Optimal. Leaf size=138 \[ -\frac{2 b (e+f x)^{3/2} (-2 a d f+b c f+b d e)}{3 d^2 f^2}+\frac{2 \sqrt{e+f x} (b c-a d)^2}{d^3}-\frac{2 (b c-a d)^2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2} \]
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Rubi [A] time = 0.12469, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 208} \[ -\frac{2 b (e+f x)^{3/2} (-2 a d f+b c f+b d e)}{3 d^2 f^2}+\frac{2 \sqrt{e+f x} (b c-a d)^2}{d^3}-\frac{2 (b c-a d)^2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 \sqrt{e+f x}}{c+d x} \, dx &=\int \left (-\frac{b (b d e+b c f-2 a d f) \sqrt{e+f x}}{d^2 f}+\frac{(-b c+a d)^2 \sqrt{e+f x}}{d^2 (c+d x)}+\frac{b^2 (e+f x)^{3/2}}{d f}\right ) \, dx\\ &=-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2}+\frac{(b c-a d)^2 \int \frac{\sqrt{e+f x}}{c+d x} \, dx}{d^2}\\ &=\frac{2 (b c-a d)^2 \sqrt{e+f x}}{d^3}-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2}+\frac{\left ((b c-a d)^2 (d e-c f)\right ) \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{d^3}\\ &=\frac{2 (b c-a d)^2 \sqrt{e+f x}}{d^3}-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2}+\frac{\left (2 (b c-a d)^2 (d e-c f)\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^3 f}\\ &=\frac{2 (b c-a d)^2 \sqrt{e+f x}}{d^3}-\frac{2 b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{3 d^2 f^2}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2}-\frac{2 (b c-a d)^2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.10317, size = 138, normalized size = 1. \[ -\frac{2 b (e+f x)^{3/2} (-2 a d f+b c f+b d e)}{3 d^2 f^2}+\frac{2 \sqrt{e+f x} (b c-a d)^2}{d^3}-\frac{2 (b c-a d)^2 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}+\frac{2 b^2 (e+f x)^{5/2}}{5 d f^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 387, normalized size = 2.8 \begin{align*}{\frac{2\,{b}^{2}}{5\,d{f}^{2}} \left ( fx+e \right ) ^{{\frac{5}{2}}}}+{\frac{4\,ab}{3\,df} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{b}^{2}c}{3\,f{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{b}^{2}e}{3\,d{f}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}\sqrt{fx+e}}{d}}-4\,{\frac{abc\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{{b}^{2}{c}^{2}\sqrt{fx+e}}{{d}^{3}}}-2\,{\frac{{a}^{2}cf}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{{a}^{2}e}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+4\,{\frac{ab{c}^{2}f}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{abce}{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}^{3}f}{{d}^{3}\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}e}{{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 = 1.40626, size = 863, normalized size = 6.25 \begin{align*} \left [\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) + 2 \,{\left (3 \, b^{2} d^{2} f^{2} x^{2} - 2 \, b^{2} d^{2} e^{2} - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} e f + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} +{\left (b^{2} d^{2} e f - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}}{15 \, d^{3} f^{2}}, -\frac{2 \,{\left (15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} \sqrt{-\frac{d e - c f}{d}} \arctan \left (-\frac{\sqrt{f x + e} d \sqrt{-\frac{d e - c f}{d}}}{d e - c f}\right ) -{\left (3 \, b^{2} d^{2} f^{2} x^{2} - 2 \, b^{2} d^{2} e^{2} - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} e f + 15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} +{\left (b^{2} d^{2} e f - 5 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}\right )}}{15 \, d^{3} f^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.01214, size = 155, normalized size = 1.12 \begin{align*} \frac{2 \left (\frac{b^{2} \left (e + f x\right )^{\frac{5}{2}}}{5 d f} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a b d f - b^{2} c f - b^{2} d e\right )}{3 d^{2} f} + \frac{\sqrt{e + f x} \left (a^{2} d^{2} f - 2 a b c d f + b^{2} c^{2} f\right )}{d^{3}} - \frac{f \left (a d - b c\right )^{2} \left (c f - d e\right ) \operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d^{4} \sqrt{\frac{c f - d e}{d}}}\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.5432, size = 338, normalized size = 2.45 \begin{align*} -\frac{2 \,{\left (b^{2} c^{3} f - 2 \, a b c^{2} d f + a^{2} c d^{2} f - b^{2} c^{2} d e + 2 \, a b c d^{2} e - a^{2} d^{3} e\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^{3}} + \frac{2 \,{\left (3 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{2} d^{4} f^{8} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} c d^{3} f^{9} + 10 \,{\left (f x + e\right )}^{\frac{3}{2}} a b d^{4} f^{9} + 15 \, \sqrt{f x + e} b^{2} c^{2} d^{2} f^{10} - 30 \, \sqrt{f x + e} a b c d^{3} f^{10} + 15 \, \sqrt{f x + e} a^{2} d^{4} f^{10} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{2} d^{4} f^{8} e\right )}}{15 \, d^{5} f^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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