Optimal. Leaf size=154 \[ -\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{3/2}}-\frac{\sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)} \]
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Rubi [A] time = 0.260734, antiderivative size = 154, 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 = {740, 826, 1166, 208} \[ -\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{3/2}}-\frac{\sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 740
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (b x+c x^2\right )^2} \, dx &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}-\frac{\int \frac{\frac{1}{2} (c d-b e) (4 c d+b e)+\frac{1}{2} c e (2 c d-b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{-\frac{1}{2} c d e (2 c d-b e)+\frac{1}{2} e (c d-b e) (4 c d+b e)+\frac{1}{2} c 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 d (c d-b e)}\\ &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}+\frac{\left (c^2 (4 c d-5 b e)\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^3 (c d-b e)}-\frac{(c (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 d}\\ &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \left (b x+c x^2\right )}+\frac{(4 c d+b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{3/2}}-\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.499211, size = 148, normalized size = 0.96 \[ \frac{\frac{b \sqrt{d+e x} \left (b^2 e+b c (e x-d)-2 c^2 d x\right )}{x (b+c x) (c d-b e)}+\frac{c^{3/2} d (5 b e-4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.227, size = 202, normalized size = 1.3 \begin{align*}{\frac{{c}^{2}e}{{b}^{2} \left ( be-cd \right ) \left ( cex+be \right ) }\sqrt{ex+d}}+5\,{\frac{{c}^{2}e}{{b}^{2} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-4\,{\frac{{c}^{3}d}{{b}^{3} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{{b}^{2}dx}\sqrt{ex+d}}+{\frac{e}{{b}^{2}}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{3}{2}}}}+4\,{\frac{c}{{b}^{3}\sqrt{d}}{\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 [B] time = 7.38875, size = 2396, normalized size = 15.56 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (b + c x\right )^{2} \sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32119, size = 342, normalized size = 2.22 \begin{align*} \frac{{\left (4 \, c^{3} d - 5 \, b c^{2} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c d - b^{4} e\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d e - 2 \, \sqrt{x e + d} c^{2} d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} b c e^{2} + 2 \, \sqrt{x e + d} b c d e^{2} - \sqrt{x e + d} b^{2} e^{3}}{{\left (b^{2} c d^{2} - b^{3} d e\right )}{\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 )}} - \frac{{\left (4 \, c d + b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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