Optimal. Leaf size=70 \[ \frac{x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{b \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{d \sqrt{c^2 d-e}} \]
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Rubi [A] time = 0.0769833, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {191, 4912, 12, 444, 63, 208} \[ \frac{x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{b \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{d \sqrt{c^2 d-e}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4912
Rule 12
Rule 444
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-(b c) \int \frac{x}{d \left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{(b c) \int \frac{x}{\left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx}{d}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\left (1+c^2 x\right ) \sqrt{d+e x}} \, dx,x,x^2\right )}{2 d}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{1-\frac{c^2 d}{e}+\frac{c^2 x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{d e}\\ &=\frac{x \left (a+b \tan ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}+\frac{b \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{d \sqrt{c^2 d-e}}\\ \end{align*}
Mathematica [C] time = 0.265082, size = 202, normalized size = 2.89 \[ \frac{\frac{2 a x}{\sqrt{d+e x^2}}+\frac{b \log \left (-\frac{4 c d \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \sqrt{c^2 d-e}}\right )}{\sqrt{c^2 d-e}}+\frac{b \log \left (-\frac{4 c d \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \sqrt{c^2 d-e}}\right )}{\sqrt{c^2 d-e}}+\frac{2 b x \tan ^{-1}(c x)}{\sqrt{d+e x^2}}}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.188, size = 0, normalized size = 0. \begin{align*} \int{(a+b\arctan \left ( cx \right ) ) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \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 [B] time = 2.35641, size = 810, normalized size = 11.57 \begin{align*} \left [\frac{{\left (b e x^{2} + b d\right )} \sqrt{c^{2} d - e} \log \left (\frac{c^{4} e^{2} x^{4} + 8 \, c^{4} d^{2} - 8 \, c^{2} d e + 2 \,{\left (4 \, c^{4} d e - 3 \, c^{2} e^{2}\right )} x^{2} + 4 \,{\left (c^{3} e x^{2} + 2 \, c^{3} d - c e\right )} \sqrt{c^{2} d - e} \sqrt{e x^{2} + d} + e^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + 4 \, \sqrt{e x^{2} + d}{\left ({\left (b c^{2} d - b e\right )} x \arctan \left (c x\right ) +{\left (a c^{2} d - a e\right )} x\right )}}{4 \,{\left (c^{2} d^{3} - d^{2} e +{\left (c^{2} d^{2} e - d e^{2}\right )} x^{2}\right )}}, \frac{{\left (b e x^{2} + b d\right )} \sqrt{-c^{2} d + e} \arctan \left (-\frac{{\left (c^{2} e x^{2} + 2 \, c^{2} d - e\right )} \sqrt{-c^{2} d + e} \sqrt{e x^{2} + d}}{2 \,{\left (c^{3} d^{2} - c d e +{\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) + 2 \, \sqrt{e x^{2} + d}{\left ({\left (b c^{2} d - b e\right )} x \arctan \left (c x\right ) +{\left (a c^{2} d - a e\right )} x\right )}}{2 \,{\left (c^{2} d^{3} - d^{2} e +{\left (c^{2} d^{2} e - d e^{2}\right )} x^{2}\right )}}\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\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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