Integrand size = 17, antiderivative size = 50 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, dx=a c^2 x+\frac {1}{4} c (b c+2 a d) x^4+\frac {1}{7} d (2 b c+a d) x^7+\frac {1}{10} b d^2 x^{10} \]
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Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {380} \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, dx=\frac {1}{7} d x^7 (a d+2 b c)+\frac {1}{4} c x^4 (2 a d+b c)+a c^2 x+\frac {1}{10} b d^2 x^{10} \]
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Rule 380
Rubi steps \begin{align*} \text {integral}& = \int \left (a c^2+c (b c+2 a d) x^3+d (2 b c+a d) x^6+b d^2 x^9\right ) \, dx \\ & = a c^2 x+\frac {1}{4} c (b c+2 a d) x^4+\frac {1}{7} d (2 b c+a d) x^7+\frac {1}{10} b d^2 x^{10} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, dx=a c^2 x+\frac {1}{4} c (b c+2 a d) x^4+\frac {1}{7} d (2 b c+a d) x^7+\frac {1}{10} b d^2 x^{10} \]
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Time = 3.87 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {b \,d^{2} x^{10}}{10}+\frac {\left (a \,d^{2}+2 b c d \right ) x^{7}}{7}+\frac {\left (2 a c d +b \,c^{2}\right ) x^{4}}{4}+a \,c^{2} x\) | \(49\) |
norman | \(\frac {b \,d^{2} x^{10}}{10}+\left (\frac {1}{7} a \,d^{2}+\frac {2}{7} b c d \right ) x^{7}+\left (\frac {1}{2} a c d +\frac {1}{4} b \,c^{2}\right ) x^{4}+a \,c^{2} x\) | \(49\) |
gosper | \(\frac {1}{10} b \,d^{2} x^{10}+\frac {1}{7} x^{7} a \,d^{2}+\frac {2}{7} x^{7} b c d +\frac {1}{2} x^{4} a c d +\frac {1}{4} x^{4} b \,c^{2}+a \,c^{2} x\) | \(51\) |
risch | \(\frac {1}{10} b \,d^{2} x^{10}+\frac {1}{7} x^{7} a \,d^{2}+\frac {2}{7} x^{7} b c d +\frac {1}{2} x^{4} a c d +\frac {1}{4} x^{4} b \,c^{2}+a \,c^{2} x\) | \(51\) |
parallelrisch | \(\frac {1}{10} b \,d^{2} x^{10}+\frac {1}{7} x^{7} a \,d^{2}+\frac {2}{7} x^{7} b c d +\frac {1}{2} x^{4} a c d +\frac {1}{4} x^{4} b \,c^{2}+a \,c^{2} x\) | \(51\) |
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, dx=\frac {1}{10} \, b d^{2} x^{10} + \frac {1}{7} \, {\left (2 \, b c d + a d^{2}\right )} x^{7} + \frac {1}{4} \, {\left (b c^{2} + 2 \, a c d\right )} x^{4} + a c^{2} x \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, dx=a c^{2} x + \frac {b d^{2} x^{10}}{10} + x^{7} \left (\frac {a d^{2}}{7} + \frac {2 b c d}{7}\right ) + x^{4} \left (\frac {a c d}{2} + \frac {b c^{2}}{4}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, dx=\frac {1}{10} \, b d^{2} x^{10} + \frac {1}{7} \, {\left (2 \, b c d + a d^{2}\right )} x^{7} + \frac {1}{4} \, {\left (b c^{2} + 2 \, a c d\right )} x^{4} + a c^{2} x \]
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Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, dx=\frac {1}{10} \, b d^{2} x^{10} + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, a d^{2} x^{7} + \frac {1}{4} \, b c^{2} x^{4} + \frac {1}{2} \, a c d x^{4} + a c^{2} x \]
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Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \left (a+b x^3\right ) \left (c+d x^3\right )^2 \, dx=x^4\,\left (\frac {b\,c^2}{4}+\frac {a\,d\,c}{2}\right )+x^7\,\left (\frac {a\,d^2}{7}+\frac {2\,b\,c\,d}{7}\right )+\frac {b\,d^2\,x^{10}}{10}+a\,c^2\,x \]
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