Optimal. Leaf size=228 \[ \frac{1}{2} i b d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^3 \text{PolyLog}(2,i c x)+\frac{3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac{3 b d^2 e \tan ^{-1}(c x)}{2 c^2}+\frac{3 b d e^2 x}{4 c^3}-\frac{3 b d e^2 \tan ^{-1}(c x)}{4 c^4}+\frac{b e^3 x^3}{18 c^3}-\frac{b e^3 x}{6 c^5}+\frac{b e^3 \tan ^{-1}(c x)}{6 c^6}-\frac{3 b d^2 e x}{2 c}-\frac{b d e^2 x^3}{4 c}-\frac{b e^3 x^5}{30 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.221279, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4980, 4848, 2391, 4852, 321, 203, 302} \[ \frac{1}{2} i b d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^3 \text{PolyLog}(2,i c x)+\frac{3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac{3 b d^2 e \tan ^{-1}(c x)}{2 c^2}+\frac{3 b d e^2 x}{4 c^3}-\frac{3 b d e^2 \tan ^{-1}(c x)}{4 c^4}+\frac{b e^3 x^3}{18 c^3}-\frac{b e^3 x}{6 c^5}+\frac{b e^3 \tan ^{-1}(c x)}{6 c^6}-\frac{3 b d^2 e x}{2 c}-\frac{b d e^2 x^3}{4 c}-\frac{b e^3 x^5}{30 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4980
Rule 4848
Rule 2391
Rule 4852
Rule 321
Rule 203
Rule 302
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{x} \, dx &=\int \left (\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \tan ^{-1}(c x)\right )+3 d e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+e^3 x^5 \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^3 \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx+\left (3 d^2 e\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (3 d e^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx+e^3 \int x^5 \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=\frac{3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac{1}{2} \left (i b d^3\right ) \int \frac{\log (1-i c x)}{x} \, dx-\frac{1}{2} \left (i b d^3\right ) \int \frac{\log (1+i c x)}{x} \, dx-\frac{1}{2} \left (3 b c d^2 e\right ) \int \frac{x^2}{1+c^2 x^2} \, dx-\frac{1}{4} \left (3 b c d e^2\right ) \int \frac{x^4}{1+c^2 x^2} \, dx-\frac{1}{6} \left (b c e^3\right ) \int \frac{x^6}{1+c^2 x^2} \, dx\\ &=-\frac{3 b d^2 e x}{2 c}+\frac{3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac{1}{2} i b d^3 \text{Li}_2(-i c x)-\frac{1}{2} i b d^3 \text{Li}_2(i c x)+\frac{\left (3 b d^2 e\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 c}-\frac{1}{4} \left (3 b c d e^2\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac{1}{6} \left (b c e^3\right ) \int \left (\frac{1}{c^6}-\frac{x^2}{c^4}+\frac{x^4}{c^2}-\frac{1}{c^6 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{3 b d^2 e x}{2 c}+\frac{3 b d e^2 x}{4 c^3}-\frac{b e^3 x}{6 c^5}-\frac{b d e^2 x^3}{4 c}+\frac{b e^3 x^3}{18 c^3}-\frac{b e^3 x^5}{30 c}+\frac{3 b d^2 e \tan ^{-1}(c x)}{2 c^2}+\frac{3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac{1}{2} i b d^3 \text{Li}_2(-i c x)-\frac{1}{2} i b d^3 \text{Li}_2(i c x)-\frac{\left (3 b d e^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 c^3}+\frac{\left (b e^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{6 c^5}\\ &=-\frac{3 b d^2 e x}{2 c}+\frac{3 b d e^2 x}{4 c^3}-\frac{b e^3 x}{6 c^5}-\frac{b d e^2 x^3}{4 c}+\frac{b e^3 x^3}{18 c^3}-\frac{b e^3 x^5}{30 c}+\frac{3 b d^2 e \tan ^{-1}(c x)}{2 c^2}-\frac{3 b d e^2 \tan ^{-1}(c x)}{4 c^4}+\frac{b e^3 \tan ^{-1}(c x)}{6 c^6}+\frac{3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac{1}{2} i b d^3 \text{Li}_2(-i c x)-\frac{1}{2} i b d^3 \text{Li}_2(i c x)\\ \end{align*}
