Optimal. Leaf size=167 \[ \frac {3 a b^2 x^{14} \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac {a^2 b x^{12} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {b^3 x^{16} \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac {a^3 x^{10} \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 167, 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} \begin {gather*} \frac {b^3 x^{16} \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 \left (a+b x^2\right )}+\frac {3 a b^2 x^{14} \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac {a^2 b x^{12} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {a^3 x^{10} \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^4 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int x^4 \left (a b+b^2 x\right )^3 \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \left (a^3 b^3 x^4+3 a^2 b^4 x^5+3 a b^5 x^6+b^6 x^7\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {a^3 x^{10} \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 \left (a+b x^2\right )}+\frac {a^2 b x^{12} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {3 a b^2 x^{14} \sqrt {a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac {b^3 x^{16} \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 \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^{10} \sqrt {\left (a+b x^2\right )^2} \left (56 a^3+140 a^2 b x^2+120 a b^2 x^4+35 b^3 x^6\right )}{560 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 12.31, size = 61, normalized size = 0.37 \begin {gather*} \frac {x^{10} \sqrt {\left (a+b x^2\right )^2} \left (56 a^3+140 a^2 b x^2+120 a b^2 x^4+35 b^3 x^6\right )}{560 \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}{16} \, b^{3} x^{16} + \frac {3}{14} \, a b^{2} x^{14} + \frac {1}{4} \, a^{2} b x^{12} + \frac {1}{10} \, a^{3} x^{10} \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}{16} \, b^{3} x^{16} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{14} \, a b^{2} x^{14} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{4} \, a^{2} b x^{12} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{10} \, a^{3} x^{10} \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 (35 b^{3} x^{6}+120 a \,b^{2} x^{4}+140 a^{2} b \,x^{2}+56 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} x^{10}}{560 \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 = 1.31, size = 35, normalized size = 0.21 \begin {gather*} \frac {1}{16} \, b^{3} x^{16} + \frac {3}{14} \, a b^{2} x^{14} + \frac {1}{4} \, a^{2} b x^{12} + \frac {1}{10} \, a^{3} x^{10} \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^9\,{\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^{9} \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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