Optimal. Leaf size=34 \[ \frac {\left (a+b x^2\right )^5}{10 b^2}-\frac {a \left (a+b x^2\right )^4}{8 b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {\left (a+b x^2\right )^5}{10 b^2}-\frac {a \left (a+b x^2\right )^4}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^3 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {a \left (a+b x^2\right )^4}{8 b^2}+\frac {\left (a+b x^2\right )^5}{10 b^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 43, normalized size = 1.26 \begin {gather*} \frac {a^3 x^4}{4}+\frac {1}{2} a^2 b x^6+\frac {3}{8} a b^2 x^8+\frac {b^3 x^{10}}{10} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 \left (a+b x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.95, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{10} x^{10} b^{3} + \frac {3}{8} x^{8} b^{2} a + \frac {1}{2} x^{6} b a^{2} + \frac {1}{4} x^{4} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{10} \, b^{3} x^{10} + \frac {3}{8} \, a b^{2} x^{8} + \frac {1}{2} \, a^{2} b x^{6} + \frac {1}{4} \, a^{3} x^{4} \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 = 1.06 \begin {gather*} \frac {1}{10} b^{3} x^{10}+\frac {3}{8} a \,b^{2} x^{8}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{4} a^{3} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{10} \, b^{3} x^{10} + \frac {3}{8} \, a b^{2} x^{8} + \frac {1}{2} \, a^{2} b x^{6} + \frac {1}{4} \, a^{3} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 35, normalized size = 1.03 \begin {gather*} \frac {a^3\,x^4}{4}+\frac {a^2\,b\,x^6}{2}+\frac {3\,a\,b^2\,x^8}{8}+\frac {b^3\,x^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 37, normalized size = 1.09 \begin {gather*} \frac {a^{3} x^{4}}{4} + \frac {a^{2} b x^{6}}{2} + \frac {3 a b^{2} x^{8}}{8} + \frac {b^{3} x^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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