Optimal. Leaf size=34 \[ -\frac {a \left (a+b x^3\right )^4}{12 b^2}+\frac {\left (a+b x^3\right )^5}{15 b^2} \]
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Rubi [A]
time = 0.02, 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 = {272, 45}
\begin {gather*} \frac {\left (a+b x^3\right )^5}{15 b^2}-\frac {a \left (a+b x^3\right )^4}{12 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^5 \left (a+b x^3\right )^3 \, dx &=\frac {1}{3} \text {Subst}\left (\int x (a+b x)^3 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac {a \left (a+b x^3\right )^4}{12 b^2}+\frac {\left (a+b x^3\right )^5}{15 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^6}{6}+\frac {1}{3} a^2 b x^9+\frac {1}{4} a b^2 x^{12}+\frac {b^3 x^{15}}{15} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 36, normalized size = 1.06
method | result | size |
gosper | \(\frac {1}{6} a^{3} x^{6}+\frac {1}{3} a^{2} b \,x^{9}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{15} b^{3} x^{15}\) | \(36\) |
default | \(\frac {1}{6} a^{3} x^{6}+\frac {1}{3} a^{2} b \,x^{9}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{15} b^{3} x^{15}\) | \(36\) |
norman | \(\frac {1}{6} a^{3} x^{6}+\frac {1}{3} a^{2} b \,x^{9}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{15} b^{3} x^{15}\) | \(36\) |
risch | \(\frac {1}{6} a^{3} x^{6}+\frac {1}{3} a^{2} b \,x^{9}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{15} b^{3} x^{15}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{15} \, b^{3} x^{15} + \frac {1}{4} \, a b^{2} x^{12} + \frac {1}{3} \, a^{2} b x^{9} + \frac {1}{6} \, a^{3} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{15} \, b^{3} x^{15} + \frac {1}{4} \, a b^{2} x^{12} + \frac {1}{3} \, a^{2} b x^{9} + \frac {1}{6} \, a^{3} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 36, normalized size = 1.06 \begin {gather*} \frac {a^{3} x^{6}}{6} + \frac {a^{2} b x^{9}}{3} + \frac {a b^{2} x^{12}}{4} + \frac {b^{3} x^{15}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{15} \, b^{3} x^{15} + \frac {1}{4} \, a b^{2} x^{12} + \frac {1}{3} \, a^{2} b x^{9} + \frac {1}{6} \, a^{3} x^{6} \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^6}{6}+\frac {a^2\,b\,x^9}{3}+\frac {a\,b^2\,x^{12}}{4}+\frac {b^3\,x^{15}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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