c^2 \left (c+d x^2\right ) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 71.6083, size = 143, normalized size = 0.92 \[ - \frac{d^{3}}{2 c^{2} \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} + \frac{d^{3} \left (a d - 2 b c\right ) \log{\left (c + d x^{2} \right )}}{c^{3} \left (a d - b c\right )^{3}} - \frac{b^{3}}{2 a^{2} \left (a + b x^{2}\right ) \left (a d - b c\right )^{2}} - \frac{1}{2 a^{2} c^{2} x^{2}} + \frac{b^{3} \left (2 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{a^{3} \left (a d - b c\right )^{3}} - \frac{\left (a d + b c\right ) \log{\left (x^{2} \right )}}{a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.368365, size = 157, normalized size = 1.01 \[ \frac{1}{2} \left (\frac{2 b^3 (2 a d-b c) \log \left (a+b x^2\right )}{a^3 (a d-b c)^3}-\frac{4 \log (x) (a d+b c)}{a^3 c^3}-\frac{b^3}{a^2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{1}{a^2 c^2 x^2}+\frac{2 d^3 (2 b c-a d) \log \left (c+d x^2\right )}{c^3 (b c-a d)^3}-\frac{d^3}{c^2 \left (c+d x^2\right ) (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^2),x]
[Out]
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Maple [A] time = 0.037, size = 254, normalized size = 1.6 \[ -{\frac{1}{2\,{a}^{2}{c}^{2}{x}^{2}}}-2\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{3}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}{c}^{2}}}-{\frac{{d}^{4}a}{2\,{c}^{2} \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{3}b}{2\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{4}\ln \left ( d{x}^{2}+c \right ) a}{{c}^{3} \left ( ad-bc \right ) ^{3}}}-2\,{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) b}{{c}^{2} \left ( ad-bc \right ) ^{3}}}+2\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) d}{{a}^{2} \left ( ad-bc \right ) ^{3}}}-{\frac{{b}^{4}\ln \left ( b{x}^{2}+a \right ) c}{{a}^{3} \left ( ad-bc \right ) ^{3}}}-{\frac{d{b}^{3}}{2\,a \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}c}{2\,{a}^{2} \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.37145, size = 514, normalized size = 3.29 \[ \frac{{\left (b^{4} c - 2 \, a b^{3} d\right )} \log \left (b x^{2} + a\right )}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}} + \frac{{\left (2 \, b c d^{3} - a d^{4}\right )} \log \left (d x^{2} + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + 2 \,{\left (b^{3} c^{2} d - a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} x^{2}}{2 \,{\left ({\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{6} +{\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x^{4} +{\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2}\right )}} - \frac{{\left (b c + a d\right )} \log \left (x^{2}\right )}{a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 9.23125, size = 900, normalized size = 5.77 \[ -\frac{a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + 2 \,{\left (a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} +{\left (2 \, a b^{4} c^{5} - 3 \, a^{2} b^{3} c^{4} d + 3 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}\right )} x^{2} - 2 \,{\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{6} +{\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x^{4} +{\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left ({\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} +{\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} +{\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 4 \,{\left ({\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{6} +{\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{4} +{\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left ({\left (a^{3} b^{4} c^{6} d - 3 \, a^{4} b^{3} c^{5} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{3} - a^{6} b c^{3} d^{4}\right )} x^{6} +{\left (a^{3} b^{4} c^{7} - 2 \, a^{4} b^{3} c^{6} d + 2 \, a^{6} b c^{4} d^{3} - a^{7} c^{3} d^{4}\right )} x^{4} +{\left (a^{4} b^{3} c^{7} - 3 \, a^{5} b^{2} c^{6} d + 3 \, a^{6} b c^{5} d^{2} - a^{7} c^{4} d^{3}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2*x^3),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**3/(b*x**2+a)**2/(d*x**2+c)**2,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)^2*(d*x^2 + c)^2*x^3),x, algorithm="giac")
[Out]