Optimal. Leaf size=359 \[ \frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 e^{3/2} \sqrt{d+e x^2}}-\frac{d^{3/2} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{3/2} \sqrt{d+e x^2}}-\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 e^{3/2} \sqrt{d+e x^2}}+\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{b n x \sqrt{d+e x^2}}{4 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.410335, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {2341, 321, 215, 2350, 12, 14, 195, 5659, 3716, 2190, 2279, 2391} \[ \frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 e^{3/2} \sqrt{d+e x^2}}-\frac{d^{3/2} \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{3/2} \sqrt{d+e x^2}}-\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 e^{3/2} \sqrt{d+e x^2}}+\frac{b d^{3/2} n \sqrt{\frac{e x^2}{d}+1} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{b n x \sqrt{d+e x^2}}{4 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2341
Rule 321
Rule 215
Rule 2350
Rule 12
Rule 14
Rule 195
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d+e x^2}} \, dx &=\frac{\sqrt{1+\frac{e x^2}{d}} \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{\sqrt{d+e x^2}}\\ &=\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\frac{d x \sqrt{1+\frac{e x^2}{d}}}{e}-\frac{d^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}}{2 x} \, dx}{\sqrt{d+e x^2}}\\ &=\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\frac{d x \sqrt{1+\frac{e x^2}{d}}}{e}-\frac{d^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2}}}{x} \, dx}{2 \sqrt{d+e x^2}}\\ &=\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b n \sqrt{1+\frac{e x^2}{d}}\right ) \int \left (\frac{d \sqrt{1+\frac{e x^2}{d}}}{e}-\frac{d^{3/2} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{3/2} x}\right ) \, dx}{2 \sqrt{d+e x^2}}\\ &=\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{\left (b d^{3/2} n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b d n \sqrt{1+\frac{e x^2}{d}}\right ) \int \sqrt{1+\frac{e x^2}{d}} \, dx}{2 e \sqrt{d+e x^2}}\\ &=-\frac{b n x \sqrt{d+e x^2}}{4 e}+\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{\left (b d^{3/2} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b d n \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1+\frac{e x^2}{d}}} \, dx}{4 e \sqrt{d+e x^2}}\\ &=-\frac{b n x \sqrt{d+e x^2}}{4 e}-\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 e^{3/2} \sqrt{d+e x^2}}-\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{3/2} \sqrt{d+e x^2}}+\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b d^{3/2} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{e^{3/2} \sqrt{d+e x^2}}\\ &=-\frac{b n x \sqrt{d+e x^2}}{4 e}-\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 e^{3/2} \sqrt{d+e x^2}}-\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{3/2} \sqrt{d+e x^2}}+\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b d^{3/2} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}\\ &=-\frac{b n x \sqrt{d+e x^2}}{4 e}-\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 e^{3/2} \sqrt{d+e x^2}}-\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{3/2} \sqrt{d+e x^2}}+\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}-\frac{\left (b d^{3/2} n \sqrt{1+\frac{e x^2}{d}}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 e^{3/2} \sqrt{d+e x^2}}\\ &=-\frac{b n x \sqrt{d+e x^2}}{4 e}-\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{4 e^{3/2} \sqrt{d+e x^2}}-\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{4 e^{3/2} \sqrt{d+e x^2}}+\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac{d^{3/2} \sqrt{1+\frac{e x^2}{d}} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^{3/2} \sqrt{d+e x^2}}+\frac{b d^{3/2} n \sqrt{1+\frac{e x^2}{d}} \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}\right )}{4 e^{3/2} \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 0.784351, size = 205, normalized size = 0.57 \[ \frac{\frac{b n \sqrt{\frac{e x^2}{d}+1} \left (2 e^2 x^3 \, _3F_2\left (\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{5}{2},\frac{5}{2};-\frac{e x^2}{d}\right )+9 d \sqrt{e} (2 \log (x)-1) \left (\sqrt{e} x \sqrt{\frac{e x^2}{d}+1}-\sqrt{d} \sinh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )\right )}{\sqrt{d+e x^2}}+18 e x \sqrt{d+e x^2} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-18 d \sqrt{e} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{36 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.406, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{\sqrt{e x^{2} + d} b x^{2} \log \left (c x^{n}\right ) + \sqrt{e x^{2} + d} a x^{2}}{e x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (a + b \log{\left (c x^{n} \right )}\right )}{\sqrt{d + e x^{2}}}\, dx \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 )} x^{2}}{\sqrt{e x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]