Optimal. Leaf size=79 \[ \frac {b x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {a x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, 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 = {1355, 14} \begin {gather*} \frac {b x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac {a x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1355
Rubi steps
\begin {align*} \int x^4 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^4 \left (a b+b^2 x^3\right ) \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a b x^4+b^2 x^7\right ) \, dx}{a b+b^2 x^3}\\ &=\frac {a x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {b x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.49 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (8 a x^5+5 b x^8\right )}{40 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 6.45, size = 39, normalized size = 0.49 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (8 a x^5+5 b x^8\right )}{40 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 13, normalized size = 0.16 \begin {gather*} \frac {1}{8} \, b x^{8} + \frac {1}{5} \, a x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 29, normalized size = 0.37 \begin {gather*} \frac {1}{8} \, b x^{8} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{5} \, a x^{5} \mathrm {sgn}\left (b x^{3} + 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 = 0.46 \begin {gather*} \frac {\left (5 b \,x^{3}+8 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}\, x^{5}}{40 b \,x^{3}+40 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 13, normalized size = 0.16 \begin {gather*} \frac {1}{8} \, b x^{8} + \frac {1}{5} \, a x^{5} \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^4\,\sqrt {{\left (b\,x^3+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 12, normalized size = 0.15 \begin {gather*} \frac {a x^{5}}{5} + \frac {b x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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