Optimal. Leaf size=112 \[ -\frac{2 b \sqrt{e+f x} (-2 a d f+b c f+b d e)}{d^2 f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2} \]
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Rubi [A] time = 0.117035, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {88, 63, 208} \[ -\frac{2 b \sqrt{e+f x} (-2 a d f+b c f+b d e)}{d^2 f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{(c+d x) \sqrt{e+f x}} \, dx &=\int \left (-\frac{b (b d e+b c f-2 a d f)}{d^2 f \sqrt{e+f x}}+\frac{(-b c+a d)^2}{d^2 (c+d x) \sqrt{e+f x}}+\frac{b^2 \sqrt{e+f x}}{d f}\right ) \, dx\\ &=-\frac{2 b (b d e+b c f-2 a d f) \sqrt{e+f x}}{d^2 f^2}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2}+\frac{(b c-a d)^2 \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{d^2}\\ &=-\frac{2 b (b d e+b c f-2 a d f) \sqrt{e+f x}}{d^2 f^2}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2}+\frac{\left (2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{c-\frac{d e}{f}+\frac{d x^2}{f}} \, dx,x,\sqrt{e+f x}\right )}{d^2 f}\\ &=-\frac{2 b (b d e+b c f-2 a d f) \sqrt{e+f x}}{d^2 f^2}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}\\ \end{align*}
Mathematica [A] time = 0.0897442, size = 112, normalized size = 1. \[ -\frac{2 b \sqrt{e+f x} (-2 a d f+b c f+b d e)}{d^2 f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 201, normalized size = 1.8 \begin{align*}{\frac{2\,{b}^{2}}{3\,d{f}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+4\,{\frac{ab\sqrt{fx+e}}{df}}-2\,{\frac{{b}^{2}c\sqrt{fx+e}}{f{d}^{2}}}-2\,{\frac{{b}^{2}e\sqrt{fx+e}}{d{f}^{2}}}+2\,{\frac{{a}^{2}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{abc}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{{b}^{2}{c}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) 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 = 1.43694, size = 779, normalized size = 6.96 \begin{align*} \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{d^{2} e - c d f} f^{2} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{d^{2} e - c d f} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (2 \, b^{2} d^{3} e^{2} +{\left (b^{2} c d^{2} - 6 \, a b d^{3}\right )} e f - 3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} f^{2} -{\left (b^{2} d^{3} e f - b^{2} c d^{2} f^{2}\right )} x\right )} \sqrt{f x + e}}{3 \,{\left (d^{4} e f^{2} - c d^{3} f^{3}\right )}}, \frac{2 \,{\left (3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-d^{2} e + c d f} f^{2} \arctan \left (\frac{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}{d f x + d e}\right ) -{\left (2 \, b^{2} d^{3} e^{2} +{\left (b^{2} c d^{2} - 6 \, a b d^{3}\right )} e f - 3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} f^{2} -{\left (b^{2} d^{3} e f - b^{2} c d^{2} f^{2}\right )} x\right )} \sqrt{f x + e}\right )}}{3 \,{\left (d^{4} e f^{2} - c d^{3} f^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.1489, size = 110, normalized size = 0.98 \begin{align*} \frac{2 b^{2} \left (e + f x\right )^{\frac{3}{2}}}{3 d f^{2}} + \frac{2 b \sqrt{e + f x} \left (2 a d f - b c f - b d e\right )}{d^{2} f^{2}} - \frac{2 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{d^{2} \sqrt{\frac{d}{c f - d e}} \left (c f - d e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.49055, size = 203, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{2}} + \frac{2 \,{\left ({\left (f x + e\right )}^{\frac{3}{2}} b^{2} d^{2} f^{4} - 3 \, \sqrt{f x + e} b^{2} c d f^{5} + 6 \, \sqrt{f x + e} a b d^{2} f^{5} - 3 \, \sqrt{f x + e} b^{2} d^{2} f^{4} e\right )}}{3 \, d^{3} f^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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