Integrand size = 15, antiderivative size = 34 \[ \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx=\frac {a^2 x^4}{4}+\frac {6}{13} a b x^{13/3}+\frac {3}{14} b^2 x^{14/3} \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx=\frac {a^2 x^4}{4}+\frac {6}{13} a b x^{13/3}+\frac {3}{14} b^2 x^{14/3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^{11} (a+b x)^2 \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (a^2 x^{11}+2 a b x^{12}+b^2 x^{13}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {a^2 x^4}{4}+\frac {6}{13} a b x^{13/3}+\frac {3}{14} b^2 x^{14/3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx=\frac {1}{364} \left (91 a^2+168 a b \sqrt [3]{x}+78 b^2 x^{2/3}\right ) x^4 \]
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Time = 10.37 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {a^{2} x^{4}}{4}+\frac {6 a b \,x^{\frac {13}{3}}}{13}+\frac {3 b^{2} x^{\frac {14}{3}}}{14}\) | \(25\) |
default | \(\frac {a^{2} x^{4}}{4}+\frac {6 a b \,x^{\frac {13}{3}}}{13}+\frac {3 b^{2} x^{\frac {14}{3}}}{14}\) | \(25\) |
trager | \(\frac {a^{2} \left (x^{3}+x^{2}+x +1\right ) \left (-1+x \right )}{4}+\frac {6 a b \,x^{\frac {13}{3}}}{13}+\frac {3 b^{2} x^{\frac {14}{3}}}{14}\) | \(34\) |
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none
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx=\frac {3}{14} \, b^{2} x^{\frac {14}{3}} + \frac {6}{13} \, a b x^{\frac {13}{3}} + \frac {1}{4} \, a^{2} x^{4} \]
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Time = 0.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx=\frac {a^{2} x^{4}}{4} + \frac {6 a b x^{\frac {13}{3}}}{13} + \frac {3 b^{2} x^{\frac {14}{3}}}{14} \]
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (24) = 48\).
Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 5.88 \[ \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx=\frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{14}}{14 \, b^{12}} - \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13} a}{13 \, b^{12}} + \frac {55 \, {\left (b x^{\frac {1}{3}} + a\right )}^{12} a^{2}}{4 \, b^{12}} - \frac {45 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{3}}{b^{12}} + \frac {99 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10} a^{4}}{b^{12}} - \frac {154 \, {\left (b x^{\frac {1}{3}} + a\right )}^{9} a^{5}}{b^{12}} + \frac {693 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} a^{6}}{4 \, b^{12}} - \frac {990 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a^{7}}{7 \, b^{12}} + \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{8}}{2 \, b^{12}} - \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{9}}{b^{12}} + \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{10}}{4 \, b^{12}} - \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{11}}{b^{12}} \]
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Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx=\frac {3}{14} \, b^{2} x^{\frac {14}{3}} + \frac {6}{13} \, a b x^{\frac {13}{3}} + \frac {1}{4} \, a^{2} x^{4} \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int \left (a+b \sqrt [3]{x}\right )^2 x^3 \, dx=\frac {a^2\,x^4}{4}+\frac {3\,b^2\,x^{14/3}}{14}+\frac {6\,a\,b\,x^{13/3}}{13} \]
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