Optimal. Leaf size=106 \[ \frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 b^3}-\frac {a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{5 b^3}+\frac {a^2 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 b^3} \]
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Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.12, 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 = {1111, 645} \[ \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^3}-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^4}{5 b^3}+\frac {a^2 \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 645
Rule 1111
Rubi steps
\begin {align*} \int x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \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 \left (\frac {a^2 \left (a b+b^2 x\right )^3}{b^2}-\frac {2 a \left (a b+b^2 x\right )^4}{b^3}+\frac {\left (a b+b^2 x\right )^5}{b^4}\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {a^2 \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 b^3}-\frac {a \left (a+b x^2\right )^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 b^3}+\frac {\left (a+b x^2\right )^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{12 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.58 \[ \frac {x^6 \sqrt {\left (a+b x^2\right )^2} \left (20 a^3+45 a^2 b x^2+36 a b^2 x^4+10 b^3 x^6\right )}{120 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 35, normalized size = 0.33 \[ \frac {1}{12} \, b^{3} x^{12} + \frac {3}{10} \, a b^{2} x^{10} + \frac {3}{8} \, a^{2} b x^{8} + \frac {1}{6} \, a^{3} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 67, normalized size = 0.63 \[ \frac {1}{12} \, b^{3} x^{12} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{10} \, a b^{2} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{8} \, a^{2} b x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {1}{6} \, a^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.55 \[ \frac {\left (10 b^{3} x^{6}+36 a \,b^{2} x^{4}+45 a^{2} b \,x^{2}+20 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} x^{6}}{120 \left (b \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 35, normalized size = 0.33 \[ \frac {1}{12} \, b^{3} x^{12} + \frac {3}{10} \, a b^{2} x^{10} + \frac {3}{8} \, a^{2} b x^{8} + \frac {1}{6} \, a^{3} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \]
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 \[ \int x^{5} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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