Optimal. Leaf size=147 \[ x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-e r x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \left (d+e \log \left (f x^r\right )\right )+2 a b e n r x+2 b e n r x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \left (d+e \log \left (f x^r\right )\right )+4 b^2 e n r x \log \left (c x^n\right )+2 b^2 n^2 x \left (d+e \log \left (f x^r\right )\right )-4 b^2 e n^2 r x \]
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Rubi [A] time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2296, 2295, 2361} \[ x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-e r x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \left (d+e \log \left (f x^r\right )\right )+2 a b e n r x+2 b e n r x (a-b n)-2 b^2 n x \log \left (c x^n\right ) \left (d+e \log \left (f x^r\right )\right )+4 b^2 e n r x \log \left (c x^n\right )+2 b^2 n^2 x \left (d+e \log \left (f x^r\right )\right )-4 b^2 e n^2 r x \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2296
Rule 2361
Rubi steps
\begin {align*} \int \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right ) \, dx &=-2 a b n x \left (d+e \log \left (f x^r\right )\right )+2 b^2 n^2 x \left (d+e \log \left (f x^r\right )\right )-2 b^2 n x \log \left (c x^n\right ) \left (d+e \log \left (f x^r\right )\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-(e r) \int \left (-2 a b n+2 b^2 n^2-2 b^2 n \log \left (c x^n\right )+\left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=2 b e n (a-b n) r x-2 a b n x \left (d+e \log \left (f x^r\right )\right )+2 b^2 n^2 x \left (d+e \log \left (f x^r\right )\right )-2 b^2 n x \log \left (c x^n\right ) \left (d+e \log \left (f x^r\right )\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )-(e r) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\left (2 b^2 e n r\right ) \int \log \left (c x^n\right ) \, dx\\ &=-2 b^2 e n^2 r x+2 b e n (a-b n) r x+2 b^2 e n r x \log \left (c x^n\right )-e r x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \left (d+e \log \left (f x^r\right )\right )+2 b^2 n^2 x \left (d+e \log \left (f x^r\right )\right )-2 b^2 n x \log \left (c x^n\right ) \left (d+e \log \left (f x^r\right )\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )+(2 b e n r) \int \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=2 a b e n r x-2 b^2 e n^2 r x+2 b e n (a-b n) r x+2 b^2 e n r x \log \left (c x^n\right )-e r x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \left (d+e \log \left (f x^r\right )\right )+2 b^2 n^2 x \left (d+e \log \left (f x^r\right )\right )-2 b^2 n x \log \left (c x^n\right ) \left (d+e \log \left (f x^r\right )\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )+\left (2 b^2 e n r\right ) \int \log \left (c x^n\right ) \, dx\\ &=2 a b e n r x-4 b^2 e n^2 r x+2 b e n (a-b n) r x+4 b^2 e n r x \log \left (c x^n\right )-e r x \left (a+b \log \left (c x^n\right )\right )^2-2 a b n x \left (d+e \log \left (f x^r\right )\right )+2 b^2 n^2 x \left (d+e \log \left (f x^r\right )\right )-2 b^2 n x \log \left (c x^n\right ) \left (d+e \log \left (f x^r\right )\right )+x \left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 141, normalized size = 0.96 \[ x \left (e \left (a^2-2 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+a^2 d-a^2 e r+2 b \log \left (c x^n\right ) \left (e (a-b n) \log \left (f x^r\right )+a d-a e r-b d n+2 b e n r\right )-2 a b d n+4 a b e n r+b^2 \log ^2\left (c x^n\right ) \left (d+e \log \left (f x^r\right )-e r\right )+2 b^2 d n^2-6 b^2 e n^2 r\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 345, normalized size = 2.35 \[ b^{2} e n^{2} r x \log \relax (x)^{3} - {\left (b^{2} e r - b^{2} d\right )} x \log \relax (c)^{2} - 2 \, {\left (b^{2} d n - a b d - {\left (2 \, b^{2} e n - a b e\right )} r\right )} x \log \relax (c) + {\left (2 \, b^{2} e n r x \log \relax (c) + b^{2} e n^{2} x \log \relax (f) + {\left (b^{2} d n^{2} - {\left (3 \, b^{2} e n^{2} - 2 \, a b e n\right )} r\right )} x\right )} \log \relax (x)^{2} + {\left (2 \, b^{2} d n^{2} - 2 \, a b d n + a^{2} d - {\left (6 \, b^{2} e n^{2} - 4 \, a b e n + a^{2} e\right )} r\right )} x + {\left (b^{2} e x \log \relax (c)^{2} - 2 \, {\left (b^{2} e n - a b e\right )} x \log \relax (c) + {\left (2 \, b^{2} e n^{2} - 2 \, a b e n + a^{2} e\right )} x\right )} \log \relax (f) + {\left (b^{2} e r x \log \relax (c)^{2} + 2 \, {\left (b^{2} d n - {\left (2 \, b^{2} e n - a b e\right )} r\right )} x \log \relax (c) - {\left (2 \, b^{2} d n^{2} - 2 \, a b d n - {\left (6 \, b^{2} e n^{2} - 4 \, a b e n + a^{2} e\right )} r\right )} x + 2 \, {\left (b^{2} e n x \log \relax (c) - {\left (b^{2} e n^{2} - a b e n\right )} x\right )} \log \relax (f)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 425, normalized size = 2.89 \[ b^{2} n^{2} r x e \log \relax (x)^{3} - 3 \, b^{2} n^{2} r x e \log \relax (x)^{2} + 2 \, b^{2} n r x e \log \relax (c) \log \relax (x)^{2} + b^{2} n^{2} x e \log \relax (f) \log \relax (x)^{2} + 6 \, b^{2} n^{2} r x e \log \relax (x) - 4 \, b^{2} n r x e \log \relax (c) \log \relax (x) + b^{2} r x e \log \relax (c)^{2} \log \relax (x) - 2 \, b^{2} n^{2} x e \log \relax (f) \log \relax (x) + 2 \, b^{2} n x e \log \relax (c) \log \relax (f) \log \relax (x) + b^{2} d n^{2} x \log \relax (x)^{2} + 2 \, a b n r x e \log \relax (x)^{2} - 6 \, b^{2} n^{2} r x e + 4 \, b^{2} n r x e \log \relax (c) - b^{2} r x e \log \relax (c)^{2} + 2 \, b^{2} n^{2} x e \log \relax (f) - 2 \, b^{2} n x e \log \relax (c) \log \relax (f) + b^{2} x e \log \relax (c)^{2} \log \relax (f) - 2 \, b^{2} d n^{2} x \log \relax (x) - 4 \, a b n r x e \log \relax (x) + 2 \, b^{2} d n x \log \relax (c) \log \relax (x) + 2 \, a b r x e \log \relax (c) \log \relax (x) + 2 \, a b n x e \log \relax (f) \log \relax (x) + 2 \, b^{2} d n^{2} x + 4 \, a b n r x e - 2 \, b^{2} d n x \log \relax (c) - 2 \, a b r x e \log \relax (c) + b^{2} d x \log \relax (c)^{2} - 2 \, a b n x e \log \relax (f) + 2 \, a b x e \log \relax (c) \log \relax (f) + 2 \, a b d n x \log \relax (x) + a^{2} r x e \log \relax (x) - 2 \, a b d n x - a^{2} r x e + 2 \, a b d x \log \relax (c) + a^{2} x e \log \relax (f) + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.83, size = 8701, normalized size = 59.19 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 213, normalized size = 1.45 \[ -{\left (r x - x \log \left (f x^{r}\right )\right )} b^{2} e \log \left (c x^{n}\right )^{2} + b^{2} d x \log \left (c x^{n}\right )^{2} + 2 \, {\left ({\left (2 \, r - \log \relax (f)\right )} x - x \log \left (x^{r}\right )\right )} a b e n - 2 \, a b d n x - a^{2} e r x - 2 \, {\left (r x - x \log \left (f x^{r}\right )\right )} a b e \log \left (c x^{n}\right ) + 2 \, a b d x \log \left (c x^{n}\right ) + a^{2} e x \log \left (f x^{r}\right ) + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} b^{2} d - 2 \, {\left ({\left ({\left (3 \, r - \log \relax (f)\right )} x - x \log \left (x^{r}\right )\right )} n^{2} - {\left ({\left (2 \, r - \log \relax (f)\right )} x - x \log \left (x^{r}\right )\right )} n \log \left (c x^{n}\right )\right )} b^{2} e + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.88, size = 165, normalized size = 1.12 \[ x\,\left (a^2\,d+2\,b^2\,d\,n^2-a^2\,e\,r-6\,b^2\,e\,n^2\,r-2\,a\,b\,d\,n+4\,a\,b\,e\,n\,r\right )+\ln \left (f\,x^r\right )\,\left (a^2\,e\,x-\ln \left (c\,x^n\right )\,\left (2\,b^2\,e\,n\,x-2\,a\,b\,e\,x\right )+2\,b^2\,e\,n^2\,x+b^2\,e\,x\,{\ln \left (c\,x^n\right )}^2-2\,a\,b\,e\,n\,x\right )+2\,b\,x\,\ln \left (c\,x^n\right )\,\left (a\,d-b\,d\,n-a\,e\,r+2\,b\,e\,n\,r\right )+b^2\,x\,{\ln \left (c\,x^n\right )}^2\,\left (d-e\,r\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 9.26, size = 534, normalized size = 3.63 \[ a^{2} d x + a^{2} e r x \log {\relax (x )} - a^{2} e r x + a^{2} e x \log {\relax (f )} + 2 a b d n x \log {\relax (x )} - 2 a b d n x + 2 a b d x \log {\relax (c )} + 2 a b e n r x \log {\relax (x )}^{2} - 4 a b e n r x \log {\relax (x )} + 4 a b e n r x + 2 a b e n x \log {\relax (f )} \log {\relax (x )} - 2 a b e n x \log {\relax (f )} + 2 a b e r x \log {\relax (c )} \log {\relax (x )} - 2 a b e r x \log {\relax (c )} + 2 a b e x \log {\relax (c )} \log {\relax (f )} + b^{2} d n^{2} x \log {\relax (x )}^{2} - 2 b^{2} d n^{2} x \log {\relax (x )} + 2 b^{2} d n^{2} x + 2 b^{2} d n x \log {\relax (c )} \log {\relax (x )} - 2 b^{2} d n x \log {\relax (c )} + b^{2} d x \log {\relax (c )}^{2} + b^{2} e n^{2} r x \log {\relax (x )}^{3} - 3 b^{2} e n^{2} r x \log {\relax (x )}^{2} + 6 b^{2} e n^{2} r x \log {\relax (x )} - 6 b^{2} e n^{2} r x + b^{2} e n^{2} x \log {\relax (f )} \log {\relax (x )}^{2} - 2 b^{2} e n^{2} x \log {\relax (f )} \log {\relax (x )} + 2 b^{2} e n^{2} x \log {\relax (f )} + 2 b^{2} e n r x \log {\relax (c )} \log {\relax (x )}^{2} - 4 b^{2} e n r x \log {\relax (c )} \log {\relax (x )} + 4 b^{2} e n r x \log {\relax (c )} + 2 b^{2} e n x \log {\relax (c )} \log {\relax (f )} \log {\relax (x )} - 2 b^{2} e n x \log {\relax (c )} \log {\relax (f )} + b^{2} e r x \log {\relax (c )}^{2} \log {\relax (x )} - b^{2} e r x \log {\relax (c )}^{2} + b^{2} e x \log {\relax (c )}^{2} \log {\relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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