Optimal. Leaf size=237 \[ -\frac {d^2 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {a d x}{e^2}+\frac {b d \log \left (c^2 x^2+1\right )}{2 c e^2}+\frac {b \tan ^{-1}(c x)}{2 c^2 e}+\frac {i b d^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}-\frac {b d x \tan ^{-1}(c x)}{e^2}-\frac {b x}{2 c e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {4876, 4846, 260, 4852, 321, 203, 4856, 2402, 2315, 2447} \[ \frac {i b d^2 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b d^2 \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^3}-\frac {d^2 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {a d x}{e^2}+\frac {b d \log \left (c^2 x^2+1\right )}{2 c e^2}+\frac {b \tan ^{-1}(c x)}{2 c^2 e}-\frac {b d x \tan ^{-1}(c x)}{e^2}-\frac {b x}{2 c e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 260
Rule 321
Rule 2315
Rule 2402
Rule 2447
Rule 4846
Rule 4852
Rule 4856
Rule 4876
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{e}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e^2}+\frac {d^2 \int \frac {a+b \tan ^{-1}(c x)}{d+e x} \, dx}{e^2}+\frac {\int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e}\\ &=-\frac {a d x}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {\left (b c d^2\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{e^3}-\frac {\left (b c d^2\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{e^3}-\frac {(b d) \int \tan ^{-1}(c x) \, dx}{e^2}-\frac {(b c) \int \frac {x^2}{1+c^2 x^2} \, dx}{2 e}\\ &=-\frac {a d x}{e^2}-\frac {b x}{2 c e}-\frac {b d x \tan ^{-1}(c x)}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}-\frac {i b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}+\frac {\left (i b d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{e^3}+\frac {(b c d) \int \frac {x}{1+c^2 x^2} \, dx}{e^2}+\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{2 c e}\\ &=-\frac {a d x}{e^2}-\frac {b x}{2 c e}+\frac {b \tan ^{-1}(c x)}{2 c^2 e}-\frac {b d x \tan ^{-1}(c x)}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {b d \log \left (1+c^2 x^2\right )}{2 c e^2}+\frac {i b d^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.68, size = 404, normalized size = 1.70 \[ \frac {2 a d^2 \log (d+e x)-2 a d e x+a e^2 x^2-\frac {b d e \sqrt {\frac {c^2 d^2}{e^2}+1} \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left (\frac {c d}{e}\right )}}{c}+\frac {1}{2} \pi b d^2 \log \left (c^2 x^2+1\right )+\frac {b d e \log \left (c^2 x^2+1\right )}{c}+\frac {b e^2 \tan ^{-1}(c x)}{c^2}-i b d^2 \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-2 i b d^2 \tan ^{-1}(c x) \tan ^{-1}\left (\frac {c d}{e}\right )+2 b d^2 \tan ^{-1}\left (\frac {c d}{e}\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+2 b d^2 \tan ^{-1}(c x) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-2 b d^2 \tan ^{-1}\left (\frac {c d}{e}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )+i b d^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+i b d^2 \tan ^{-1}(c x)^2+i \pi b d^2 \tan ^{-1}(c x)+\pi b d^2 \log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )-2 b d^2 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+\frac {b d e \tan ^{-1}(c x)^2}{c}-2 b d e x \tan ^{-1}(c x)+b e^2 x^2 \tan ^{-1}(c x)-\frac {b e^2 x}{c}}{2 e^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \arctan \left (c x\right ) + a x^{2}}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 305, normalized size = 1.29 \[ \frac {a \,x^{2}}{2 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \ln \left (c e x +d c \right )}{e^{3}}+\frac {b \arctan \left (c x \right ) x^{2}}{2 e}-\frac {b d x \arctan \left (c x \right )}{e^{2}}+\frac {b \arctan \left (c x \right ) d^{2} \ln \left (c e x +d c \right )}{e^{3}}+\frac {i b \,d^{2} \ln \left (c e x +d c \right ) \ln \left (\frac {-c e x +i e}{d c +i e}\right )}{2 e^{3}}-\frac {i b \,d^{2} \ln \left (c e x +d c \right ) \ln \left (\frac {c e x +i e}{-d c +i e}\right )}{2 e^{3}}+\frac {i b \,d^{2} \dilog \left (\frac {-c e x +i e}{d c +i e}\right )}{2 e^{3}}-\frac {i b \,d^{2} \dilog \left (\frac {c e x +i e}{-d c +i e}\right )}{2 e^{3}}+\frac {b d \ln \left (c^{2} d^{2}-2 \left (c e x +d c \right ) c d +\left (c e x +d c \right )^{2}+e^{2}\right )}{2 c \,e^{2}}+\frac {b \arctan \left (c x \right )}{2 c^{2} e}-\frac {b x}{2 c e}-\frac {b d}{2 c \,e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + 2 \, b \int \frac {x^{2} \arctan \left (c x\right )}{2 \, {\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________