Integrand size = 26, antiderivative size = 79 \[ \int x^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {a x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {b x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \]
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Time = 0.01 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 14} \[ \int x^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {b x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {a x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]
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Rule 14
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int x^3 \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^3+b^2 x^6\right ) \, dx}{a b+b^2 x^3} \\ & = \frac {a x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {b x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int x^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {\sqrt {\left (a+b x^3\right )^2} \left (7 a x^4+4 b x^7\right )}{28 \left (a+b x^3\right )} \]
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Time = 3.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(\frac {x^{4} \left (4 b \,x^{3}+7 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{28 b \,x^{3}+28 a}\) | \(36\) |
default | \(\frac {x^{4} \left (4 b \,x^{3}+7 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{28 b \,x^{3}+28 a}\) | \(36\) |
risch | \(\frac {a \,x^{4} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 b \,x^{3}+4 a}+\frac {b \,x^{7} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{7 b \,x^{3}+7 a}\) | \(54\) |
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.16 \[ \int x^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {1}{7} \, b x^{7} + \frac {1}{4} \, a x^{4} \]
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Timed out. \[ \int x^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\text {Timed out} \]
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none
Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.16 \[ \int x^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {1}{7} \, b x^{7} + \frac {1}{4} \, a x^{4} \]
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Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.37 \[ \int x^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {1}{7} \, b x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{4} \, a x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) \]
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Timed out. \[ \int x^3 \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\int x^3\,\sqrt {{\left (b\,x^3+a\right )}^2} \,d x \]
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