Optimal. Leaf size=196 \[ 2 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+d^2 f^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{1}{4} b d^2 f^2 n \log ^2(x)+b d^2 f^2 n \log \left (d f \sqrt{x}+1\right )-\frac{1}{2} b d^2 f^2 n \log (x)-\frac{3 b d f n}{\sqrt{x}}-\frac{b n \log \left (d f \sqrt{x}+1\right )}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.151145, antiderivative size = 196, normalized size of antiderivative = 1., 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 = {2454, 2395, 44, 2376, 2391, 2301} \[ 2 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+d^2 f^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{1}{4} b d^2 f^2 n \log ^2(x)+b d^2 f^2 n \log \left (d f \sqrt{x}+1\right )-\frac{1}{2} b d^2 f^2 n \log (x)-\frac{3 b d f n}{\sqrt{x}}-\frac{b n \log \left (d f \sqrt{x}+1\right )}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2395
Rule 44
Rule 2376
Rule 2391
Rule 2301
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) \operatorname{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) \operatorname{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) \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{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*}
Mathematica [A] time = 0.191486, size = 124, normalized size = 0.63 \[ 2 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )+b n\right )+\frac{\left (d^2 f^2 x-1\right ) \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{x}-\frac{d f \left (a+b \log \left (c x^n\right )+3 b n\right )}{\sqrt{x}}+\frac{1}{4} b d^2 f^2 n \log ^2(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}}\ln \left ( d \left ({d}^{-1}+f\sqrt{x} \right ) \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*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{2}}\,{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 (d f \sqrt{x} + 1\right )}{x^{2}}, 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} + \frac{1}{d}\right )} d\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]