Optimal. Leaf size=289 \[ -\frac {7 b d f n}{36 x^{3/2}}+\frac {3 b d^2 f^2 n}{8 x}-\frac {5 b d^3 f^3 n}{4 \sqrt {x}}+\frac {1}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right )-\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b d^4 f^4 n \log (x)+\frac {1}{8} b d^4 f^4 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+b d^4 f^4 n \text {Li}_2\left (-d f \sqrt {x}\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 289, 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*} b d^4 f^4 n \text {PolyLog}\left (2,-d f \sqrt {x}\right )+\frac {1}{2} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac {1}{8} b d^4 f^4 n \log ^2(x)+\frac {1}{4} b d^4 f^4 n \log \left (d f \sqrt {x}+1\right )-\frac {1}{8} b d^4 f^4 n \log (x)-\frac {5 b d^3 f^3 n}{4 \sqrt {x}}+\frac {3 b d^2 f^2 n}{8 x}-\frac {7 b d f n}{36 x^{3/2}}-\frac {b n \log \left (d f \sqrt {x}+1\right )}{4 x^2} \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^3} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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 )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d f}{6 x^{5/2}}+\frac {d^2 f^2}{4 x^2}-\frac {d^3 f^3}{2 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{2 x^3}+\frac {d^4 f^4 \log \left (1+d f \sqrt {x}\right )}{2 x}-\frac {d^4 f^4 \log (x)}{4 x}\right ) \, dx\\ &=-\frac {b d f n}{9 x^{3/2}}+\frac {b d^2 f^2 n}{4 x}-\frac {b d^3 f^3 n}{\sqrt {x}}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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 )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} (b n) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^3} \, dx+\frac {1}{4} \left (b d^4 f^4 n\right ) \int \frac {\log (x)}{x} \, dx-\frac {1}{2} \left (b d^4 f^4 n\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {b d f n}{9 x^{3/2}}+\frac {b d^2 f^2 n}{4 x}-\frac {b d^3 f^3 n}{\sqrt {x}}+\frac {1}{8} b d^4 f^4 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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 )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+b d^4 f^4 n \text {Li}_2\left (-d f \sqrt {x}\right )+(b n) \text {Subst}\left (\int \frac {\log (1+d f x)}{x^5} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b d f n}{9 x^{3/2}}+\frac {b d^2 f^2 n}{4 x}-\frac {b d^3 f^3 n}{\sqrt {x}}-\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 x^2}+\frac {1}{8} b d^4 f^4 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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 )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+b d^4 f^4 n \text {Li}_2\left (-d f \sqrt {x}\right )+\frac {1}{4} (b d f n) \text {Subst}\left (\int \frac {1}{x^4 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b d f n}{9 x^{3/2}}+\frac {b d^2 f^2 n}{4 x}-\frac {b d^3 f^3 n}{\sqrt {x}}-\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 x^2}+\frac {1}{8} b d^4 f^4 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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 )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+b d^4 f^4 n \text {Li}_2\left (-d f \sqrt {x}\right )+\frac {1}{4} (b d f n) \text {Subst}\left (\int \left (\frac {1}{x^4}-\frac {d f}{x^3}+\frac {d^2 f^2}{x^2}-\frac {d^3 f^3}{x}+\frac {d^4 f^4}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {7 b d f n}{36 x^{3/2}}+\frac {3 b d^2 f^2 n}{8 x}-\frac {5 b d^3 f^3 n}{4 \sqrt {x}}+\frac {1}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right )-\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b d^4 f^4 n \log (x)+\frac {1}{8} b d^4 f^4 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \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 )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+b d^4 f^4 n \text {Li}_2\left (-d f \sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 207, normalized size = 0.72 \begin {gather*} \frac {\left (-1+d^4 f^4 x^2\right ) \log \left (1+d f \sqrt {x}\right ) \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{4 x^2}-\frac {d f \left (12 a+14 b n-18 a d f \sqrt {x}-27 b d f n \sqrt {x}+36 a d^2 f^2 x+90 b d^2 f^2 n x-9 b d^3 f^3 n x^{3/2} \log ^2(x)+6 b \left (2-3 d f \sqrt {x}+6 d^2 f^2 x\right ) \log \left (c x^n\right )+9 d^3 f^3 x^{3/2} \log (x) \left (2 a+b n+2 b \log \left (c x^n\right )\right )\right )}{72 x^{3/2}}+b d^4 f^4 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 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{3}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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