Optimal. Leaf size=119 \[ \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^3}-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{7 b^3}+\frac {a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^3} \]
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Rubi [A] time = 0.10, antiderivative size = 119, 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 {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^3}-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{7 b^3}+\frac {a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rule 1111
Rubi steps
\begin {align*} \int x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int x^2 \left (a b+b^2 x\right )^5 \, dx,x,x^2\right )}{2 b^4 \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 (\frac {a^2 \left (a b+b^2 x\right )^5}{b^2}-\frac {2 a \left (a b+b^2 x\right )^6}{b^3}+\frac {\left (a b+b^2 x\right )^7}{b^4}\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {a^2 \left (a+b x^2\right )^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 b^3}-\frac {a \left (a+b x^2\right )^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 b^3}+\frac {\left (a+b x^2\right )^7 \sqrt {a^2+2 a b x^2+b^2 x^4}}{16 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.70 \begin {gather*} \frac {x^6 \sqrt {\left (a+b x^2\right )^2} \left (56 a^5+210 a^4 b x^2+336 a^3 b^2 x^4+280 a^2 b^3 x^6+120 a b^4 x^8+21 b^5 x^{10}\right )}{336 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 13.03, size = 83, normalized size = 0.70 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (56 a^5 x^6+210 a^4 b x^8+336 a^3 b^2 x^{10}+280 a^2 b^3 x^{12}+120 a b^4 x^{14}+21 b^5 x^{16}\right )}{336 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 56, normalized size = 0.47 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} + \frac {5}{14} \, a b^{4} x^{14} + \frac {5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac {5}{8} \, a^{4} b x^{8} + \frac {1}{6} \, a^{5} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 104, normalized size = 0.87 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{14} \, a b^{4} x^{14} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{6} \, a^{2} b^{3} x^{12} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} b^{2} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{8} \, a^{4} b x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{6} \, a^{5} x^{6} \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 = 80, normalized size = 0.67 \begin {gather*} \frac {\left (21 b^{5} x^{10}+120 a \,b^{4} x^{8}+280 a^{2} b^{3} x^{6}+336 a^{3} b^{2} x^{4}+210 a^{4} b \,x^{2}+56 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x^{6}}{336 \left (b \,x^{2}+a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 56, normalized size = 0.47 \begin {gather*} \frac {1}{16} \, b^{5} x^{16} + \frac {5}{14} \, a b^{4} x^{14} + \frac {5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac {5}{8} \, a^{4} b x^{8} + \frac {1}{6} \, a^{5} x^{6} \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^5\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,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^{5} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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