Optimal. Leaf size=167 \[ \frac {3 a b^2 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {3 a^2 b x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {b^3 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{15 \left (a+b x^2\right )}+\frac {a^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1112, 270} \begin {gather*} \frac {b^3 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{15 \left (a+b x^2\right )}+\frac {3 a b^2 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {3 a^2 b x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {a^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rubi steps
\begin {align*} \int x^8 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int x^8 \left (a b+b^2 x^2\right )^3 \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 x^8+3 a^2 b^4 x^{10}+3 a b^5 x^{12}+b^6 x^{14}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {a^3 x^9 \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac {3 a^2 b x^{11} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac {3 a b^2 x^{13} \sqrt {a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac {b^3 x^{15} \sqrt {a^2+2 a b x^2+b^2 x^4}}{15 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.37 \begin {gather*} \frac {x^9 \sqrt {\left (a+b x^2\right )^2} \left (715 a^3+1755 a^2 b x^2+1485 a b^2 x^4+429 b^3 x^6\right )}{6435 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 7.30, size = 61, normalized size = 0.37 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (715 a^3 x^9+1755 a^2 b x^{11}+1485 a b^2 x^{13}+429 b^3 x^{15}\right )}{6435 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 35, normalized size = 0.21 \begin {gather*} \frac {1}{15} \, b^{3} x^{15} + \frac {3}{13} \, a b^{2} x^{13} + \frac {3}{11} \, a^{2} b x^{11} + \frac {1}{9} \, a^{3} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 67, normalized size = 0.40 \begin {gather*} \frac {1}{15} \, b^{3} x^{15} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{13} \, a b^{2} x^{13} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{11} \, a^{2} b x^{11} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{9} \, a^{3} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.35 \begin {gather*} \frac {\left (429 b^{3} x^{6}+1485 a \,b^{2} x^{4}+1755 a^{2} b \,x^{2}+715 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} x^{9}}{6435 \left (b \,x^{2}+a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 35, normalized size = 0.21 \begin {gather*} \frac {1}{15} \, b^{3} x^{15} + \frac {3}{13} \, a b^{2} x^{13} + \frac {3}{11} \, a^{2} b x^{11} + \frac {1}{9} \, a^{3} x^{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^8\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \end {gather*}
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 \begin {gather*} \int x^{8} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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