Integrand size = 15, antiderivative size = 40 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx=a c x+\frac {(b c+a d) x^{1+n}}{1+n}+\frac {b d x^{1+2 n}}{1+2 n} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {380} \[ \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx=\frac {x^{n+1} (a d+b c)}{n+1}+a c x+\frac {b d x^{2 n+1}}{2 n+1} \]
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Rule 380
Rubi steps \begin{align*} \text {integral}& = \int \left (a c+(b c+a d) x^n+b d x^{2 n}\right ) \, dx \\ & = a c x+\frac {(b c+a d) x^{1+n}}{1+n}+\frac {b d x^{1+2 n}}{1+2 n} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx=x \left (a c+\frac {(b c+a d) x^n}{1+n}+\frac {b d x^{2 n}}{1+2 n}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98
method | result | size |
risch | \(a c x +\frac {\left (a d +b c \right ) x \,x^{n}}{1+n}+\frac {b d x \,x^{2 n}}{1+2 n}\) | \(39\) |
norman | \(a c x +\frac {\left (a d +b c \right ) x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}+\frac {b d x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}\) | \(43\) |
parallelrisch | \(\frac {x \,x^{2 n} b d n +b d x \,x^{2 n}+2 x \,x^{n} a d n +2 x \,x^{n} b c n +2 x a c \,n^{2}+x \,x^{n} a d +x \,x^{n} b c +3 x a c n +a c x}{\left (1+n \right ) \left (1+2 n \right )}\) | \(84\) |
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Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.72 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx=\frac {{\left (b d n + b d\right )} x x^{2 \, n} + {\left (b c + a d + 2 \, {\left (b c + a d\right )} n\right )} x x^{n} + {\left (2 \, a c n^{2} + 3 \, a c n + a c\right )} x}{2 \, n^{2} + 3 \, n + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (34) = 68\).
Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 5.90 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx=\begin {cases} a c x + a d \log {\left (x \right )} + b c \log {\left (x \right )} - \frac {b d}{x} & \text {for}\: n = -1 \\a c x + 2 a d \sqrt {x} + 2 b c \sqrt {x} + b d \log {\left (x \right )} & \text {for}\: n = - \frac {1}{2} \\\frac {2 a c n^{2} x}{2 n^{2} + 3 n + 1} + \frac {3 a c n x}{2 n^{2} + 3 n + 1} + \frac {a c x}{2 n^{2} + 3 n + 1} + \frac {2 a d n x x^{n}}{2 n^{2} + 3 n + 1} + \frac {a d x x^{n}}{2 n^{2} + 3 n + 1} + \frac {2 b c n x x^{n}}{2 n^{2} + 3 n + 1} + \frac {b c x x^{n}}{2 n^{2} + 3 n + 1} + \frac {b d n x x^{2 n}}{2 n^{2} + 3 n + 1} + \frac {b d x x^{2 n}}{2 n^{2} + 3 n + 1} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx=a c x + \frac {b d x^{2 \, n + 1}}{2 \, n + 1} + \frac {b c x^{n + 1}}{n + 1} + \frac {a d x^{n + 1}}{n + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (40) = 80\).
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.08 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx=\frac {2 \, a c n^{2} x + b d n x x^{2 \, n} + 2 \, b c n x x^{n} + 2 \, a d n x x^{n} + 3 \, a c n x + b d x x^{2 \, n} + b c x x^{n} + a d x x^{n} + a c x}{2 \, n^{2} + 3 \, n + 1} \]
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Time = 5.58 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \left (a+b x^n\right ) \left (c+d x^n\right ) \, dx=a\,c\,x+\frac {x\,x^n\,\left (a\,d+b\,c\right )}{n+1}+\frac {b\,d\,x\,x^{2\,n}}{2\,n+1} \]
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