Integrand size = 37, antiderivative size = 20 \[ \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx=\frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2641, 2624} \[ \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx=\frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \]
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Rule 2624
Rule 2641
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right )}{x^2} \, dx \\ & = \int \frac {\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x+b \log ^2\left (c x^n\right )\right )}{x} \, dx \\ & = \frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx=\frac {a^2 x^2}{2}+a b x \log ^2\left (c x^n\right )+\frac {1}{2} b^2 \log ^4\left (c x^n\right ) \]
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Time = 0.86 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75
method | result | size |
parallelrisch | \(\frac {b^{2} \ln \left (c \,x^{n}\right )^{4}}{2}+a x b \ln \left (c \,x^{n}\right )^{2}+\frac {x^{2} a^{2}}{2}\) | \(35\) |
default | \(\frac {x^{2} a^{2}}{2}+a b x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-2 a b n x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )+\frac {b^{2} \ln \left (c \,x^{n}\right )^{4}}{2}+2 \ln \left (c \,x^{n}\right ) a b n x\) | \(63\) |
parts | \(\frac {x^{2} a^{2}}{2}+a b x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-2 a b n x \ln \left (c \,{\mathrm e}^{n \ln \left (x \right )}\right )+\frac {b^{2} \ln \left (c \,x^{n}\right )^{4}}{2}+2 \ln \left (c \,x^{n}\right ) a b n x\) | \(63\) |
risch | \(\text {Expression too large to display}\) | \(2698\) |
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (18) = 36\).
Time = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 4.45 \[ \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx=\frac {1}{2} \, b^{2} n^{4} \log \left (x\right )^{4} + 2 \, b^{2} n^{3} \log \left (c\right ) \log \left (x\right )^{3} + a b x \log \left (c\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (3 \, b^{2} n^{2} \log \left (c\right )^{2} + a b n^{2} x\right )} \log \left (x\right )^{2} + 2 \, {\left (b^{2} n \log \left (c\right )^{3} + a b n x \log \left (c\right )\right )} \log \left (x\right ) \]
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Time = 2.52 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx=\frac {a^{2} x^{2}}{2} + a b x \log {\left (c x^{n} \right )}^{2} - 2 b^{2} n \left (\begin {cases} - \log {\left (c \right )}^{3} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{4}}{4 n} & \text {otherwise} \end {cases}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (18) = 36\).
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.70 \[ \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx=\frac {1}{2} \, b^{2} \log \left (c x^{n}\right )^{4} - 2 \, a b n^{2} x + 2 \, a b n x \log \left (c x^{n}\right ) + a b x \log \left (c x^{n}\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} a b \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (18) = 36\).
Time = 0.50 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.50 \[ \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx=\frac {1}{2} \, b^{2} n^{4} \log \left (x\right )^{4} + 2 \, b^{2} n^{3} \log \left (c\right ) \log \left (x\right )^{3} + 2 \, b^{2} n \log \left (c\right )^{3} \log \left (x\right ) + 2 \, a b n x \log \left (c\right ) \log \left (x\right ) + a b x \log \left (c\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (3 \, b^{2} n^{2} \log \left (c\right )^{2} + a b n^{2} x\right )} \log \left (x\right )^{2} \]
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Time = 1.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx=\frac {{\left (b\,{\ln \left (c\,x^n\right )}^2+a\,x\right )}^2}{2} \]
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