Optimal. Leaf size=389 \[ 4 b d^2 f^2 n \text {Li}_2\left (-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+d^2 f^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b d^2 f^2 n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}-\frac {2 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)+2 b^2 d^2 f^2 n^2 \log \left (d f \sqrt {x}+1\right )-b^2 d^2 f^2 n^2 \log (x)-\frac {14 b^2 d f n^2}{\sqrt {x}}-\frac {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{x} \]
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Rubi [A] time = 0.41, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2454, 2395, 44, 2377, 2304, 2376, 2391, 2301, 2374, 6589, 2366, 12, 2302, 30} \[ 4 b d^2 f^2 n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+4 b^2 d^2 f^2 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right )-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+d^2 f^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+2 b d^2 f^2 n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}-\frac {2 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)+2 b^2 d^2 f^2 n^2 \log \left (d f \sqrt {x}+1\right )-b^2 d^2 f^2 n^2 \log (x)-\frac {14 b^2 d f n^2}{\sqrt {x}}-\frac {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 44
Rule 2301
Rule 2302
Rule 2304
Rule 2366
Rule 2374
Rule 2376
Rule 2377
Rule 2391
Rule 2395
Rule 2454
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 x}\right ) \, dx\\ &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2+(2 b n) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx+(2 b d f n) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\left (b d^2 f^2 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (2 b d^2 f^2 n\right ) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=-\frac {8 b^2 d f n^2}{\sqrt {x}}-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-\left (b d^2 f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (2 b^2 n^2\right ) \int \left (-\frac {d f}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right )}{x}-\frac {d^2 f^2 \log (x)}{2 x}\right ) \, dx-\left (4 b^2 d^2 f^2 n^2\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {12 b^2 d f n^2}{\sqrt {x}}-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (d^2 f^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\left (2 b^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^2} \, dx+\left (b^2 d^2 f^2 n^2\right ) \int \frac {\log (x)}{x} \, dx-\left (2 b^2 d^2 f^2 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {12 b^2 d f n^2}{\sqrt {x}}+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {\left (d^2 f^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}+\left (4 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+d f x)}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {12 b^2 d f n^2}{\sqrt {x}}-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{x}+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\left (2 b^2 d f n^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {12 b^2 d f n^2}{\sqrt {x}}-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{x}+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )+\left (2 b^2 d f n^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {d f}{x}+\frac {d^2 f^2}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {14 b^2 d f n^2}{\sqrt {x}}+2 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{x}-b^2 d^2 f^2 n^2 \log (x)+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \text {Li}_2\left (-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \text {Li}_3\left (-d f \sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.41, size = 627, normalized size = 1.61 \[ -\frac {3 a^2 d^2 f^2 x \log (x)-6 a^2 d^2 f^2 x \log \left (d f \sqrt {x}+1\right )+6 a^2 d f \sqrt {x}+6 a^2 \log \left (d f \sqrt {x}+1\right )-24 b d^2 f^2 n x \text {Li}_2\left (-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )+b n\right )+6 a b d^2 f^2 x \log (x) \log \left (c x^n\right )-12 a b d^2 f^2 x \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+12 a b \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+12 a b d f \sqrt {x} \log \left (c x^n\right )-3 a b d^2 f^2 n x \log ^2(x)+6 a b d^2 f^2 n x \log (x)-12 a b d^2 f^2 n x \log \left (d f \sqrt {x}+1\right )+36 a b d f n \sqrt {x}+12 a b n \log \left (d f \sqrt {x}+1\right )-3 b^2 d^2 f^2 n x \log ^2(x) \log \left (c x^n\right )+3 b^2 d^2 f^2 x \log (x) \log ^2\left (c x^n\right )-6 b^2 d^2 f^2 x \log ^2\left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+6 b^2 d^2 f^2 n x \log (x) \log \left (c x^n\right )-12 b^2 d^2 f^2 n x \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+6 b^2 \log ^2\left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+6 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )+12 b^2 n \log \left (c x^n\right ) \log \left (d f \sqrt {x}+1\right )+36 b^2 d f n \sqrt {x} \log \left (c x^n\right )+48 b^2 d^2 f^2 n^2 x \text {Li}_3\left (-d f \sqrt {x}\right )+b^2 d^2 f^2 n^2 x \log ^3(x)-3 b^2 d^2 f^2 n^2 x \log ^2(x)+6 b^2 d^2 f^2 n^2 x \log (x)-12 b^2 d^2 f^2 n^2 x \log \left (d f \sqrt {x}+1\right )+84 b^2 d f n^2 \sqrt {x}+12 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{6 x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left (d f \sqrt {x} + 1\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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