Optimal. Leaf size=196 \[ -\frac {3 b d f n}{\sqrt {x}}+b d^2 f^2 n \log \left (1+d f \sqrt {x}\right )-\frac {b n \log \left (1+d f \sqrt {x}\right )}{x}-\frac {1}{2} b d^2 f^2 n \log (x)+\frac {1}{4} b d^2 f^2 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+2 b d^2 f^2 n \text {Li}_2\left (-d f \sqrt {x}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2504, 2442, 46,
2423, 2438, 2338} \begin {gather*} 2 b d^2 f^2 n \text {PolyLog}\left (2,-d f \sqrt {x}\right )+d^2 f^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} d^2 f^2 \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 )}{\sqrt {x}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {1}{4} b d^2 f^2 n \log ^2(x)+b d^2 f^2 n \log \left (d f \sqrt {x}+1\right )-\frac {1}{2} b d^2 f^2 n \log (x)-\frac {3 b d f n}{\sqrt {x}}-\frac {b n \log \left (d f \sqrt {x}+1\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2338
Rule 2423
Rule 2438
Rule 2442
Rule 2504
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 )}{x^2} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-(b n) \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\\ &=-\frac {2 b d f n}{\sqrt {x}}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+(b n) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^2} \, dx+\frac {1}{2} \left (b d^2 f^2 n\right ) \int \frac {\log (x)}{x} \, dx-\left (b d^2 f^2 n\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {2 b d f n}{\sqrt {x}}+\frac {1}{4} b d^2 f^2 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+2 b d^2 f^2 n \text {Li}_2\left (-d f \sqrt {x}\right )+(2 b n) \text {Subst}\left (\int \frac {\log (1+d f x)}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b d f n}{\sqrt {x}}-\frac {b n \log \left (1+d f \sqrt {x}\right )}{x}+\frac {1}{4} b d^2 f^2 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+2 b d^2 f^2 n \text {Li}_2\left (-d f \sqrt {x}\right )+(b d f n) \text {Subst}\left (\int \frac {1}{x^2 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 b d f n}{\sqrt {x}}-\frac {b n \log \left (1+d f \sqrt {x}\right )}{x}+\frac {1}{4} b d^2 f^2 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+2 b d^2 f^2 n \text {Li}_2\left (-d f \sqrt {x}\right )+(b d f n) \text {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 {3 b d f n}{\sqrt {x}}+b d^2 f^2 n \log \left (1+d f \sqrt {x}\right )-\frac {b n \log \left (1+d f \sqrt {x}\right )}{x}-\frac {1}{2} b d^2 f^2 n \log (x)+\frac {1}{4} b d^2 f^2 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+2 b d^2 f^2 n \text {Li}_2\left (-d f \sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 124, normalized size = 0.63 \begin {gather*} \frac {1}{4} b d^2 f^2 n \log ^2(x)+\frac {\left (-1+d^2 f^2 x\right ) \log \left (1+d f \sqrt {x}\right ) \left (a+b n+b \log \left (c x^n\right )\right )}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b n+b \log \left (c x^n\right )\right )-\frac {d f \left (a+3 b n+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \text {Li}_2\left (-d f \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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