Optimal. Leaf size=118 \[ -\frac{a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac{3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac{5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac{5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac{15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{b^7}+\frac{x^2}{2 b^6} \]
[Out]
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Rubi [A] time = 0.255735, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^6}{10 b^7 \left (a+b x^2\right )^5}+\frac{3 a^5}{4 b^7 \left (a+b x^2\right )^4}-\frac{5 a^4}{2 b^7 \left (a+b x^2\right )^3}+\frac{5 a^3}{b^7 \left (a+b x^2\right )^2}-\frac{15 a^2}{2 b^7 \left (a+b x^2\right )}-\frac{3 a \log \left (a+b x^2\right )}{b^7}+\frac{x^2}{2 b^6} \]
Antiderivative was successfully verified.
[In] Int[x^13/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{6}}{10 b^{7} \left (a + b x^{2}\right )^{5}} + \frac{3 a^{5}}{4 b^{7} \left (a + b x^{2}\right )^{4}} - \frac{5 a^{4}}{2 b^{7} \left (a + b x^{2}\right )^{3}} + \frac{5 a^{3}}{b^{7} \left (a + b x^{2}\right )^{2}} - \frac{15 a^{2}}{2 b^{7} \left (a + b x^{2}\right )} - \frac{3 a \log{\left (a + b x^{2} \right )}}{b^{7}} + \frac{b^{6} \int ^{x^{2}} \frac{1}{b^{12}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**13/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.0507, size = 101, normalized size = 0.86 \[ -\frac{87 a^6+375 a^5 b x^2+600 a^4 b^2 x^4+400 a^3 b^3 x^6+50 a^2 b^4 x^8-50 a b^5 x^{10}+60 a \left (a+b x^2\right )^5 \log \left (a+b x^2\right )-10 b^6 x^{12}}{20 b^7 \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
[In] Integrate[x^13/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
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Maple [A] time = 0.02, size = 109, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,{b}^{6}}}-{\frac{{a}^{6}}{10\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3\,{a}^{5}}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{5\,{a}^{4}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{3}}}+5\,{\frac{{a}^{3}}{{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,{a}^{2}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) }}-3\,{\frac{a\ln \left ( b{x}^{2}+a \right ) }{{b}^{7}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^13/(b^2*x^4+2*a*b*x^2+a^2)^3,x)
[Out]
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Maxima [A] time = 0.699634, size = 178, normalized size = 1.51 \[ -\frac{150 \, a^{2} b^{4} x^{8} + 500 \, a^{3} b^{3} x^{6} + 650 \, a^{4} b^{2} x^{4} + 385 \, a^{5} b x^{2} + 87 \, a^{6}}{20 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} + \frac{x^{2}}{2 \, b^{6}} - \frac{3 \, a \log \left (b x^{2} + a\right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26371, size = 257, normalized size = 2.18 \[ \frac{10 \, b^{6} x^{12} + 50 \, a b^{5} x^{10} - 50 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 600 \, a^{4} b^{2} x^{4} - 375 \, a^{5} b x^{2} - 87 \, a^{6} - 60 \,{\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{20 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.14548, size = 136, normalized size = 1.15 \[ - \frac{3 a \log{\left (a + b x^{2} \right )}}{b^{7}} - \frac{87 a^{6} + 385 a^{5} b x^{2} + 650 a^{4} b^{2} x^{4} + 500 a^{3} b^{3} x^{6} + 150 a^{2} b^{4} x^{8}}{20 a^{5} b^{7} + 100 a^{4} b^{8} x^{2} + 200 a^{3} b^{9} x^{4} + 200 a^{2} b^{10} x^{6} + 100 a b^{11} x^{8} + 20 b^{12} x^{10}} + \frac{x^{2}}{2 b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**13/(b**2*x**4+2*a*b*x**2+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.271691, size = 128, normalized size = 1.08 \[ \frac{x^{2}}{2 \, b^{6}} - \frac{3 \, a{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{7}} + \frac{137 \, a b^{5} x^{10} + 535 \, a^{2} b^{4} x^{8} + 870 \, a^{3} b^{3} x^{6} + 720 \, a^{4} b^{2} x^{4} + 300 \, a^{5} b x^{2} + 50 \, a^{6}}{20 \,{\left (b x^{2} + a\right )}^{5} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")
[Out]