Optimal. Leaf size=204 \[ -\frac {e r \left (2 a^2+2 a b n+b^2 n^2\right )}{8 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {b e r (2 a+b n) \log \left (c x^n\right )}{4 x^2}-\frac {b e n r (2 a+b n)}{8 x^2}-\frac {b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac {b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b^2 e n^2 r}{8 x^2} \]
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Rubi [A] time = 0.21, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2305, 2304, 2366, 12, 14} \[ -\frac {e r \left (2 a^2+2 a b n+b^2 n^2\right )}{8 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {b e r (2 a+b n) \log \left (c x^n\right )}{4 x^2}-\frac {b e n r (2 a+b n)}{8 x^2}-\frac {b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac {b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b^2 e n^2 r}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 12
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^3} \, dx &=-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-(e r) \int \frac {-2 a^2 \left (1+\frac {b n (2 a+b n)}{2 a^2}\right )-2 b (2 a+b n) \log \left (c x^n\right )-2 b^2 \log ^2\left (c x^n\right )}{4 x^3} \, dx\\ &=-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {1}{4} (e r) \int \frac {-2 a^2 \left (1+\frac {b n (2 a+b n)}{2 a^2}\right )-2 b (2 a+b n) \log \left (c x^n\right )-2 b^2 \log ^2\left (c x^n\right )}{x^3} \, dx\\ &=-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {1}{4} (e r) \int \left (\frac {-2 a^2-2 a b n-b^2 n^2}{x^3}-\frac {2 b (2 a+b n) \log \left (c x^n\right )}{x^3}-\frac {2 b^2 \log ^2\left (c x^n\right )}{x^3}\right ) \, dx\\ &=-\frac {e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}+\frac {1}{2} \left (b^2 e r\right ) \int \frac {\log ^2\left (c x^n\right )}{x^3} \, dx+\frac {1}{2} (b e (2 a+b n) r) \int \frac {\log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac {b e n (2 a+b n) r}{8 x^2}-\frac {e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac {b e (2 a+b n) r \log \left (c x^n\right )}{4 x^2}-\frac {b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}+\frac {1}{2} \left (b^2 e n r\right ) \int \frac {\log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac {b^2 e n^2 r}{8 x^2}-\frac {b e n (2 a+b n) r}{8 x^2}-\frac {e \left (2 a^2+2 a b n+b^2 n^2\right ) r}{8 x^2}-\frac {b^2 e n r \log \left (c x^n\right )}{4 x^2}-\frac {b e (2 a+b n) r \log \left (c x^n\right )}{4 x^2}-\frac {b^2 e r \log ^2\left (c x^n\right )}{4 x^2}-\frac {b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 151, normalized size = 0.74 \[ -\frac {2 e \left (2 a^2+2 a b n+b^2 n^2\right ) \log \left (f x^r\right )+4 a^2 d+2 a^2 e r+4 b \log \left (c x^n\right ) \left (e (2 a+b n) \log \left (f x^r\right )+2 a d+a e r+b d n+b e n r\right )+4 a b d n+4 a b e n r+2 b^2 \log ^2\left (c x^n\right ) \left (2 d+2 e \log \left (f x^r\right )+e r\right )+2 b^2 d n^2+3 b^2 e n^2 r}{8 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 326, normalized size = 1.60 \[ -\frac {4 \, b^{2} e n^{2} r \log \relax (x)^{3} + 2 \, b^{2} d n^{2} + 4 \, a b d n + 4 \, a^{2} d + 2 \, {\left (b^{2} e r + 2 \, b^{2} d\right )} \log \relax (c)^{2} + 2 \, {\left (4 \, b^{2} e n r \log \relax (c) + 2 \, b^{2} e n^{2} \log \relax (f) + 2 \, b^{2} d n^{2} + {\left (3 \, b^{2} e n^{2} + 4 \, a b e n\right )} r\right )} \log \relax (x)^{2} + {\left (3 \, b^{2} e n^{2} + 4 \, a b e n + 2 \, a^{2} e\right )} r + 4 \, {\left (b^{2} d n + 2 \, a b d + {\left (b^{2} e n + a b e\right )} r\right )} \log \relax (c) + 2 \, {\left (b^{2} e n^{2} + 2 \, b^{2} e \log \relax (c)^{2} + 2 \, a b e n + 2 \, a^{2} e + 2 \, {\left (b^{2} e n + 2 \, a b e\right )} \log \relax (c)\right )} \log \relax (f) + 2 \, {\left (2 \, b^{2} e r \log \relax (c)^{2} + 2 \, b^{2} d n^{2} + 4 \, a b d n + {\left (3 \, b^{2} e n^{2} + 4 \, a b e n + 2 \, a^{2} e\right )} r + 4 \, {\left (b^{2} d n + {\left (b^{2} e n + a b e\right )} r\right )} \log \relax (c) + 2 \, {\left (b^{2} e n^{2} + 2 \, b^{2} e n \log \relax (c) + 2 \, a b e n\right )} \log \relax (f)\right )} \log \relax (x)}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 403, normalized size = 1.98 \[ -\frac {4 \, b^{2} n^{2} r e \log \relax (x)^{3} + 6 \, b^{2} n^{2} r e \log \relax (x)^{2} + 8 \, b^{2} n r e \log \relax (c) \log \relax (x)^{2} + 4 \, b^{2} n^{2} e \log \relax (f) \log \relax (x)^{2} + 6 \, b^{2} n^{2} r e \log \relax (x) + 8 \, b^{2} n r e \log \relax (c) \log \relax (x) + 4 \, b^{2} r e \log \relax (c)^{2} \log \relax (x) + 4 \, b^{2} n^{2} e \log \relax (f) \log \relax (x) + 8 \, b^{2} n e \log \relax (c) \log \relax (f) \log \relax (x) + 4 \, b^{2} d n^{2} \log \relax (x)^{2} + 8 \, a b n r e \log \relax (x)^{2} + 3 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \relax (c) + 2 \, b^{2} r e \log \relax (c)^{2} + 2 \, b^{2} n^{2} e \log \relax (f) + 4 \, b^{2} n e \log \relax (c) \log \relax (f) + 4 \, b^{2} e \log \relax (c)^{2} \log \relax (f) + 4 \, b^{2} d n^{2} \log \relax (x) + 8 \, a b n r e \log \relax (x) + 8 \, b^{2} d n \log \relax (c) \log \relax (x) + 8 \, a b r e \log \relax (c) \log \relax (x) + 8 \, a b n e \log \relax (f) \log \relax (x) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 4 \, b^{2} d n \log \relax (c) + 4 \, a b r e \log \relax (c) + 4 \, b^{2} d \log \relax (c)^{2} + 4 \, a b n e \log \relax (f) + 8 \, a b e \log \relax (c) \log \relax (f) + 8 \, a b d n \log \relax (x) + 4 \, a^{2} r e \log \relax (x) + 4 \, a b d n + 2 \, a^{2} r e + 8 \, a b d \log \relax (c) + 4 \, a^{2} e \log \relax (f) + 4 \, a^{2} d}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.84, size = 8407, normalized size = 41.21 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 224, normalized size = 1.10 \[ -\frac {1}{4} \, b^{2} e {\left (\frac {r}{x^{2}} + \frac {2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} \log \left (c x^{n}\right )^{2} - \frac {1}{2} \, a b e {\left (\frac {r}{x^{2}} + \frac {2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} \log \left (c x^{n}\right ) - \frac {1}{8} \, b^{2} e {\left (\frac {{\left (2 \, r \log \relax (x) + 3 \, r + 2 \, \log \relax (f)\right )} n^{2}}{x^{2}} + \frac {4 \, n {\left (r + \log \relax (f) + \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {1}{4} \, b^{2} d {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {a b e n {\left (r + \log \relax (f) + \log \left (x^{r}\right )\right )}}{2 \, x^{2}} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b d n}{2 \, x^{2}} - \frac {a^{2} e r}{4 \, x^{2}} - \frac {a b d \log \left (c x^{n}\right )}{x^{2}} - \frac {a^{2} e \log \left (f x^{r}\right )}{2 \, x^{2}} - \frac {a^{2} d}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.08, size = 186, normalized size = 0.91 \[ -\ln \left (f\,x^r\right )\,\left (\ln \left (c\,x^n\right )\,\left (\frac {a\,b\,e}{x^2}+\frac {b^2\,e\,n}{2\,x^2}\right )+\frac {a^2\,e}{2\,x^2}+\frac {b^2\,e\,n^2}{4\,x^2}+\frac {b^2\,e\,{\ln \left (c\,x^n\right )}^2}{2\,x^2}+\frac {a\,b\,e\,n}{2\,x^2}\right )-\frac {\frac {a^2\,d}{2}+\frac {b^2\,d\,n^2}{4}+\frac {a^2\,e\,r}{4}+\frac {3\,b^2\,e\,n^2\,r}{8}+\frac {a\,b\,d\,n}{2}+\frac {a\,b\,e\,n\,r}{2}}{x^2}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2\,\left (2\,d+e\,r\right )}{4\,x^2}-\frac {b\,\ln \left (c\,x^n\right )\,\left (2\,a\,d+b\,d\,n+a\,e\,r+b\,e\,n\,r\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 9.01, size = 602, normalized size = 2.95 \[ - \frac {a^{2} d}{2 x^{2}} - \frac {a^{2} e r \log {\relax (x )}}{2 x^{2}} - \frac {a^{2} e r}{4 x^{2}} - \frac {a^{2} e \log {\relax (f )}}{2 x^{2}} - \frac {a b d n \log {\relax (x )}}{x^{2}} - \frac {a b d n}{2 x^{2}} - \frac {a b d \log {\relax (c )}}{x^{2}} - \frac {a b e n r \log {\relax (x )}^{2}}{x^{2}} - \frac {a b e n r \log {\relax (x )}}{x^{2}} - \frac {a b e n r}{2 x^{2}} - \frac {a b e n \log {\relax (f )} \log {\relax (x )}}{x^{2}} - \frac {a b e n \log {\relax (f )}}{2 x^{2}} - \frac {a b e r \log {\relax (c )} \log {\relax (x )}}{x^{2}} - \frac {a b e r \log {\relax (c )}}{2 x^{2}} - \frac {a b e \log {\relax (c )} \log {\relax (f )}}{x^{2}} - \frac {b^{2} d n^{2} \log {\relax (x )}^{2}}{2 x^{2}} - \frac {b^{2} d n^{2} \log {\relax (x )}}{2 x^{2}} - \frac {b^{2} d n^{2}}{4 x^{2}} - \frac {b^{2} d n \log {\relax (c )} \log {\relax (x )}}{x^{2}} - \frac {b^{2} d n \log {\relax (c )}}{2 x^{2}} - \frac {b^{2} d \log {\relax (c )}^{2}}{2 x^{2}} - \frac {b^{2} e n^{2} r \log {\relax (x )}^{3}}{2 x^{2}} - \frac {3 b^{2} e n^{2} r \log {\relax (x )}^{2}}{4 x^{2}} - \frac {3 b^{2} e n^{2} r \log {\relax (x )}}{4 x^{2}} - \frac {3 b^{2} e n^{2} r}{8 x^{2}} - \frac {b^{2} e n^{2} \log {\relax (f )} \log {\relax (x )}^{2}}{2 x^{2}} - \frac {b^{2} e n^{2} \log {\relax (f )} \log {\relax (x )}}{2 x^{2}} - \frac {b^{2} e n^{2} \log {\relax (f )}}{4 x^{2}} - \frac {b^{2} e n r \log {\relax (c )} \log {\relax (x )}^{2}}{x^{2}} - \frac {b^{2} e n r \log {\relax (c )} \log {\relax (x )}}{x^{2}} - \frac {b^{2} e n r \log {\relax (c )}}{2 x^{2}} - \frac {b^{2} e n \log {\relax (c )} \log {\relax (f )} \log {\relax (x )}}{x^{2}} - \frac {b^{2} e n \log {\relax (c )} \log {\relax (f )}}{2 x^{2}} - \frac {b^{2} e r \log {\relax (c )}^{2} \log {\relax (x )}}{2 x^{2}} - \frac {b^{2} e r \log {\relax (c )}^{2}}{4 x^{2}} - \frac {b^{2} e \log {\relax (c )}^{2} \log {\relax (f )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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