Optimal. Leaf size=134 \[ -\frac{\sqrt{d+e x} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac{\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c}} \]
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Rubi [A] time = 0.183317, antiderivative size = 134, 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 = {738, 826, 1166, 208} \[ -\frac{\sqrt{d+e x} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac{\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 738
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac{\sqrt{d+e x} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{\int \frac{\frac{1}{2} d (4 c d-3 b e)+\frac{1}{2} e (2 c d-b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{b^2}\\ &=-\frac{\sqrt{d+e x} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} d e (4 c d-3 b e)-\frac{1}{2} d e (2 c d-b e)+\frac{1}{2} 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 )}{b^2}\\ &=-\frac{\sqrt{d+e x} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac{(c d (4 c d-3 b e)) \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^3}+\frac{((c d-b e) (4 c d-b e)) \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^3}\\ &=-\frac{\sqrt{d+e x} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac{\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.214338, size = 127, normalized size = 0.95 \[ \frac{\frac{b \sqrt{d+e x} (-b d+b e x-2 c d x)}{x (b+c x)}+\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c}}}{b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.264, size = 237, normalized size = 1.8 \begin{align*}{\frac{{e}^{2}}{b \left ( cex+be \right ) }\sqrt{ex+d}}-{\frac{ced}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{{e}^{2}}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-5\,{\frac{ced}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}{d}^{2}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{d}{{b}^{2}x}\sqrt{ex+d}}-3\,{\frac{e\sqrt{d}}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{3/2}c}{{b}^{3}}{\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 = 2.21837, size = 1679, normalized size = 12.53 \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 [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 [A] time = 1.38327, size = 285, normalized size = 2.13 \begin{align*} \frac{{\left (4 \, c^{2} d^{2} - 5 \, 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^{3}} - \frac{{\left (4 \, c d^{2} - 3 \, b d e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c d e - 2 \, \sqrt{x e + d} c d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} + 2 \, \sqrt{x e + d} b d e^{2}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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