Optimal. Leaf size=943 \[ \frac {i c \sqrt {-d} \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac {i c \sqrt {-d} \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac {\text {Li}_3\left (1-\frac {2}{1-i c x}\right ) b^2}{2 e^2}+\frac {\text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 e^2}+\frac {\text {Li}_3\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 e^2}-\frac {c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) b}{e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )}-\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2} \]
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Rubi [A] time = 1.76, antiderivative size = 943, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4980, 4978, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854, 4858} \[ \frac {i c \sqrt {-d} \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac {i c \sqrt {-d} \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac {\text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right ) b^2}{2 e^2}+\frac {\text {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 e^2}+\frac {\text {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 e^2}-\frac {c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b}{e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 e^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )}-\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 2447
Rule 4854
Rule 4856
Rule 4858
Rule 4864
Rule 4884
Rule 4920
Rule 4978
Rule 4980
Rule 4984
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {d x \left (a+b \tan ^{-1}(c x)\right )^2}{e \left (d+e x^2\right )^2}+\frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{d+e x^2} \, dx}{e}-\frac {d \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 \sqrt {-d} e^{3/2}}-\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 \sqrt {-d} e^{3/2}}+\frac {\int \left (-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(b c) \int \left (\frac {\sqrt {-d} e \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d-e\right ) \left (-\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 d \left (\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {-d} \left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e^2}-\frac {(b c) \int \left (\frac {\sqrt {-d} e \left (a+b \tan ^{-1}(c x)\right )}{\left (-c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 \left (d+\sqrt {-d} \sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e^2}-\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{3/2}}+\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{3/2}}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}-\frac {\left (b c^3\right ) \int \frac {\left (d+\sqrt {-d} \sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}+\frac {\left (b c^3 \sqrt {-d}\right ) \int \frac {\left (\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}-\frac {\left (b c \sqrt {-d}\right ) \int \frac {a+b \tan ^{-1}(c x)}{-\sqrt {-d}+\sqrt {e} x} \, dx}{2 \left (c^2 d-e\right ) e}+\frac {\left (b c \sqrt {-d}\right ) \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \left (c^2 d-e\right ) e}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}-\frac {b c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {b c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}-\frac {\left (b c^3\right ) \int \left (\frac {d \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac {\sqrt {-d} \sqrt {e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 \left (c^2 d-e\right ) e^2}+\frac {\left (b c^3 \sqrt {-d}\right ) \int \left (\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac {\sqrt {e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 \left (c^2 d-e\right ) e^2}+\frac {\left (b^2 c^2 \sqrt {-d}\right ) \int \frac {\log \left (\frac {2 c \left (-\sqrt {-d}+\sqrt {e} x\right )}{\left (-c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^{3/2}}-\frac {\left (b^2 c^2 \sqrt {-d}\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^{3/2}}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}-\frac {b c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {b c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^2}+\frac {i b^2 c \sqrt {-d} \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}-\frac {i b^2 c \sqrt {-d} \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}-2 \frac {\left (b c^3 d\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}\\ &=-\frac {c^2 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}-\frac {b c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {b c \sqrt {-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{e^2}+\frac {i b^2 c \sqrt {-d} \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}-\frac {i b^2 c \sqrt {-d} \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 e^2}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 e^2}\\ \end {align*}
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Mathematica [F] time = 19.75, size = 0, normalized size = 0.00 \[ \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{3} \arctan \left (c x\right )^{2} + 2 \, a b x^{3} \arctan \left (c x\right ) + a^{2} x^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
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 [F] time = 26.50, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a +b \arctan \left (c x \right )\right )^{2}}{\left (e \,x^{2}+d \right )^{2}}\, dx \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {d}{e^{3} x^{2} + d e^{2}} + \frac {\log \left (e x^{2} + d\right )}{e^{2}}\right )} + \int \frac {b^{2} x^{3} \arctan \left (c x\right )^{2} + 2 \, a b x^{3} \arctan \left (c x\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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