Optimal. Leaf size=893 \[ \frac {3 i b \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \log \left (-\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b \log \left (\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac {3 i b \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b \text {Li}_2\left (\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \text {Li}_2\left (\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c}{8 d \left (c^2 d-e\right ) \left (e x^2+d\right )}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (e x^2+d\right )}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (e x^2+d\right )^2}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}} \]
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Rubi [A] time = 0.95, antiderivative size = 893, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391} \[ \frac {3 i b \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \log \left (-\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b \log \left (\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d^2 \left (c^2 d-e\right )^2}+\frac {3 i b \text {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \text {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b \text {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b \text {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) c}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c}{8 d \left (c^2 d-e\right ) \left (e x^2+d\right )}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (e x^2+d\right )}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (e x^2+d\right )^2}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 199
Rule 205
Rule 571
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 4908
Rule 4912
Rule 6725
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-(b c) \int \frac {\frac {x}{4 d \left (d+e x^2\right )^2}+\frac {3 x}{8 d^2 \left (d+e x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}}{1+c^2 x^2} \, dx\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-(b c) \int \left (\frac {x \left (5 d+3 e x^2\right )}{8 d^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-\frac {(b c) \int \frac {x \left (5 d+3 e x^2\right )}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{8 d^2}-\frac {(3 b c) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{8 d^{5/2} \sqrt {e}}\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {5 d+3 e x}{\left (1+c^2 x\right ) (d+e x)^2} \, dx,x,x^2\right )}{16 d^2}-\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{16 d^{5/2} \sqrt {e}}+\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{16 d^{5/2} \sqrt {e}}\\ &=\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-\frac {(b c) \operatorname {Subst}\left (\int \left (\frac {5 c^4 d-3 c^2 e}{\left (c^2 d-e\right )^2 \left (1+c^2 x\right )}-\frac {2 d e}{\left (c^2 d-e\right ) (d+e x)^2}+\frac {e \left (-5 c^2 d+3 e\right )}{\left (-c^2 d+e\right )^2 (d+e x)}\right ) \, dx,x,x^2\right )}{16 d^2}-\frac {(3 i b c) \int \left (\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{16 d^{5/2} \sqrt {e}}+\frac {(3 i b c) \int \left (\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{16 d^{5/2} \sqrt {e}}\\ &=-\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}-\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{32 d^{5/2} \sqrt {e}}-\frac {(3 i b c) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{32 d^{5/2} \sqrt {e}}+\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{32 d^{5/2} \sqrt {e}}+\frac {(3 i b c) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{32 d^{5/2} \sqrt {e}}\\ &=-\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}-\frac {(3 b c) \int \frac {\log \left (-\frac {i \sqrt {e} \left (1-\sqrt {-c^2} x\right )}{\sqrt {d} \left (\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1-\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^3}-\frac {(3 b c) \int \frac {\log \left (\frac {i \sqrt {e} \left (1-\sqrt {-c^2} x\right )}{\sqrt {d} \left (\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1+\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^3}+\frac {(3 b c) \int \frac {\log \left (-\frac {i \sqrt {e} \left (1+\sqrt {-c^2} x\right )}{\sqrt {d} \left (-\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1-\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^3}+\frac {(3 b c) \int \frac {\log \left (\frac {i \sqrt {e} \left (1+\sqrt {-c^2} x\right )}{\sqrt {d} \left (-\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1+\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^3}\\ &=-\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {(3 i b c) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c^2} x}{-\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {(3 i b c) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c^2} x}{\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {(3 i b c) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c^2} x}{-\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {(3 i b c) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c^2} x}{\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}\\ &=-\frac {b c}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {3 x \left (a+b \tan ^{-1}(c x)\right )}{8 d^2 \left (d+e x^2\right )}+\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d^2 \left (c^2 d-e\right )^2}+\frac {3 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}+\frac {3 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}-\frac {3 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{5/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 12.82, size = 1745, normalized size = 1.95 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [B] time = 1.99, size = 4027, normalized size = 4.51 \[ \text {output too large to display} \]
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 \[ \frac {1}{8} \, a {\left (\frac {3 \, e x^{3} + 5 \, d x}{d^{2} e^{2} x^{4} + 2 \, d^{3} e x^{2} + d^{4}} + \frac {3 \, \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d^{2}}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}\right )}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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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