Optimal. Leaf size=181 \[ -\frac {e r \left (a^2+2 a b n+2 b^2 n^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b e r (a+b n) \log \left (c x^n\right )}{x}-\frac {2 b e n r (a+b n)}{x}-\frac {b^2 e r \log ^2\left (c x^n\right )}{x}-\frac {2 b^2 e n r \log \left (c x^n\right )}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b^2 e n^2 r}{x} \]
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Rubi [A] time = 0.19, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2305, 2304, 2366, 14} \[ -\frac {e r \left (a^2+2 a b n+2 b^2 n^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b e r (a+b n) \log \left (c x^n\right )}{x}-\frac {2 b e n r (a+b n)}{x}-\frac {b^2 e r \log ^2\left (c x^n\right )}{x}-\frac {2 b^2 e n r \log \left (c x^n\right )}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b^2 e n^2 r}{x} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2304
Rule 2305
Rule 2366
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-(e r) \int \frac {-a^2 \left (1+\frac {2 b n (a+b n)}{a^2}\right )-2 b (a+b n) \log \left (c x^n\right )-b^2 \log ^2\left (c x^n\right )}{x^2} \, dx\\ &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-(e r) \int \left (\frac {-a^2-2 a b n-2 b^2 n^2}{x^2}-\frac {2 b (a+b n) \log \left (c x^n\right )}{x^2}-\frac {b^2 \log ^2\left (c x^n\right )}{x^2}\right ) \, dx\\ &=-\frac {e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}+\left (b^2 e r\right ) \int \frac {\log ^2\left (c x^n\right )}{x^2} \, dx+(2 b e (a+b n) r) \int \frac {\log \left (c x^n\right )}{x^2} \, dx\\ &=-\frac {2 b e n (a+b n) r}{x}-\frac {e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac {2 b e (a+b n) r \log \left (c x^n\right )}{x}-\frac {b^2 e r \log ^2\left (c x^n\right )}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}+\left (2 b^2 e n r\right ) \int \frac {\log \left (c x^n\right )}{x^2} \, dx\\ &=-\frac {2 b^2 e n^2 r}{x}-\frac {2 b e n (a+b n) r}{x}-\frac {e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac {2 b^2 e n r \log \left (c x^n\right )}{x}-\frac {2 b e (a+b n) r \log \left (c x^n\right )}{x}-\frac {b^2 e r \log ^2\left (c x^n\right )}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 138, normalized size = 0.76 \[ -\frac {e \left (a^2+2 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+a^2 d+a^2 e r+2 b \log \left (c x^n\right ) \left (e (a+b n) \log \left (f x^r\right )+a (d+e r)+b n (d+2 e r)\right )+2 a b d n+4 a b e n r+b^2 \log ^2\left (c x^n\right ) \left (d+e \log \left (f x^r\right )+e r\right )+2 b^2 d n^2+6 b^2 e n^2 r}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 311, normalized size = 1.72 \[ -\frac {b^{2} e n^{2} r \log \relax (x)^{3} + 2 \, b^{2} d n^{2} + 2 \, a b d n + a^{2} d + {\left (b^{2} e r + b^{2} d\right )} \log \relax (c)^{2} + {\left (2 \, b^{2} e n r \log \relax (c) + b^{2} e n^{2} \log \relax (f) + b^{2} d n^{2} + {\left (3 \, b^{2} e n^{2} + 2 \, a b e n\right )} r\right )} \log \relax (x)^{2} + {\left (6 \, b^{2} e n^{2} + 4 \, a b e n + a^{2} e\right )} r + 2 \, {\left (b^{2} d n + a b d + {\left (2 \, b^{2} e n + a b e\right )} r\right )} \log \relax (c) + {\left (2 \, b^{2} e n^{2} + b^{2} e \log \relax (c)^{2} + 2 \, a b e n + a^{2} e + 2 \, {\left (b^{2} e n + a b e\right )} \log \relax (c)\right )} \log \relax (f) + {\left (b^{2} e r \log \relax (c)^{2} + 2 \, b^{2} d n^{2} + 2 \, a b d n + {\left (6 \, b^{2} e n^{2} + 4 \, a b e n + a^{2} e\right )} r + 2 \, {\left (b^{2} d n + {\left (2 \, b^{2} e n + a b e\right )} r\right )} \log \relax (c) + 2 \, {\left (b^{2} e n^{2} + b^{2} e n \log \relax (c) + a b e n\right )} \log \relax (f)\right )} \log \relax (x)}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 392, normalized size = 2.17 \[ -\frac {b^{2} n^{2} r e \log \relax (x)^{3} + 3 \, b^{2} n^{2} r e \log \relax (x)^{2} + 2 \, b^{2} n r e \log \relax (c) \log \relax (x)^{2} + b^{2} n^{2} e \log \relax (f) \log \relax (x)^{2} + 6 \, b^{2} n^{2} r e \log \relax (x) + 4 \, b^{2} n r e \log \relax (c) \log \relax (x) + b^{2} r e \log \relax (c)^{2} \log \relax (x) + 2 \, b^{2} n^{2} e \log \relax (f) \log \relax (x) + 2 \, b^{2} n e \log \relax (c) \log \relax (f) \log \relax (x) + b^{2} d n^{2} \log \relax (x)^{2} + 2 \, a b n r e \log \relax (x)^{2} + 6 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \relax (c) + b^{2} r e \log \relax (c)^{2} + 2 \, b^{2} n^{2} e \log \relax (f) + 2 \, b^{2} n e \log \relax (c) \log \relax (f) + b^{2} e \log \relax (c)^{2} \log \relax (f) + 2 \, b^{2} d n^{2} \log \relax (x) + 4 \, a b n r e \log \relax (x) + 2 \, b^{2} d n \log \relax (c) \log \relax (x) + 2 \, a b r e \log \relax (c) \log \relax (x) + 2 \, a b n e \log \relax (f) \log \relax (x) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 2 \, b^{2} d n \log \relax (c) + 2 \, a b r e \log \relax (c) + b^{2} d \log \relax (c)^{2} + 2 \, a b n e \log \relax (f) + 2 \, a b e \log \relax (c) \log \relax (f) + 2 \, a b d n \log \relax (x) + a^{2} r e \log \relax (x) + 2 \, a b d n + a^{2} r e + 2 \, a b d \log \relax (c) + a^{2} e \log \relax (f) + a^{2} d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.85, size = 8407, normalized size = 46.45 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 221, normalized size = 1.22 \[ -b^{2} e {\left (\frac {r}{x} + \frac {\log \left (f x^{r}\right )}{x}\right )} \log \left (c x^{n}\right )^{2} - 2 \, a b e {\left (\frac {r}{x} + \frac {\log \left (f x^{r}\right )}{x}\right )} \log \left (c x^{n}\right ) - 2 \, {\left (\frac {{\left (r \log \relax (x) + 3 \, r + \log \relax (f)\right )} n^{2}}{x} + \frac {n {\left (2 \, r + \log \relax (f) + \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x}\right )} b^{2} e - 2 \, b^{2} d {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} - \frac {2 \, a b e n {\left (2 \, r + \log \relax (f) + \log \left (x^{r}\right )\right )}}{x} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{x} - \frac {2 \, a b d n}{x} - \frac {a^{2} e r}{x} - \frac {2 \, a b d \log \left (c x^{n}\right )}{x} - \frac {a^{2} e \log \left (f x^{r}\right )}{x} - \frac {a^{2} d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.01, size = 181, normalized size = 1.00 \[ -\ln \left (f\,x^r\right )\,\left (\ln \left (c\,x^n\right )\,\left (\frac {2\,a\,b\,e}{x}+\frac {2\,b^2\,e\,n}{x}\right )+\frac {a^2\,e}{x}+\frac {2\,b^2\,e\,n^2}{x}+\frac {b^2\,e\,{\ln \left (c\,x^n\right )}^2}{x}+\frac {2\,a\,b\,e\,n}{x}\right )-\frac {a^2\,d+2\,b^2\,d\,n^2+a^2\,e\,r+6\,b^2\,e\,n^2\,r+2\,a\,b\,d\,n+4\,a\,b\,e\,n\,r}{x}-\frac {2\,b\,\ln \left (c\,x^n\right )\,\left (a\,d+b\,d\,n+a\,e\,r+2\,b\,e\,n\,r\right )}{x}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2\,\left (d+e\,r\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 9.40, size = 536, normalized size = 2.96 \[ - \frac {a^{2} d}{x} - \frac {a^{2} e r \log {\relax (x )}}{x} - \frac {a^{2} e r}{x} - \frac {a^{2} e \log {\relax (f )}}{x} - \frac {2 a b d n \log {\relax (x )}}{x} - \frac {2 a b d n}{x} - \frac {2 a b d \log {\relax (c )}}{x} - \frac {2 a b e n r \log {\relax (x )}^{2}}{x} - \frac {4 a b e n r \log {\relax (x )}}{x} - \frac {4 a b e n r}{x} - \frac {2 a b e n \log {\relax (f )} \log {\relax (x )}}{x} - \frac {2 a b e n \log {\relax (f )}}{x} - \frac {2 a b e r \log {\relax (c )} \log {\relax (x )}}{x} - \frac {2 a b e r \log {\relax (c )}}{x} - \frac {2 a b e \log {\relax (c )} \log {\relax (f )}}{x} - \frac {b^{2} d n^{2} \log {\relax (x )}^{2}}{x} - \frac {2 b^{2} d n^{2} \log {\relax (x )}}{x} - \frac {2 b^{2} d n^{2}}{x} - \frac {2 b^{2} d n \log {\relax (c )} \log {\relax (x )}}{x} - \frac {2 b^{2} d n \log {\relax (c )}}{x} - \frac {b^{2} d \log {\relax (c )}^{2}}{x} - \frac {b^{2} e n^{2} r \log {\relax (x )}^{3}}{x} - \frac {3 b^{2} e n^{2} r \log {\relax (x )}^{2}}{x} - \frac {6 b^{2} e n^{2} r \log {\relax (x )}}{x} - \frac {6 b^{2} e n^{2} r}{x} - \frac {b^{2} e n^{2} \log {\relax (f )} \log {\relax (x )}^{2}}{x} - \frac {2 b^{2} e n^{2} \log {\relax (f )} \log {\relax (x )}}{x} - \frac {2 b^{2} e n^{2} \log {\relax (f )}}{x} - \frac {2 b^{2} e n r \log {\relax (c )} \log {\relax (x )}^{2}}{x} - \frac {4 b^{2} e n r \log {\relax (c )} \log {\relax (x )}}{x} - \frac {4 b^{2} e n r \log {\relax (c )}}{x} - \frac {2 b^{2} e n \log {\relax (c )} \log {\relax (f )} \log {\relax (x )}}{x} - \frac {2 b^{2} e n \log {\relax (c )} \log {\relax (f )}}{x} - \frac {b^{2} e r \log {\relax (c )}^{2} \log {\relax (x )}}{x} - \frac {b^{2} e r \log {\relax (c )}^{2}}{x} - \frac {b^{2} e \log {\relax (c )}^{2} \log {\relax (f )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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