Optimal. Leaf size=270 \[ \frac{i b^2 \left (3 c^2 d^2-e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^3}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{2 b \left (3 c^2 d^2-e^2\right ) \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}-\frac{2 a b d e x}{c}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{b^2 d e \log \left (c^2 x^2+1\right )}{c^2}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.398677, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4864, 4846, 260, 4852, 321, 203, 4984, 4884, 4920, 4854, 2402, 2315} \[ \frac{i b^2 \left (3 c^2 d^2-e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^3}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{2 b \left (3 c^2 d^2-e^2\right ) \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}-\frac{2 a b d e x}{c}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{b^2 d e \log \left (c^2 x^2+1\right )}{c^2}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 4864
Rule 4846
Rule 260
Rule 4852
Rule 321
Rule 203
Rule 4984
Rule 4884
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac{(2 b c) \int \left (\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{c^2}+\frac{e^3 x \left (a+b \tan ^{-1}(c x)\right )}{c^2}+\frac{\left (c^2 d^3-3 d e^2+e \left (3 c^2 d^2-e^2\right ) x\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 e}\\ &=\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac{(2 b) \int \frac{\left (c^2 d^3-3 d e^2+e \left (3 c^2 d^2-e^2\right ) x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c e}-\frac{(2 b d e) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}-\frac{\left (2 b e^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}\\ &=-\frac{2 a b d e x}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}-\frac{(2 b) \int \left (\frac{c^2 d^3 \left (1-\frac{3 e^2}{c^2 d^2}\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}-\frac{e \left (-3 c^2 d^2+e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{3 c e}-\frac{\left (2 b^2 d e\right ) \int \tan ^{-1}(c x) \, dx}{c}+\frac{1}{3} \left (b^2 e^2\right ) \int \frac{x^2}{1+c^2 x^2} \, dx\\ &=-\frac{2 a b d e x}{c}+\frac{b^2 e^2 x}{3 c^2}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\left (2 b^2 d e\right ) \int \frac{x}{1+c^2 x^2} \, dx-\frac{\left (b^2 e^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{3 c^2}-\frac{1}{3} \left (2 b d \left (\frac{c d^2}{e}-\frac{3 e}{c}\right )\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx-\frac{\left (2 b \left (3 c^2 d^2-e^2\right )\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac{2 a b d e x}{c}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{b^2 d e \log \left (1+c^2 x^2\right )}{c^2}+\frac{\left (2 b \left (3 c^2 d^2-e^2\right )\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}\\ &=-\frac{2 a b d e x}{c}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{2 b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{b^2 d e \log \left (1+c^2 x^2\right )}{c^2}-\frac{\left (2 b^2 \left (3 c^2 d^2-e^2\right )\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}\\ &=-\frac{2 a b d e x}{c}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{2 b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{b^2 d e \log \left (1+c^2 x^2\right )}{c^2}+\frac{\left (2 i b^2 \left (3 c^2 d^2-e^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{3 c^3}\\ &=-\frac{2 a b d e x}{c}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 e}+\frac{2 b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{b^2 d e \log \left (1+c^2 x^2\right )}{c^2}+\frac{i b^2 \left (3 c^2 d^2-e^2\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^3}\\ \end{align*}
Mathematica [A] time = 0.573217, size = 312, normalized size = 1.16 \[ \frac{-i b^2 \left (3 c^2 d^2-e^2\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+3 a^2 c^3 d^2 x+3 a^2 c^3 d e x^2+a^2 c^3 e^2 x^3+b \tan ^{-1}(c x) \left (2 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+6 a c d e+2 b \left (3 c^2 d^2-e^2\right ) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-b e \left (6 c^2 d x+c^2 e x^2+e\right )\right )-3 a b c^2 d^2 \log \left (c^2 x^2+1\right )-6 a b c^2 d e x-a b c^2 e^2 x^2+a b e^2 \log \left (c^2 x^2+1\right )+b^2 \tan ^{-1}(c x)^2 \left (c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-3 i c^2 d^2+3 c d e+i e^2\right )+3 b^2 c d e \log \left (c^2 x^2+1\right )+b^2 c e^2 x}{3 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.069, size = 750, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (a^{2} e^{2} x^{2} + 2 \, a^{2} d e x + a^{2} d^{2} +{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \arctan \left (c x\right )^{2} + 2 \,{\left (a b e^{2} x^{2} + 2 \, a b d e x + a b d^{2}\right )} \arctan \left (c x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atan}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (b \arctan \left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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