Optimal. Leaf size=130 \[ -\frac{2 (e+f x)^{3/2} (b c-a d)}{3 d^2}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)}{d^3}+\frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}+\frac{2 b (e+f x)^{5/2}}{5 d f} \]
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Rubi [A] time = 0.107757, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {80, 50, 63, 208} \[ -\frac{2 (e+f x)^{3/2} (b c-a d)}{3 d^2}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)}{d^3}+\frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}+\frac{2 b (e+f x)^{5/2}}{5 d f} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) (e+f x)^{3/2}}{c+d x} \, dx &=\frac{2 b (e+f x)^{5/2}}{5 d f}+\frac{\left (2 \left (-\frac{5}{2} b c f+\frac{5 a d f}{2}\right )\right ) \int \frac{(e+f x)^{3/2}}{c+d x} \, dx}{5 d f}\\ &=-\frac{2 (b c-a d) (e+f x)^{3/2}}{3 d^2}+\frac{2 b (e+f x)^{5/2}}{5 d f}-\frac{((b c-a d) (d e-c f)) \int \frac{\sqrt{e+f x}}{c+d x} \, dx}{d^2}\\ &=-\frac{2 (b c-a d) (d e-c f) \sqrt{e+f x}}{d^3}-\frac{2 (b c-a d) (e+f x)^{3/2}}{3 d^2}+\frac{2 b (e+f x)^{5/2}}{5 d f}-\frac{\left ((b c-a d) (d e-c f)^2\right ) \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{d^3}\\ &=-\frac{2 (b c-a d) (d e-c f) \sqrt{e+f x}}{d^3}-\frac{2 (b c-a d) (e+f x)^{3/2}}{3 d^2}+\frac{2 b (e+f x)^{5/2}}{5 d f}-\frac{\left (2 (b c-a d) (d e-c f)^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^3 f}\\ &=-\frac{2 (b c-a d) (d e-c f) \sqrt{e+f x}}{d^3}-\frac{2 (b c-a d) (e+f x)^{3/2}}{3 d^2}+\frac{2 b (e+f x)^{5/2}}{5 d f}+\frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.150638, size = 112, normalized size = 0.86 \[ \frac{2 \left (\frac{5 (a d f-b c f) \left (\sqrt{d} \sqrt{e+f x} (-3 c f+4 d e+d f x)-3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )\right )}{3 d^{5/2}}+b (e+f x)^{5/2}\right )}{5 d f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 370, normalized size = 2.9 \begin{align*}{\frac{2\,b}{5\,df} \left ( fx+e \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3\,d} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,bc}{3\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-2\,{\frac{acf\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{ae\sqrt{fx+e}}{d}}+2\,{\frac{b{c}^{2}f\sqrt{fx+e}}{{d}^{3}}}-2\,{\frac{bce\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{a{c}^{2}{f}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{acfe}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{a{e}^{2}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{b{c}^{3}{f}^{2}}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+4\,{\frac{b{c}^{2}fe}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{bc{e}^{2}}{d\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 = 1.38147, size = 795, normalized size = 6.12 \begin{align*} \left [-\frac{15 \,{\left ({\left (b c d - a d^{2}\right )} e f -{\left (b c^{2} - a c d\right )} f^{2}\right )} \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 d^{2} f^{2} x^{2} + 3 \, b d^{2} e^{2} - 20 \,{\left (b c d - a d^{2}\right )} e f + 15 \,{\left (b c^{2} - a c d\right )} f^{2} +{\left (6 \, b d^{2} e f - 5 \,{\left (b c d - a d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}}{15 \, d^{3} f}, \frac{2 \,{\left (15 \,{\left ({\left (b c d - a d^{2}\right )} e f -{\left (b c^{2} - a c d\right )} f^{2}\right )} \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 d^{2} f^{2} x^{2} + 3 \, b d^{2} e^{2} - 20 \,{\left (b c d - a d^{2}\right )} e f + 15 \,{\left (b c^{2} - a c d\right )} f^{2} +{\left (6 \, b d^{2} e f - 5 \,{\left (b c d - a d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}\right )}}{15 \, d^{3} f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 29.2673, size = 139, normalized size = 1.07 \begin{align*} \frac{2 b \left (e + f x\right )^{\frac{5}{2}}}{5 d f} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a d - 2 b c\right )}{3 d^{2}} + \frac{\sqrt{e + f x} \left (- 2 a c d f + 2 a d^{2} e + 2 b c^{2} f - 2 b c d e\right )}{d^{3}} + \frac{2 \left (a d - b c\right ) \left (c f - d e\right )^{2} \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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.98881, size = 321, normalized size = 2.47 \begin{align*} -\frac{2 \,{\left (b c^{3} f^{2} - a c^{2} d f^{2} - 2 \, b c^{2} d f e + 2 \, a c d^{2} f e + b c d^{2} e^{2} - a d^{3} e^{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^{3}} + \frac{2 \,{\left (3 \,{\left (f x + e\right )}^{\frac{5}{2}} b d^{4} f^{4} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b c d^{3} f^{5} + 5 \,{\left (f x + e\right )}^{\frac{3}{2}} a d^{4} f^{5} + 15 \, \sqrt{f x + e} b c^{2} d^{2} f^{6} - 15 \, \sqrt{f x + e} a c d^{3} f^{6} - 15 \, \sqrt{f x + e} b c d^{3} f^{5} e + 15 \, \sqrt{f x + e} a d^{4} f^{5} e\right )}}{15 \, d^{5} f^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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