Optimal. Leaf size=39 \[ \frac {x^8}{40 a^2 \left (a+b x^2\right )^4}+\frac {x^8}{10 a \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 266, 45, 37} \begin {gather*} \frac {x^8}{40 a^2 \left (a+b x^2\right )^4}+\frac {x^8}{10 a \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^7}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \frac {x^3}{\left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac {x^8}{10 a \left (a+b x^2\right )^5}+\frac {b^5 \operatorname {Subst}\left (\int \frac {x^3}{\left (a b+b^2 x\right )^5} \, dx,x,x^2\right )}{10 a}\\ &=\frac {x^8}{10 a \left (a+b x^2\right )^5}+\frac {x^8}{40 a^2 \left (a+b x^2\right )^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.18 \begin {gather*} -\frac {a^3+5 a^2 b x^2+10 a b^2 x^4+10 b^3 x^6}{40 b^4 \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.69, size = 91, normalized size = 2.33 \begin {gather*} -\frac {10 \, b^{3} x^{6} + 10 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{40 \, {\left (b^{9} x^{10} + 5 \, a b^{8} x^{8} + 10 \, a^{2} b^{7} x^{6} + 10 \, a^{3} b^{6} x^{4} + 5 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 44, normalized size = 1.13 \begin {gather*} -\frac {10 \, b^{3} x^{6} + 10 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{40 \, {\left (b x^{2} + a\right )}^{5} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 1.67 \begin {gather*} \frac {a^{3}}{10 \left (b \,x^{2}+a \right )^{5} b^{4}}-\frac {3 a^{2}}{8 \left (b \,x^{2}+a \right )^{4} b^{4}}+\frac {a}{2 \left (b \,x^{2}+a \right )^{3} b^{4}}-\frac {1}{4 \left (b \,x^{2}+a \right )^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 91, normalized size = 2.33 \begin {gather*} -\frac {10 \, b^{3} x^{6} + 10 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{40 \, {\left (b^{9} x^{10} + 5 \, a b^{8} x^{8} + 10 \, a^{2} b^{7} x^{6} + 10 \, a^{3} b^{6} x^{4} + 5 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 93, normalized size = 2.38 \begin {gather*} -\frac {a^3+5\,a^2\,b\,x^2+10\,a\,b^2\,x^4+10\,b^3\,x^6}{40\,a^5\,b^4+200\,a^4\,b^5\,x^2+400\,a^3\,b^6\,x^4+400\,a^2\,b^7\,x^6+200\,a\,b^8\,x^8+40\,b^9\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.67, size = 95, normalized size = 2.44 \begin {gather*} \frac {- a^{3} - 5 a^{2} b x^{2} - 10 a b^{2} x^{4} - 10 b^{3} x^{6}}{40 a^{5} b^{4} + 200 a^{4} b^{5} x^{2} + 400 a^{3} b^{6} x^{4} + 400 a^{2} b^{7} x^{6} + 200 a b^{8} x^{8} + 40 b^{9} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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