Integrand size = 22, antiderivative size = 162 \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {a^3 x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{\left (a+b x^3\right )^3}+\frac {3 a^2 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{4 \left (a+b x^3\right )^3}+\frac {3 a b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{7 \left (a+b x^3\right )^3}+\frac {b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{10 \left (a+b x^3\right )^3} \]
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Time = 0.02 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1357, 200} \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {3 a b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{7 \left (a+b x^3\right )^3}+\frac {3 a^2 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{4 \left (a+b x^3\right )^3}+\frac {b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{10 \left (a+b x^3\right )^3}+\frac {a^3 x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{\left (a+b x^3\right )^3} \]
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Rule 200
Rule 1357
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \int \left (2 a b+2 b^2 x^3\right )^3 \, dx}{\left (2 a b+2 b^2 x^3\right )^3} \\ & = \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \int \left (8 a^3 b^3+24 a^2 b^4 x^3+24 a b^5 x^6+8 b^6 x^9\right ) \, dx}{\left (2 a b+2 b^2 x^3\right )^3} \\ & = \frac {a^3 x \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{\left (a+b x^3\right )^3}+\frac {3 a^2 b x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{4 \left (a+b x^3\right )^3}+\frac {3 a b^2 x^7 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{7 \left (a+b x^3\right )^3}+\frac {b^3 x^{10} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{10 \left (a+b x^3\right )^3} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.36 \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {x \sqrt {\left (a+b x^3\right )^2} \left (140 a^3+105 a^2 b x^3+60 a b^2 x^6+14 b^3 x^9\right )}{140 \left (a+b x^3\right )} \]
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Time = 1.52 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(\frac {x \left (14 b^{3} x^{9}+60 b^{2} x^{6} a +105 a^{2} b \,x^{3}+140 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{140 \left (b \,x^{3}+a \right )^{3}}\) | \(56\) |
default | \(\frac {x \left (14 b^{3} x^{9}+60 b^{2} x^{6} a +105 a^{2} b \,x^{3}+140 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{140 \left (b \,x^{3}+a \right )^{3}}\) | \(56\) |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{3} x^{10}}{10 b \,x^{3}+10 a}+\frac {3 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a \,b^{2} x^{7}}{7 \left (b \,x^{3}+a \right )}+\frac {3 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{2} b \,x^{4}}{4 \left (b \,x^{3}+a \right )}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{3} x}{b \,x^{3}+a}\) | \(113\) |
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.20 \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{10} \, b^{3} x^{10} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{4} \, a^{2} b x^{4} + a^{3} x \]
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\[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\int \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.20 \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{10} \, b^{3} x^{10} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{4} \, a^{2} b x^{4} + a^{3} x \]
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Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.40 \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\frac {1}{10} \, b^{3} x^{10} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{7} \, a b^{2} x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {3}{4} \, a^{2} b x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{3} x \mathrm {sgn}\left (b x^{3} + a\right ) \]
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Timed out. \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx=\int {\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2} \,d x \]
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