Optimal. Leaf size=346 \[ -\frac{b f^4 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e^4}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{2 x^2}+\frac{f^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac{f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac{f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{x}}+\frac{f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{4 x^2}-\frac{5 b f^3 k n}{4 e^3 \sqrt{x}}+\frac{3 b f^2 k n}{8 e^2 x}+\frac{b f^4 k n \log ^2(x)}{8 e^4}+\frac{b f^4 k n \log \left (e+f \sqrt{x}\right )}{4 e^4}-\frac{b f^4 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{b f^4 k n \log (x)}{8 e^4}-\frac{7 b f k n}{36 e x^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.279793, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2454, 2395, 44, 2376, 2394, 2315, 2301} \[ -\frac{b f^4 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e^4}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{2 x^2}+\frac{f^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac{f^4 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac{f^3 k \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{x}}+\frac{f^2 k \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{6 e x^{3/2}}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{4 x^2}-\frac{5 b f^3 k n}{4 e^3 \sqrt{x}}+\frac{3 b f^2 k n}{8 e^2 x}+\frac{b f^4 k n \log ^2(x)}{8 e^4}+\frac{b f^4 k n \log \left (e+f \sqrt{x}\right )}{4 e^4}-\frac{b f^4 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^4}-\frac{b f^4 k n \log (x)}{8 e^4}-\frac{7 b f k n}{36 e x^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2395
Rule 44
Rule 2376
Rule 2394
Rule 2315
Rule 2301
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) \operatorname{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 ) \operatorname{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) \operatorname{Subst}\left (\int \frac{1}{x^4 (e+f x)} \, dx,x,\sqrt{x}\right )+\frac{\left (b f^5 k n\right ) \operatorname{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) \operatorname{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*}
Mathematica [A] time = 0.409932, size = 359, normalized size = 1.04 \[ -\frac{-72 b f^4 k n x^2 \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )-18 f^4 k x^2 \log \left (e+f \sqrt{x}\right ) \left (2 a+2 b \log \left (c x^n\right )-2 b n \log (x)+b n\right )+36 a e^4 \log \left (d \left (e+f \sqrt{x}\right )^k\right )-18 a e^2 f^2 k x+12 a e^3 f k \sqrt{x}+36 a e f^3 k x^{3/2}+18 a f^4 k x^2 \log (x)+36 b e^4 \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-18 b e^2 f^2 k x \log \left (c x^n\right )+12 b e^3 f k \sqrt{x} \log \left (c x^n\right )+36 b e f^3 k x^{3/2} \log \left (c x^n\right )+18 b f^4 k x^2 \log (x) \log \left (c x^n\right )+18 b e^4 n \log \left (d \left (e+f \sqrt{x}\right )^k\right )-27 b e^2 f^2 k n x+14 b e^3 f k n \sqrt{x}+90 b e f^3 k n x^{3/2}-36 b f^4 k n x^2 \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )-9 b f^4 k n x^2 \log ^2(x)+9 b f^4 k n x^2 \log (x)}{72 e^4 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3}}\ln \left ( d \left ( e+f\sqrt{x} \right ) ^{k} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{18 \, b e \log \left (d\right ) \log \left (x^{n}\right ) + 18 \, a e \log \left (d\right ) + 9 \,{\left (e n \log \left (d\right ) + 2 \, e \log \left (c\right ) \log \left (d\right )\right )} b + 9 \,{\left (2 \, b e \log \left (x^{n}\right ) +{\left (e n + 2 \, e \log \left (c\right )\right )} b + 2 \, a e\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right ) + \frac{6 \, b f k x \log \left (x^{n}\right ) +{\left (6 \, a f k +{\left (7 \, f k n + 6 \, f k \log \left (c\right )\right )} b\right )} x}{\sqrt{x}}}{36 \, e x^{2}} - \int \frac{2 \, b f^{2} k \log \left (x^{n}\right ) + 2 \, a f^{2} k +{\left (f^{2} k n + 2 \, f^{2} k \log \left (c\right )\right )} b}{8 \,{\left (e f x^{\frac{5}{2}} + e^{2} x^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]