Optimal. Leaf size=67 \[ \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{6 b^2}-\frac {a \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 b^2} \]
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1111, 640, 609} \begin {gather*} \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{6 b^2}-\frac {a \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 609
Rule 640
Rule 1111
Rubi steps
\begin {align*} \int x^3 \sqrt {a^2+2 a b x^2+b^2 x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \sqrt {a^2+2 a b x+b^2 x^2} \, dx,x,x^2\right )\\ &=\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{6 b^2}-\frac {a \operatorname {Subst}\left (\int \sqrt {a^2+2 a b x+b^2 x^2} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {a \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 b^2}+\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{6 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.58 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (3 a x^4+2 b x^6\right )}{12 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 6.19, size = 39, normalized size = 0.58 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (3 a x^4+2 b x^6\right )}{12 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 13, normalized size = 0.19 \begin {gather*} \frac {1}{6} \, b x^{6} + \frac {1}{4} \, a x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 23, normalized size = 0.34 \begin {gather*} \frac {1}{12} \, {\left (2 \, b x^{6} + 3 \, a x^{4}\right )} \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 = 0.00, size = 36, normalized size = 0.54 \begin {gather*} \frac {\left (2 b \,x^{2}+3 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x^{4}}{12 b \,x^{2}+12 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 13, normalized size = 0.19 \begin {gather*} \frac {1}{6} \, b x^{6} + \frac {1}{4} \, a x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 59, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (8\,b^2\,\left (a^2+b^2\,x^4\right )-12\,a^2\,b^2+4\,a\,b^3\,x^2\right )}{48\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 12, normalized size = 0.18 \begin {gather*} \frac {a x^{4}}{4} + \frac {b x^{6}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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