Mathematica [A] time = 0.169533, size = 190, normalized size = 0.83 \[ \frac{1}{2} i b d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^3 \text{PolyLog}(2,i c x)+\frac{3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)-\frac{3 b d^2 e \left (c x-\tan ^{-1}(c x)\right )}{2 c^2}-\frac{b d e^2 \left (c^3 x^3-3 c x+3 \tan ^{-1}(c x)\right )}{4 c^4}-\frac{b e^3 \left (3 c^5 x^5-5 c^3 x^3+15 c x-15 \tan ^{-1}(c x)\right )}{90 c^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.052, size = 272, normalized size = 1.2 \begin{align*}{\frac{a{x}^{6}{e}^{3}}{6}}+{\frac{3\,a{x}^{4}d{e}^{2}}{4}}+{\frac{3\,a{x}^{2}{d}^{2}e}{2}}+a{d}^{3}\ln \left ( cx \right ) +{\frac{b\arctan \left ( cx \right ){x}^{6}{e}^{3}}{6}}+{\frac{3\,b\arctan \left ( cx \right ){x}^{4}d{e}^{2}}{4}}+{\frac{3\,b\arctan \left ( cx \right ){x}^{2}{d}^{2}e}{2}}+b\arctan \left ( cx \right ){d}^{3}\ln \left ( cx \right ) -{\frac{b{e}^{3}{x}^{5}}{30\,c}}-{\frac{bd{e}^{2}{x}^{3}}{4\,c}}-{\frac{3\,b{d}^{2}ex}{2\,c}}+{\frac{b{e}^{3}{x}^{3}}{18\,{c}^{3}}}+{\frac{3\,bd{e}^{2}x}{4\,{c}^{3}}}-{\frac{b{e}^{3}x}{6\,{c}^{5}}}+{\frac{3\,b{d}^{2}e\arctan \left ( cx \right ) }{2\,{c}^{2}}}-{\frac{3\,bd{e}^{2}\arctan \left ( cx \right ) }{4\,{c}^{4}}}+{\frac{b{e}^{3}\arctan \left ( cx \right ) }{6\,{c}^{6}}}-{\frac{i}{2}}b{d}^{3}\ln \left ( cx \right ) \ln \left ( 1-icx \right ) +{\frac{i}{2}}b{d}^{3}\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -{\frac{i}{2}}b{d}^{3}{\it dilog} \left ( 1-icx \right ) +{\frac{i}{2}}b{d}^{3}{\it dilog} \left ( 1+icx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.22607, size = 356, normalized size = 1.56 \begin{align*} \frac{1}{6} \, a e^{3} x^{6} + \frac{3}{4} \, a d e^{2} x^{4} + \frac{3}{2} \, a d^{2} e x^{2} + a d^{3} \log \left (x\right ) - \frac{6 \, b c^{5} e^{3} x^{5} + 45 \, \pi b c^{6} d^{3} \log \left (c^{2} x^{2} + 1\right ) - 180 \, b c^{6} d^{3} \arctan \left (c x\right ) \log \left (x{\left | c \right |}\right ) + 90 i \, b c^{6} d^{3}{\rm Li}_2\left (i \, c x + 1\right ) - 90 i \, b c^{6} d^{3}{\rm Li}_2\left (-i \, c x + 1\right ) + 5 \,{\left (9 \, b c^{5} d e^{2} - 2 \, b c^{3} e^{3}\right )} x^{3} + 15 \,{\left (18 \, b c^{5} d^{2} e - 9 \, b c^{3} d e^{2} + 2 \, b c e^{3}\right )} x -{\left (30 \, b c^{6} e^{3} x^{6} + 135 \, b c^{6} d e^{2} x^{4} + 270 \, b c^{6} d^{2} e x^{2} + 180 i \, b c^{6} d^{3} \arctan \left (0, c\right ) + 270 \, b c^{4} d^{2} e - 135 \, b c^{2} d e^{2} + 30 \, b e^{3}\right )} \arctan \left (c x\right )}{180 \, c^{6}} \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{a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} +{\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \arctan \left (c x\right )}{x}, 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{\left (a + b \operatorname{atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x}\, 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 (e x^{2} + d\right )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]