Optimal. Leaf size=346 \[ -\frac {7 b f k n}{36 e x^{3/2}}+\frac {3 b f^2 k n}{8 e^2 x}-\frac {5 b f^3 k n}{4 e^3 \sqrt {x}}+\frac {b f^4 k n \log \left (e+f \sqrt {x}\right )}{4 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{4 x^2}-\frac {b f^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b f^4 k n \log (x)}{8 e^4}+\frac {b f^4 k n \log ^2(x)}{8 e^4}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {b f^4 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e^4} \]
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Rubi [A]
time = 0.18, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2442,
46, 2423, 2441, 2352, 2338} \begin {gather*} -\frac {b f^4 k n \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{e^4}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{2 x^2}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{4 x^2}+\frac {b f^4 k n \log ^2(x)}{8 e^4}+\frac {b f^4 k n \log \left (e+f \sqrt {x}\right )}{4 e^4}-\frac {b f^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b f^4 k n \log (x)}{8 e^4}-\frac {5 b f^3 k n}{4 e^3 \sqrt {x}}+\frac {3 b f^2 k n}{8 e^2 x}-\frac {7 b f k n}{36 e x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2338
Rule 2352
Rule 2423
Rule 2441
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-(b n) \int \left (-\frac {f k}{6 e x^{5/2}}+\frac {f^2 k}{4 e^2 x^2}-\frac {f^3 k}{2 e^3 x^{3/2}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right )}{2 e^4 x}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right )}{2 x^3}-\frac {f^4 k \log (x)}{4 e^4 x}\right ) \, dx\\ &=-\frac {b f k n}{9 e x^{3/2}}+\frac {b f^2 k n}{4 e^2 x}-\frac {b f^3 k n}{e^3 \sqrt {x}}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}+\frac {1}{2} (b n) \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right )}{x^3} \, dx+\frac {\left (b f^4 k n\right ) \int \frac {\log (x)}{x} \, dx}{4 e^4}-\frac {\left (b f^4 k n\right ) \int \frac {\log \left (e+f \sqrt {x}\right )}{x} \, dx}{2 e^4}\\ &=-\frac {b f k n}{9 e x^{3/2}}+\frac {b f^2 k n}{4 e^2 x}-\frac {b f^3 k n}{e^3 \sqrt {x}}+\frac {b f^4 k n \log ^2(x)}{8 e^4}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}+(b n) \text {Subst}\left (\int \frac {\log \left (d (e+f x)^k\right )}{x^5} \, dx,x,\sqrt {x}\right )-\frac {\left (b f^4 k n\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,\sqrt {x}\right )}{e^4}\\ &=-\frac {b f k n}{9 e x^{3/2}}+\frac {b f^2 k n}{4 e^2 x}-\frac {b f^3 k n}{e^3 \sqrt {x}}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{4 x^2}-\frac {b f^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {b f^4 k n \log ^2(x)}{8 e^4}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}+\frac {1}{4} (b f k n) \text {Subst}\left (\int \frac {1}{x^4 (e+f x)} \, dx,x,\sqrt {x}\right )+\frac {\left (b f^5 k n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,\sqrt {x}\right )}{e^4}\\ &=-\frac {b f k n}{9 e x^{3/2}}+\frac {b f^2 k n}{4 e^2 x}-\frac {b f^3 k n}{e^3 \sqrt {x}}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{4 x^2}-\frac {b f^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {b f^4 k n \log ^2(x)}{8 e^4}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {b f^4 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {1}{4} (b f k n) \text {Subst}\left (\int \left (\frac {1}{e x^4}-\frac {f}{e^2 x^3}+\frac {f^2}{e^3 x^2}-\frac {f^3}{e^4 x}+\frac {f^4}{e^4 (e+f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {7 b f k n}{36 e x^{3/2}}+\frac {3 b f^2 k n}{8 e^2 x}-\frac {5 b f^3 k n}{4 e^3 \sqrt {x}}+\frac {b f^4 k n \log \left (e+f \sqrt {x}\right )}{4 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{4 x^2}-\frac {b f^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b f^4 k n \log (x)}{8 e^4}+\frac {b f^4 k n \log ^2(x)}{8 e^4}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {f^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {b f^4 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{e^4}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 359, normalized size = 1.04 \begin {gather*} -\frac {12 a e^3 f k \sqrt {x}+14 b e^3 f k n \sqrt {x}-18 a e^2 f^2 k x-27 b e^2 f^2 k n x+36 a e f^3 k x^{3/2}+90 b e f^3 k n x^{3/2}+36 a e^4 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 b e^4 n \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 a f^4 k x^2 \log (x)+9 b f^4 k n x^2 \log (x)-36 b f^4 k n x^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-9 b f^4 k n x^2 \log ^2(x)+12 b e^3 f k \sqrt {x} \log \left (c x^n\right )-18 b e^2 f^2 k x \log \left (c x^n\right )+36 b e f^3 k x^{3/2} \log \left (c x^n\right )+36 b e^4 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+18 b f^4 k x^2 \log (x) \log \left (c x^n\right )-18 f^4 k x^2 \log \left (e+f \sqrt {x}\right ) \left (2 a+b n-2 b n \log (x)+2 b \log \left (c x^n\right )\right )-72 b f^4 k n x^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{72 e^4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )}{x^{3}}\, dx\]
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 (e+f\,\sqrt {x}\right )}^k\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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