Optimal. Leaf size=430 \[ \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e}+\frac {i b d^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^3}-\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}-\frac {d^2 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}-\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c e^2}-\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {a b x}{c e}+\frac {b^2 \log \left (c^2 x^2+1\right )}{2 c^2 e}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}-\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c e^2}-\frac {b^2 x \tan ^{-1}(c x)}{c e} \]
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Rubi [A] time = 0.42, antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 4858} \[ \frac {i b d^2 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^3}-\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}-\frac {b^2 d^2 \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^3}+\frac {b^2 d^2 \text {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e^3}-\frac {i b^2 d \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e}-\frac {d^2 \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}-\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c e^2}-\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {a b x}{c e}+\frac {b^2 \log \left (c^2 x^2+1\right )}{2 c^2 e}-\frac {b^2 x \tan ^{-1}(c x)}{c e} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4858
Rule 4876
Rule 4884
Rule 4916
Rule 4920
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx &=\int \left (-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}+\frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{e}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{e^2}+\frac {d^2 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx}{e^2}+\frac {\int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{e}\\ &=-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^3}-\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}+\frac {(2 b c d) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{e^2}-\frac {(b c) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{e}\\ &=-\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c e^2}-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^3}-\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}-\frac {(2 b d) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{e^2}-\frac {b \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c e}+\frac {b \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c e}\\ &=-\frac {a b x}{c e}-\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e}-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^3}-\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c e^2}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^3}-\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}+\frac {\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{e^2}-\frac {b^2 \int \tan ^{-1}(c x) \, dx}{c e}\\ &=-\frac {a b x}{c e}-\frac {b^2 x \tan ^{-1}(c x)}{c e}-\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e}-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^3}-\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c e^2}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^3}-\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}-\frac {\left (2 i b^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c e^2}+\frac {b^2 \int \frac {x}{1+c^2 x^2} \, dx}{e}\\ &=-\frac {a b x}{c e}-\frac {b^2 x \tan ^{-1}(c x)}{c e}-\frac {i d \left (a+b \tan ^{-1}(c x)\right )^2}{c e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 e}-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^3}-\frac {2 b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c e^2}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {b^2 \log \left (1+c^2 x^2\right )}{2 c^2 e}+\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^3}-\frac {i b^2 d \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c e^2}-\frac {i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}-\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^3}+\frac {b^2 d^2 \text {Li}_3\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}\\ \end {align*}
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Mathematica [B] time = 17.98, size = 1035, normalized size = 2.41 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{2} \arctan \left (c x\right )^{2} + 2 \, a b x^{2} \arctan \left (c x\right ) + a^{2} x^{2}}{e x + d}, x\right ) \]
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 [C] time = 40.44, size = 1784, normalized size = 4.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a^{2} {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + \frac {4 \, {\left (b^{2} e x^{2} - 2 \, b^{2} d x\right )} \arctan \left (c x\right )^{2} + 2 \, e^{2} \int \frac {12 \, {\left (b^{2} c^{2} e^{2} x^{4} + b^{2} e^{2} x^{2}\right )} \arctan \left (c x\right )^{2} + {\left (b^{2} c^{2} e^{2} x^{4} + b^{2} e^{2} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 4 \, {\left (8 \, a b c^{2} e^{2} x^{4} - b^{2} c e^{2} x^{3} + 2 \, b^{2} c d^{2} x + {\left (b^{2} c d e + 8 \, a b e^{2}\right )} x^{2}\right )} \arctan \left (c x\right ) + 2 \, {\left (b^{2} c^{2} e^{2} x^{4} - b^{2} c^{2} d e x^{3} - 2 \, b^{2} c^{2} d^{2} x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{2} e^{3} x^{3} + c^{2} d e^{2} x^{2} + e^{3} x + d e^{2}}\,{d x} - {\left (b^{2} e x^{2} - 2 \, b^{2} d x\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{32 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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