Optimal. Leaf size=92 \[ \frac{2 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}-\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e \sqrt{d+e x}}{c} \]
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Rubi [A] time = 0.195558, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {703, 826, 1166, 208} \[ \frac{2 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}-\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 e \sqrt{d+e x}}{c} \]
Antiderivative was successfully verified.
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Rule 703
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{b x+c x^2} \, dx &=\frac{2 e \sqrt{d+e x}}{c}+\frac{\int \frac{c d^2+e (2 c d-b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{c}\\ &=\frac{2 e \sqrt{d+e x}}{c}+\frac{2 \operatorname{Subst}\left (\int \frac{c d^2 e-d e (2 c d-b e)+e (2 c d-b e) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{c}\\ &=\frac{2 e \sqrt{d+e x}}{c}+\frac{\left (2 c d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b}-\frac{\left (2 (c d-b e)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b c}\\ &=\frac{2 e \sqrt{d+e x}}{c}-\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0605512, size = 97, normalized size = 1.05 \[ \frac{2 \left (b \sqrt{c} e \sqrt{d+e x}+(c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )-c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{b c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.223, size = 159, normalized size = 1.7 \begin{align*} 2\,{\frac{e\sqrt{ex+d}}{c}}-2\,{\frac{b{e}^{2}}{c\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{de}{\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{c{d}^{2}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{{d}^{3/2}}{b}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \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 [A] time = 2.60947, size = 1007, normalized size = 10.95 \begin{align*} \left [\frac{c d^{\frac{3}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 2 \, \sqrt{e x + d} b e -{\left (c d - b e\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e - 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right )}{b c}, \frac{c d^{\frac{3}{2}} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right ) + 2 \, \sqrt{e x + d} b e + 2 \,{\left (c d - b e\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (-\frac{\sqrt{e x + d} c \sqrt{-\frac{c d - b e}{c}}}{c d - b e}\right )}{b c}, \frac{2 \, c \sqrt{-d} d \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) + 2 \, \sqrt{e x + d} b e -{\left (c d - b e\right )} \sqrt{\frac{c d - b e}{c}} \log \left (\frac{c e x + 2 \, c d - b e - 2 \, \sqrt{e x + d} c \sqrt{\frac{c d - b e}{c}}}{c x + b}\right )}{b c}, \frac{2 \,{\left (c \sqrt{-d} d \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right ) + \sqrt{e x + d} b e +{\left (c d - b e\right )} \sqrt{-\frac{c d - b e}{c}} \arctan \left (-\frac{\sqrt{e x + d} c \sqrt{-\frac{c d - b e}{c}}}{c d - b e}\right )\right )}}{b c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 33.7844, size = 92, normalized size = 1. \begin{align*} \frac{2 e \sqrt{d + e x}}{c} + \frac{2 d^{2} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{b \sqrt{- d}} - \frac{2 \left (b e - c d\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{b c^{2} \sqrt{\frac{b e - c d}{c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37459, size = 151, normalized size = 1.64 \begin{align*} \frac{2 \, d^{2} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} + \frac{2 \, \sqrt{x e + d} e}{c} - \frac{2 \,{\left (c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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