Optimal. Leaf size=34 \[ \frac {3}{13} b^2 x^{13/3}+\frac {3}{7} a b x^{14/3}+\frac {a^2 x^5}{5} \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {269, 272, 45}
\begin {gather*} \frac {a^2 x^5}{5}+\frac {3}{7} a b x^{14/3}+\frac {3}{13} b^2 x^{13/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^2 x^4 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^2 x^{10/3} \, dx\\ &=3 \text {Subst}\left (\int x^{12} (b+a x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (b^2 x^{12}+2 a b x^{13}+a^2 x^{14}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{13} b^2 x^{13/3}+\frac {3}{7} a b x^{14/3}+\frac {a^2 x^5}{5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 1.00 \begin {gather*} \frac {1}{455} \left (105 b^2+195 a b \sqrt [3]{x}+91 a^2 x^{2/3}\right ) x^{13/3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 25, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {3 b^{2} x^{\frac {13}{3}}}{13}+\frac {3 a b \,x^{\frac {14}{3}}}{7}+\frac {a^{2} x^{5}}{5}\) | \(25\) |
default | \(\frac {3 b^{2} x^{\frac {13}{3}}}{13}+\frac {3 a b \,x^{\frac {14}{3}}}{7}+\frac {a^{2} x^{5}}{5}\) | \(25\) |
trager | \(\frac {a^{2} \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (x -1\right )}{5}+\frac {3 b^{2} x^{\frac {13}{3}}}{13}+\frac {3 a b \,x^{\frac {14}{3}}}{7}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 26, normalized size = 0.76 \begin {gather*} \frac {1}{455} \, {\left (91 \, a^{2} + \frac {195 \, a b}{x^{\frac {1}{3}}} + \frac {105 \, b^{2}}{x^{\frac {2}{3}}}\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 24, normalized size = 0.71 \begin {gather*} \frac {1}{5} \, a^{2} x^{5} + \frac {3}{7} \, a b x^{\frac {14}{3}} + \frac {3}{13} \, b^{2} x^{\frac {13}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 31, normalized size = 0.91 \begin {gather*} \frac {a^{2} x^{5}}{5} + \frac {3 a b x^{\frac {14}{3}}}{7} + \frac {3 b^{2} x^{\frac {13}{3}}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 24, normalized size = 0.71 \begin {gather*} \frac {1}{5} \, a^{2} x^{5} + \frac {3}{7} \, a b x^{\frac {14}{3}} + \frac {3}{13} \, b^{2} x^{\frac {13}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 24, normalized size = 0.71 \begin {gather*} \frac {a^2\,x^5}{5}+\frac {3\,b^2\,x^{13/3}}{13}+\frac {3\,a\,b\,x^{14/3}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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