Optimal. Leaf size=238 \[ -\frac {5 a^2}{b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^5}{8 b^6 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a^4}{6 b^6 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 a^3}{2 b^6 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.19, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 646, 43} \[ \frac {a^5}{8 b^6 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a^4}{6 b^6 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 a^3}{2 b^6 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a^2}{b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,x^2\right )\\ &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^5}{\left (a b+b^2 x\right )^5} \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \left (\frac {1}{b^{10}}-\frac {a^5}{b^{10} (a+b x)^5}+\frac {5 a^4}{b^{10} (a+b x)^4}-\frac {10 a^3}{b^{10} (a+b x)^3}+\frac {10 a^2}{b^{10} (a+b x)^2}-\frac {5 a}{b^{10} (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {5 a^2}{b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {a^5}{8 b^6 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a^4}{6 b^6 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 a^3}{2 b^6 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x^2 \left (a+b x^2\right )}{2 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {5 a \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 103, normalized size = 0.43 \[ \frac {-77 a^5-248 a^4 b x^2-252 a^3 b^2 x^4-48 a^2 b^3 x^6+48 a b^4 x^8-60 a \left (a+b x^2\right )^4 \log \left (a+b x^2\right )+12 b^5 x^{10}}{24 b^6 \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 157, normalized size = 0.66 \[ \frac {12 \, b^{5} x^{10} + 48 \, a b^{4} x^{8} - 48 \, a^{2} b^{3} x^{6} - 252 \, a^{3} b^{2} x^{4} - 248 \, a^{4} b x^{2} - 77 \, a^{5} - 60 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{24 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 105, normalized size = 0.44 \[ \frac {x^{2}}{2 \, b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {5 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {120 \, a^{2} b^{3} x^{6} + 300 \, a^{3} b^{2} x^{4} + 260 \, a^{4} b x^{2} + 77 \, a^{5}}{24 \, {\left (b x^{2} + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 163, normalized size = 0.68 \[ -\frac {\left (-12 b^{5} x^{10}+60 a \,b^{4} x^{8} \ln \left (b \,x^{2}+a \right )-48 a \,b^{4} x^{8}+240 a^{2} b^{3} x^{6} \ln \left (b \,x^{2}+a \right )+48 a^{2} b^{3} x^{6}+360 a^{3} b^{2} x^{4} \ln \left (b \,x^{2}+a \right )+252 a^{3} b^{2} x^{4}+240 a^{4} b \,x^{2} \ln \left (b \,x^{2}+a \right )+248 a^{4} b \,x^{2}+60 a^{5} \ln \left (b \,x^{2}+a \right )+77 a^{5}\right ) \left (b \,x^{2}+a \right )}{24 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 110, normalized size = 0.46 \[ -\frac {120 \, a^{2} b^{3} x^{6} + 300 \, a^{3} b^{2} x^{4} + 260 \, a^{4} b x^{2} + 77 \, a^{5}}{24 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}} + \frac {x^{2}}{2 \, b^{5}} - \frac {5 \, a \log \left (b x^{2} + a\right )}{2 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{11}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{11}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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