Optimal. Leaf size=177 \[ \frac {2 b^3 n \log \left (1-\frac {a}{a+b x}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}-\frac {2 b^3 n^2 \text {Li}_2\left (\frac {a}{a+b x}\right )}{3 a^3}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac {b^2 n^2}{3 a^2 x}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2} \]
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Rubi [A] time = 0.31, antiderivative size = 193, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac {2 b^3 n^2 \text {PolyLog}\left (2,\frac {b x}{a}+1\right )}{3 a^3}-\frac {b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}+\frac {2 b^3 n \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac {b^2 n^2}{3 a^2 x}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 2301
Rule 2314
Rule 2317
Rule 2319
Rule 2344
Rule 2347
Rule 2391
Rule 2398
Rule 2411
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c (a+b x)^n\right )}{x^4} \, dx &=-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {1}{3} (2 b n) \int \frac {\log \left (c (a+b x)^n\right )}{x^3 (a+b x)} \, dx\\ &=-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {1}{3} (2 n) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )\\ &=-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {(2 n) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right )}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )}{3 a}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{3 a}\\ &=-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right )}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{3 a^2}+\frac {\left (2 b^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x\right )}{3 a^2}+\frac {\left (b n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{3 a}\\ &=-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {\left (2 b^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right )}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{3 a^3}-\frac {\left (2 b^3 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right )}{x} \, dx,x,a+b x\right )}{3 a^3}+\frac {\left (b n^2\right ) \operatorname {Subst}\left (\int \left (\frac {b^2}{a (a-x)^2}+\frac {b^2}{a^2 (a-x)}+\frac {b^2}{a^2 x}\right ) \, dx,x,a+b x\right )}{3 a}-\frac {\left (2 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{3 a^3}\\ &=-\frac {b^2 n^2}{3 a^2 x}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}+\frac {2 b^3 n \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}-\frac {b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac {\left (2 b^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{a}\right )}{x} \, dx,x,a+b x\right )}{3 a^3}\\ &=-\frac {b^2 n^2}{3 a^2 x}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{3 a^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2}+\frac {2 b^2 n (a+b x) \log \left (c (a+b x)^n\right )}{3 a^3 x}+\frac {2 b^3 n \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}-\frac {b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}+\frac {2 b^3 n^2 \text {Li}_2\left (1+\frac {b x}{a}\right )}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 186, normalized size = 1.05 \[ -\frac {b^3 \log ^2\left (c (a+b x)^n\right )}{3 a^3}+\frac {2 b^3 n \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^n\right )}{3 a^3}+\frac {2 b^3 n^2 \text {Li}_2\left (\frac {a+b x}{a}\right )}{3 a^3}-\frac {b^3 n^2 \log (x)}{a^3}+\frac {b^3 n^2 \log (a+b x)}{a^3}+\frac {2 b^2 n \log \left (c (a+b x)^n\right )}{3 a^2 x}-\frac {b^2 n^2}{3 a^2 x}-\frac {\log ^2\left (c (a+b x)^n\right )}{3 x^3}-\frac {b n \log \left (c (a+b x)^n\right )}{3 a x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 1102, normalized size = 6.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 150, normalized size = 0.85 \[ -\frac {1}{3} \, b^{2} n^{2} {\left (\frac {2 \, {\left (\log \left (\frac {b x}{a} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x}{a}\right )\right )} b}{a^{3}} - \frac {3 \, b \log \left (b x + a\right )}{a^{3}} - \frac {b x \log \left (b x + a\right )^{2} - 3 \, b x \log \relax (x) - a}{a^{3} x}\right )} - \frac {1}{3} \, b n {\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \relax (x)}{a^{3}} - \frac {2 \, b x - a}{a^{2} x^{2}}\right )} \log \left ({\left (b x + a\right )}^{n} c\right ) - \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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