Optimal. Leaf size=164 \[ -\frac{2 (e+f x)^{5/2} (b c-a d)}{5 d^2}-\frac{2 (e+f x)^{3/2} (b c-a d) (d e-c f)}{3 d^3}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)^2}{d^4}+\frac{2 (b c-a d) (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}+\frac{2 b (e+f x)^{7/2}}{7 d f} \]
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Rubi [A] time = 0.192452, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {80, 50, 63, 208} \[ -\frac{2 (e+f x)^{5/2} (b c-a d)}{5 d^2}-\frac{2 (e+f x)^{3/2} (b c-a d) (d e-c f)}{3 d^3}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)^2}{d^4}+\frac{2 (b c-a d) (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}+\frac{2 b (e+f x)^{7/2}}{7 d f} \]
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)^{5/2}}{c+d x} \, dx &=\frac{2 b (e+f x)^{7/2}}{7 d f}+\frac{\left (2 \left (-\frac{7}{2} b c f+\frac{7 a d f}{2}\right )\right ) \int \frac{(e+f x)^{5/2}}{c+d x} \, dx}{7 d f}\\ &=-\frac{2 (b c-a d) (e+f x)^{5/2}}{5 d^2}+\frac{2 b (e+f x)^{7/2}}{7 d f}-\frac{((b c-a d) (d e-c f)) \int \frac{(e+f x)^{3/2}}{c+d x} \, dx}{d^2}\\ &=-\frac{2 (b c-a d) (d e-c f) (e+f x)^{3/2}}{3 d^3}-\frac{2 (b c-a d) (e+f x)^{5/2}}{5 d^2}+\frac{2 b (e+f x)^{7/2}}{7 d f}-\frac{\left ((b c-a d) (d e-c f)^2\right ) \int \frac{\sqrt{e+f x}}{c+d x} \, dx}{d^3}\\ &=-\frac{2 (b c-a d) (d e-c f)^2 \sqrt{e+f x}}{d^4}-\frac{2 (b c-a d) (d e-c f) (e+f x)^{3/2}}{3 d^3}-\frac{2 (b c-a d) (e+f x)^{5/2}}{5 d^2}+\frac{2 b (e+f x)^{7/2}}{7 d f}-\frac{\left ((b c-a d) (d e-c f)^3\right ) \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{d^4}\\ &=-\frac{2 (b c-a d) (d e-c f)^2 \sqrt{e+f x}}{d^4}-\frac{2 (b c-a d) (d e-c f) (e+f x)^{3/2}}{3 d^3}-\frac{2 (b c-a d) (e+f x)^{5/2}}{5 d^2}+\frac{2 b (e+f x)^{7/2}}{7 d f}-\frac{\left (2 (b c-a d) (d e-c f)^3\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^4 f}\\ &=-\frac{2 (b c-a d) (d e-c f)^2 \sqrt{e+f x}}{d^4}-\frac{2 (b c-a d) (d e-c f) (e+f x)^{3/2}}{3 d^3}-\frac{2 (b c-a d) (e+f x)^{5/2}}{5 d^2}+\frac{2 b (e+f x)^{7/2}}{7 d f}+\frac{2 (b c-a d) (d e-c f)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.298791, size = 136, normalized size = 0.83 \[ \frac{2 b (e+f x)^{7/2}}{7 d f}-\frac{2 (b c-a d) \left (5 (d e-c f) \left (\sqrt{d} \sqrt{e+f x} (-3 c f+4 d e+d f x)-3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )\right )+3 d^{5/2} (e+f x)^{5/2}\right )}{15 d^{9/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 573, normalized size = 3.5 \begin{align*}{\frac{2\,b}{7\,df} \left ( fx+e \right ) ^{{\frac{7}{2}}}}+{\frac{2\,a}{5\,d} \left ( fx+e \right ) ^{{\frac{5}{2}}}}-{\frac{2\,bc}{5\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{5}{2}}}}-{\frac{2\,acf}{3\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+{\frac{2\,ae}{3\,d} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+{\frac{2\,b{c}^{2}f}{3\,{d}^{3}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,bce}{3\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+2\,{\frac{a{c}^{2}{f}^{2}\sqrt{fx+e}}{{d}^{3}}}-4\,{\frac{acfe\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{a{e}^{2}\sqrt{fx+e}}{d}}-2\,{\frac{b{c}^{3}{f}^{2}\sqrt{fx+e}}{{d}^{4}}}+4\,{\frac{b{c}^{2}fe\sqrt{fx+e}}{{d}^{3}}}-2\,{\frac{bc{e}^{2}\sqrt{fx+e}}{{d}^{2}}}-2\,{\frac{a{c}^{3}{f}^{3}}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{a{c}^{2}{f}^{2}e}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{acf{e}^{2}}{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}^{3}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{b{c}^{4}{f}^{3}}{{d}^{4}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{b{c}^{3}{f}^{2}e}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{b{c}^{2}f{e}^{2}}{{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}^{3}}{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 [B] time = 1.33619, size = 1247, normalized size = 7.6 \begin{align*} \left [\frac{105 \,{\left ({\left (b c d^{2} - a d^{3}\right )} e^{2} f - 2 \,{\left (b c^{2} d - a c d^{2}\right )} e f^{2} +{\left (b c^{3} - a c^{2} d\right )} f^{3}\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 (15 \, b d^{3} f^{3} x^{3} + 15 \, b d^{3} e^{3} - 161 \,{\left (b c d^{2} - a d^{3}\right )} e^{2} f + 245 \,{\left (b c^{2} d - a c d^{2}\right )} e f^{2} - 105 \,{\left (b c^{3} - a c^{2} d\right )} f^{3} + 3 \,{\left (15 \, b d^{3} e f^{2} - 7 \,{\left (b c d^{2} - a d^{3}\right )} f^{3}\right )} x^{2} +{\left (45 \, b d^{3} e^{2} f - 77 \,{\left (b c d^{2} - a d^{3}\right )} e f^{2} + 35 \,{\left (b c^{2} d - a c d^{2}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}}{105 \, d^{4} f}, \frac{2 \,{\left (105 \,{\left ({\left (b c d^{2} - a d^{3}\right )} e^{2} f - 2 \,{\left (b c^{2} d - a c d^{2}\right )} e f^{2} +{\left (b c^{3} - a c^{2} d\right )} f^{3}\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 (15 \, b d^{3} f^{3} x^{3} + 15 \, b d^{3} e^{3} - 161 \,{\left (b c d^{2} - a d^{3}\right )} e^{2} f + 245 \,{\left (b c^{2} d - a c d^{2}\right )} e f^{2} - 105 \,{\left (b c^{3} - a c^{2} d\right )} f^{3} + 3 \,{\left (15 \, b d^{3} e f^{2} - 7 \,{\left (b c d^{2} - a d^{3}\right )} f^{3}\right )} x^{2} +{\left (45 \, b d^{3} e^{2} f - 77 \,{\left (b c d^{2} - a d^{3}\right )} e f^{2} + 35 \,{\left (b c^{2} d - a c d^{2}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}\right )}}{105 \, d^{4} f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 51.3313, size = 221, normalized size = 1.35 \begin{align*} \frac{2 b \left (e + f x\right )^{\frac{7}{2}}}{7 d f} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (2 a d - 2 b c\right )}{5 d^{2}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (- 2 a c d f + 2 a d^{2} e + 2 b c^{2} f - 2 b c d e\right )}{3 d^{3}} + \frac{\sqrt{e + f x} \left (2 a c^{2} d f^{2} - 4 a c d^{2} e f + 2 a d^{3} e^{2} - 2 b c^{3} f^{2} + 4 b c^{2} d e f - 2 b c d^{2} e^{2}\right )}{d^{4}} - \frac{2 \left (a d - b c\right ) \left (c f - d e\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d^{5} \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 = 2.47742, size = 522, normalized size = 3.18 \begin{align*} \frac{2 \,{\left (b c^{4} f^{3} - a c^{3} d f^{3} - 3 \, b c^{3} d f^{2} e + 3 \, a c^{2} d^{2} f^{2} e + 3 \, b c^{2} d^{2} f e^{2} - 3 \, a c d^{3} f e^{2} - b c d^{3} e^{3} + a d^{4} e^{3}\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^{4}} + \frac{2 \,{\left (15 \,{\left (f x + e\right )}^{\frac{7}{2}} b d^{6} f^{6} - 21 \,{\left (f x + e\right )}^{\frac{5}{2}} b c d^{5} f^{7} + 21 \,{\left (f x + e\right )}^{\frac{5}{2}} a d^{6} f^{7} + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b c^{2} d^{4} f^{8} - 35 \,{\left (f x + e\right )}^{\frac{3}{2}} a c d^{5} f^{8} - 105 \, \sqrt{f x + e} b c^{3} d^{3} f^{9} + 105 \, \sqrt{f x + e} a c^{2} d^{4} f^{9} - 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b c d^{5} f^{7} e + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} a d^{6} f^{7} e + 210 \, \sqrt{f x + e} b c^{2} d^{4} f^{8} e - 210 \, \sqrt{f x + e} a c d^{5} f^{8} e - 105 \, \sqrt{f x + e} b c d^{5} f^{7} e^{2} + 105 \, \sqrt{f x + e} a d^{6} f^{7} e^{2}\right )}}{105 \, d^{7} f^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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