Optimal. Leaf size=215 \[ \frac{b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac{1}{2 a^2 c^3 x^2}-\frac{d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.612914, antiderivative size = 215, 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 \[ \frac{b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac{1}{2 a^2 c^3 x^2}-\frac{d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 124.796, size = 202, normalized size = 0.94 \[ - \frac{d^{3}}{4 c^{2} \left (c + d x^{2}\right )^{2} \left (a d - b c\right )^{2}} - \frac{d^{3} \left (a d - 2 b c\right )}{c^{3} \left (c + d x^{2}\right ) \left (a d - b c\right )^{3}} + \frac{d^{3} \left (3 a^{2} d^{2} - 10 a b c d + 10 b^{2} c^{2}\right ) \log{\left (c + d x^{2} \right )}}{2 c^{4} \left (a d - b c\right )^{4}} + \frac{b^{4}}{2 a^{2} \left (a + b x^{2}\right ) \left (a d - b c\right )^{3}} - \frac{1}{2 a^{2} c^{3} x^{2}} - \frac{b^{4} \left (5 a d - 2 b c\right ) \log{\left (a + b x^{2} \right )}}{2 a^{3} \left (a d - b c\right )^{4}} - \frac{\left (3 a d + 2 b c\right ) \log{\left (x^{2} \right )}}{2 a^{3} c^{4}} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.62441, size = 208, normalized size = 0.97 \[ \frac{1}{4} \left (\frac{2 b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^4}-\frac{4 \log (x) (3 a d+2 b c)}{a^3 c^4}+\frac{2 b^4}{a^2 \left (a+b x^2\right ) (a d-b c)^3}+\frac{2 d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^4}-\frac{2}{a^2 c^3 x^2}+\frac{4 d^3 (a d-2 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
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Maple [A] time = 0.04, size = 405, normalized size = 1.9 \[ -{\frac{1}{2\,{a}^{2}{c}^{3}{x}^{2}}}-3\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{4}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}{c}^{3}}}-{\frac{{d}^{5}{a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}+3\,{\frac{{d}^{4}ab}{{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}-2\,{\frac{{b}^{2}{d}^{3}}{c \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{5}{a}^{2}}{4\,{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{d}^{4}ab}{2\,c \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{d}^{3}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{d}^{5}\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{c}^{4} \left ( ad-bc \right ) ^{4}}}-5\,{\frac{{d}^{4}\ln \left ( d{x}^{2}+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{4}}}+5\,{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{4}}}-{\frac{5\,{b}^{4}\ln \left ( b{x}^{2}+a \right ) d}{2\,{a}^{2} \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{5}\ln \left ( b{x}^{2}+a \right ) c}{{a}^{3} \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{4}d}{2\,a \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{5}c}{2\,{a}^{2} \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^3,x)
[Out]
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Maxima [A] time = 1.40466, size = 879, normalized size = 4.09 \[ \frac{{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}} + \frac{{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}} - \frac{2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{6} +{\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{4} +{\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x^{2}}{4 \,{\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{8} +{\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{6} +{\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{4} +{\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x^{2}\right )}} - \frac{{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 29.6754, size = 1656, normalized size = 7.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.275722, size = 861, normalized size = 4. \[ \frac{{\left (2 \, b^{6} c - 5 \, a b^{5} d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )}} + \frac{{\left (10 \, b^{2} c^{2} d^{4} - 10 \, a b c d^{5} + 3 \, a^{2} d^{6}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )}} + \frac{10 \, a^{2} b^{3} c^{2} d^{3} x^{4} - 10 \, a^{3} b^{2} c d^{4} x^{4} + 3 \, a^{4} b d^{5} x^{4} - 4 \, b^{5} c^{5} x^{2} + 10 \, a b^{4} c^{4} d x^{2} - 12 \, a^{2} b^{3} c^{3} d^{2} x^{2} + 18 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 12 \, a^{4} b c d^{4} x^{2} + 3 \, a^{5} d^{5} x^{2} - 2 \, a b^{4} c^{5} + 8 \, a^{2} b^{3} c^{4} d - 12 \, a^{3} b^{2} c^{3} d^{2} + 8 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}}{4 \,{\left (a^{2} b^{4} c^{8} - 4 \, a^{3} b^{3} c^{7} d + 6 \, a^{4} b^{2} c^{6} d^{2} - 4 \, a^{5} b c^{5} d^{3} + a^{6} c^{4} d^{4}\right )}{\left (b x^{4} + a x^{2}\right )}} - \frac{30 \, b^{2} c^{2} d^{5} x^{4} - 30 \, a b c d^{6} x^{4} + 9 \, a^{2} d^{7} x^{4} + 68 \, b^{2} c^{3} d^{4} x^{2} - 72 \, a b c^{2} d^{5} x^{2} + 22 \, a^{2} c d^{6} x^{2} + 39 \, b^{2} c^{4} d^{3} - 44 \, a b c^{3} d^{4} + 14 \, a^{2} c^{2} d^{5}}{4 \,{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}{\left (d x^{2} + c\right )}^{2}} - \frac{{\left (2 \, b c + 3 \, a d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^3),x, algorithm="giac")
[Out]