Optimal. Leaf size=501 \[ \frac{b c \text{PolyLog}\left (2,\frac{\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d-e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \text{PolyLog}\left (2,\frac{\sqrt{-\sqrt{-c^2}} (d+e x)}{\sqrt{-\sqrt{-c^2}} d-e}\right )}{2 \sqrt{-c^2} e}+\frac{b c \text{PolyLog}\left (2,\frac{\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d+e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \text{PolyLog}\left (2,\frac{\sqrt{-\sqrt{-c^2}} (d+e x)}{\sqrt{-\sqrt{-c^2}} d+e}\right )}{2 \sqrt{-c^2} e}+\frac{\log (d+e x) \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{e}+\frac{b c \log (d+e x) \log \left (\frac{e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right )}{2 \sqrt{-c^2} e}+\frac{b c \log (d+e x) \log \left (-\frac{e \left (\sqrt [4]{-c^2} x+1\right )}{\sqrt [4]{-c^2} d-e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left (\frac{e \left (1-\sqrt{-\sqrt{-c^2}} x\right )}{\sqrt{-\sqrt{-c^2}} d+e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left (-\frac{e \left (\sqrt{-\sqrt{-c^2}} x+1\right )}{\sqrt{-\sqrt{-c^2}} d-e}\right )}{2 \sqrt{-c^2} e} \]
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Rubi [F] time = 0.0637799, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{a+b \tan ^{-1}\left (c x^2\right )}{d+e x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}\left (c x^2\right )}{d+e x} \, dx &=\int \left (\frac{a}{d+e x}+\frac{b \tan ^{-1}\left (c x^2\right )}{d+e x}\right ) \, dx\\ &=\frac{a \log (d+e x)}{e}+b \int \frac{\tan ^{-1}\left (c x^2\right )}{d+e x} \, dx\\ \end{align*}
Mathematica [C] time = 32.123, size = 326, normalized size = 0.65 \[ \frac{a \log (d+e x)}{e}+\frac{b \left (2 \tan ^{-1}\left (c x^2\right ) \log (d+e x)+i \left (\text{PolyLog}\left (2,\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt [4]{-1} e}\right )+\text{PolyLog}\left (2,\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt [4]{-1} e}\right )-\text{PolyLog}\left (2,\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-(-1)^{3/4} e}\right )-\text{PolyLog}\left (2,\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+(-1)^{3/4} e}\right )+\log (d+e x) \log \left (1-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt [4]{-1} e}\right )+\log (d+e x) \log \left (1-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt [4]{-1} e}\right )-\log (d+e x) \log \left (1-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-(-1)^{3/4} e}\right )-\log (d+e x) \log \left (1-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+(-1)^{3/4} e}\right )\right )\right )}{2 e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.125, size = 138, normalized size = 0.3 \begin{align*}{\frac{a\ln \left ( ex+d \right ) }{e}}+{\frac{b\ln \left ( ex+d \right ) \arctan \left ( c{x}^{2} \right ) }{e}}-{\frac{be}{2\,c}\sum _{{\it \_R1}={\it RootOf} \left ({c}^{2}{{\it \_Z}}^{4}-4\,{c}^{2}d{{\it \_Z}}^{3}+6\,{c}^{2}{d}^{2}{{\it \_Z}}^{2}-4\,{c}^{2}{d}^{3}{\it \_Z}+{c}^{2}{d}^{4}+{e}^{4} \right ) }{\frac{1}{{{\it \_R1}}^{2}-2\,{\it \_R1}\,d+{d}^{2}} \left ( \ln \left ( ex+d \right ) \ln \left ({\frac{-ex+{\it \_R1}-d}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{-ex+{\it \_R1}-d}{{\it \_R1}}} \right ) \right ) }} \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*} 2 \, b \int \frac{\arctan \left (c x^{2}\right )}{2 \,{\left (e x + d\right )}}\,{d x} + \frac{a \log \left (e x + d\right )}{e} \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 \arctan \left (c x^{2}\right ) + a}{e x + d}, 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{b \arctan \left (c x^{2}\right ) + a}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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