Optimal. Leaf size=966 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.26016, antiderivative size = 966, normalized size of antiderivative = 1., number of steps used = 49, number of rules used = 15, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {4980, 199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391, 444, 36, 31} \[ \frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}-\frac{b \log \left (c^2 x^2+1\right ) c}{4 d \left (c^2 d-e\right ) e}+\frac{b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac{b \log \left (e x^2+d\right ) c}{4 d \left (c^2 d-e\right ) e}-\frac{b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}+\frac{b c}{8 \left (c^2 d-e\right ) e \left (e x^2+d\right )}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (e x^2+d\right )}-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (e x^2+d\right )^2}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4980
Rule 199
Rule 205
Rule 4912
Rule 6725
Rule 571
Rule 77
Rule 4908
Rule 2409
Rule 2394
Rule 2393
Rule 2391
Rule 444
Rule 36
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\int \left (-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{e \left (d+e x^2\right )^3}+\frac{a+b \tan ^{-1}(c x)}{e \left (d+e x^2\right )^2}\right ) \, dx\\ &=\frac{\int \frac{a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{e}-\frac{d \int \frac{a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx}{e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}-\frac{(b c) \int \frac{\frac{x}{2 d \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e}}}{1+c^2 x^2} \, dx}{e}+\frac{(b c d) \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}{e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}-\frac{(b c) \int \left (\frac{x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2} \sqrt{e} \left (1+c^2 x^2\right )}\right ) \, dx}{e}+\frac{(b c d) \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}{e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}+\frac{(3 b c) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{8 d^{3/2} e^{3/2}}-\frac{(b c) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{2 d^{3/2} e^{3/2}}+\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 e}-\frac{(b c) \int \frac{x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}+\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^{3/2} e^{3/2}}-\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^{3/2} e^{3/2}}-\frac{(i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{4 d^{3/2} e^{3/2}}+\frac{(i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+c^2 x^2} \, dx}{4 d^{3/2} e^{3/2}}+\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 e}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 d e}\\ &=-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^2\right )}{4 d \left (c^2 d-e\right )}+\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^{3/2} e^{3/2}}-\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^{3/2} e^{3/2}}-\frac{(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}{4 d^{3/2} e^{3/2}}+\frac{(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}{4 d^{3/2} e^{3/2}}+\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 e}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )}{4 d \left (c^2 d-e\right ) e}\\ &=\frac{b c}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}+\frac{b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d \left (c^2 d-e\right )^2 e}-\frac{b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right ) e}-\frac{b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d \left (c^2 d-e\right )^2 e}+\frac{b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right ) 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^{3/2} e^{3/2}}+\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^{3/2} e^{3/2}}-\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^{3/2} e^{3/2}}-\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^{3/2} e^{3/2}}-\frac{(i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1-\sqrt{-c^2} x} \, dx}{8 d^{3/2} e^{3/2}}-\frac{(i b c) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+\sqrt{-c^2} x} \, dx}{8 d^{3/2} e^{3/2}}+\frac{(i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1-\sqrt{-c^2} x} \, dx}{8 d^{3/2} e^{3/2}}+\frac{(i b c) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{1+\sqrt{-c^2} x} \, dx}{8 d^{3/2} e^{3/2}}\\ &=\frac{b c}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}+\frac{b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d \left (c^2 d-e\right )^2 e}-\frac{b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right ) e}-\frac{b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d \left (c^2 d-e\right )^2 e}+\frac{b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right ) e}+\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^2 e}+\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^2 e}-\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^2 e}-\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^2 e}-\frac{(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}{8 \sqrt{-c^2} d^2 e}-\frac{(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}{8 \sqrt{-c^2} d^2 e}+\frac{(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}{8 \sqrt{-c^2} d^2 e}+\frac{(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}{8 \sqrt{-c^2} d^2 e}\\ &=\frac{b c}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}+\frac{b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d \left (c^2 d-e\right )^2 e}-\frac{b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right ) e}-\frac{b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d \left (c^2 d-e\right )^2 e}+\frac{b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right ) 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^{3/2} e^{3/2}}+\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^{3/2} e^{3/2}}+\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^{3/2} e^{3/2}}-\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^{3/2} e^{3/2}}+\frac{(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 )}{8 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{(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 )}{8 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{(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 )}{8 \sqrt{-c^2} d^{3/2} e^{3/2}}+\frac{(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 )}{8 \sqrt{-c^2} d^{3/2} e^{3/2}}\\ &=\frac{b c}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac{x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (d+e x^2\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}+\frac{b c \left (5 c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )}{16 d \left (c^2 d-e\right )^2 e}-\frac{b c \log \left (1+c^2 x^2\right )}{4 d \left (c^2 d-e\right ) e}-\frac{b c \left (5 c^2 d-3 e\right ) \log \left (d+e x^2\right )}{16 d \left (c^2 d-e\right )^2 e}+\frac{b c \log \left (d+e x^2\right )}{4 d \left (c^2 d-e\right ) e}+\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}+\frac{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^{3/2} e^{3/2}}-\frac{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^{3/2} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 12.9692, size = 1894, normalized size = 1.96 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.75, size = 3801, normalized size = 3.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (\frac{b x^{2} \arctan \left (c x\right ) + a x^{2}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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