Optimal. Leaf size=228 \[ \frac {3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac {b e^3 \tan ^{-1}(c x)}{6 c^6}-\frac {b e^3 x}{6 c^5}-\frac {3 b d e^2 \tan ^{-1}(c x)}{4 c^4}+\frac {3 b d e^2 x}{4 c^3}+\frac {b e^3 x^3}{18 c^3}+\frac {3 b d^2 e \tan ^{-1}(c x)}{2 c^2}+\frac {1}{2} i b d^3 \text {Li}_2(-i c x)-\frac {1}{2} i b d^3 \text {Li}_2(i c x)-\frac {3 b d^2 e x}{2 c}-\frac {b d e^2 x^3}{4 c}-\frac {b e^3 x^5}{30 c} \]
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Rubi [A] time = 0.22, antiderivative size = 228, normalized size of antiderivative = 1.00, 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.
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Rule 203
Rule 302
Rule 321
Rule 2391
Rule 4848
Rule 4852
Rule 4980
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*}
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Mathematica [A] time = 0.18, size = 190, normalized size = 0.83 \[ \frac {3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)-\frac {3 b d^2 e \left (c x-\tan ^{-1}(c x)\right )}{2 c^2}-\frac {b d e^2 \left (c^3 x^3-3 c x+3 \tan ^{-1}(c x)\right )}{4 c^4}-\frac {b e^3 \left (3 c^5 x^5-5 c^3 x^3+15 c x-15 \tan ^{-1}(c x)\right )}{90 c^6}+\frac {1}{2} i b d^3 \text {Li}_2(-i c x)-\frac {1}{2} i b d^3 \text {Li}_2(i c x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e^{3} x^{6} + 3 \, a d e^{2} x^{4} + 3 \, a d^{2} e x^{2} + a d^{3} + {\left (b e^{3} x^{6} + 3 \, b d e^{2} x^{4} + 3 \, b d^{2} e x^{2} + b d^{3}\right )} \arctan \left (c x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 272, normalized size = 1.19 \[ \frac {a \,x^{6} e^{3}}{6}+\frac {3 a \,x^{4} d \,e^{2}}{4}+\frac {3 a \,x^{2} d^{2} e}{2}+d^{3} a \ln \left (c x \right )+\frac {b \arctan \left (c x \right ) x^{6} e^{3}}{6}+\frac {3 b \arctan \left (c x \right ) x^{4} d \,e^{2}}{4}+\frac {3 b \arctan \left (c x \right ) x^{2} d^{2} e}{2}+b \arctan \left (c x \right ) d^{3} \ln \left (c x \right )-\frac {b \,e^{3} x^{5}}{30 c}-\frac {b d \,e^{2} x^{3}}{4 c}-\frac {3 b \,d^{2} e x}{2 c}+\frac {b \,e^{3} x^{3}}{18 c^{3}}+\frac {3 b d \,e^{2} x}{4 c^{3}}-\frac {b \,e^{3} x}{6 c^{5}}+\frac {3 b \,d^{2} e \arctan \left (c x \right )}{2 c^{2}}-\frac {3 b d \,e^{2} \arctan \left (c x \right )}{4 c^{4}}+\frac {b \,e^{3} \arctan \left (c x \right )}{6 c^{6}}+\frac {i b \,d^{3} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i b \,d^{3} \dilog \left (-i c x +1\right )}{2}+\frac {i b \,d^{3} \dilog \left (i c x +1\right )}{2}-\frac {i b \,d^{3} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 251, normalized size = 1.10 \[ \frac {1}{6} \, a e^{3} x^{6} + \frac {3}{4} \, a d e^{2} x^{4} + \frac {3}{2} \, a d^{2} e x^{2} + a d^{3} \log \relax (x) - \frac {6 \, b c^{5} e^{3} x^{5} + 45 \, \pi b c^{6} d^{3} \log \left (c^{2} x^{2} + 1\right ) - 180 \, b c^{6} d^{3} \arctan \left (c x\right ) \log \left (c x\right ) + 90 i \, b c^{6} d^{3} {\rm Li}_2\left (i \, c x + 1\right ) - 90 i \, b c^{6} d^{3} {\rm Li}_2\left (-i \, c x + 1\right ) + 5 \, {\left (9 \, b c^{5} d e^{2} - 2 \, b c^{3} e^{3}\right )} x^{3} + 15 \, {\left (18 \, b c^{5} d^{2} e - 9 \, b c^{3} d e^{2} + 2 \, b c e^{3}\right )} x - 15 \, {\left (2 \, b c^{6} e^{3} x^{6} + 9 \, b c^{6} d e^{2} x^{4} + 18 \, b c^{6} d^{2} e x^{2} + 18 \, b c^{4} d^{2} e - 9 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} \arctan \left (c x\right )}{180 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 232, normalized size = 1.02 \[ \left \{\begin {array}{cl} \frac {a\,e^3\,x^6}{6}+a\,d^3\,\ln \relax (x)+\frac {3\,a\,d^2\,e\,x^2}{2}+\frac {3\,a\,d\,e^2\,x^4}{4} & \text {\ if\ \ }c=0\\ \frac {a\,e^3\,x^6}{6}+a\,d^3\,\ln \relax (x)-\frac {b\,e^3\,\left (\frac {x}{c^4}-\frac {\mathrm {atan}\left (c\,x\right )}{c^5}+\frac {x^5}{5}-\frac {x^3}{3\,c^2}\right )}{6\,c}-3\,b\,d^2\,e\,\left (\frac {x}{2\,c}-\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )\right )+\frac {3\,a\,d^2\,e\,x^2}{2}+\frac {3\,a\,d\,e^2\,x^4}{4}-3\,b\,d\,e^2\,\left (\frac {3\,\mathrm {atan}\left (c\,x\right )-3\,c\,x+c^3\,x^3}{12\,c^4}-\frac {x^4\,\mathrm {atan}\left (c\,x\right )}{4}\right )+\frac {b\,e^3\,x^6\,\mathrm {atan}\left (c\,x\right )}{6}-\frac {b\,d^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {b\,d^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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