Optimal. Leaf size=118 \[ \frac{2 (B d-A e)}{d \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{c} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}} \]
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Rubi [A] time = 0.301936, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {828, 826, 1166, 208} \[ \frac{2 (B d-A e)}{d \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{c} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 828
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx &=\frac{2 (B d-A e)}{d (c d-b e) \sqrt{d+e x}}+\frac{\int \frac{A (c d-b e)+c (B d-A e) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{d (c d-b e)}\\ &=\frac{2 (B d-A e)}{d (c d-b e) \sqrt{d+e x}}+\frac{2 \operatorname{Subst}\left (\int \frac{-c d (B d-A e)+A e (c d-b e)+c (B d-A 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 )}{d (c d-b e)}\\ &=\frac{2 (B d-A e)}{d (c d-b e) \sqrt{d+e x}}+\frac{(2 A c) \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 d}+\frac{(2 c (b B-A c)) \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 d-b e)}\\ &=\frac{2 (B d-A e)}{d (c d-b e) \sqrt{d+e x}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}}-\frac{2 \sqrt{c} (b B-A c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0341058, size = 89, normalized size = 0.75 \[ \frac{2 \left (d (b B-A c) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right )+A (c d-b e) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{e x}{d}+1\right )\right )}{b d \sqrt{d+e x} (c d-b e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 168, normalized size = 1.4 \begin{align*} -2\,{\frac{A}{b{d}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+2\,{\frac{Ae}{d \left ( be-cd \right ) \sqrt{ex+d}}}-2\,{\frac{B}{ \left ( be-cd \right ) \sqrt{ex+d}}}+2\,{\frac{A{c}^{2}}{ \left ( be-cd \right ) b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{Bc}{ \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \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 [B] time = 3.59409, size = 1777, normalized size = 15.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.2482, size = 107, normalized size = 0.91 \begin{align*} \frac{2 A \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{- d}} \right )}}{b d \sqrt{- d}} - \frac{2 \left (- A e + B d\right )}{d \sqrt{d + e x} \left (b e - c d\right )} - \frac{2 \left (- A c + B b\right ) \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{b e - c d}{c}}} \right )}}{b \sqrt{\frac{b e - c d}{c}} \left (b e - c d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25249, size = 174, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (B b c - A c^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b c d - b^{2} e\right )} \sqrt{-c^{2} d + b c e}} + \frac{2 \,{\left (B d - A e\right )}}{{\left (c d^{2} - b d e\right )} \sqrt{x e + d}} + \frac{2 \, A \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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