Optimal. Leaf size=163 \[ \frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{f^3 \sqrt{e+f x} (d e-c f)^2}-\frac{2 (b e-a f)^3}{3 f^3 (e+f x)^{3/2} (d e-c f)}+\frac{2 b^3 \sqrt{e+f x}}{d f^3} \]
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Rubi [A] time = 0.212572, antiderivative size = 163, 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 = {87, 63, 208} \[ \frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}}+\frac{2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{f^3 \sqrt{e+f x} (d e-c f)^2}-\frac{2 (b e-a f)^3}{3 f^3 (e+f x)^{3/2} (d e-c f)}+\frac{2 b^3 \sqrt{e+f x}}{d f^3} \]
Antiderivative was successfully verified.
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Rule 87
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{(c+d x) (e+f x)^{5/2}} \, dx &=\int \left (\frac{(-b e+a f)^3}{f^2 (-d e+c f) (e+f x)^{5/2}}+\frac{(-b e+a f)^2 (-2 b d e+3 b c f-a d f)}{f^2 (-d e+c f)^2 (e+f x)^{3/2}}+\frac{b^3}{d f^2 \sqrt{e+f x}}+\frac{(-b c+a d)^3}{d (d e-c f)^2 (c+d x) \sqrt{e+f x}}\right ) \, dx\\ &=-\frac{2 (b e-a f)^3}{3 f^3 (d e-c f) (e+f x)^{3/2}}+\frac{2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{f^3 (d e-c f)^2 \sqrt{e+f x}}+\frac{2 b^3 \sqrt{e+f x}}{d f^3}-\frac{(b c-a d)^3 \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{d (d e-c f)^2}\\ &=-\frac{2 (b e-a f)^3}{3 f^3 (d e-c f) (e+f x)^{3/2}}+\frac{2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{f^3 (d e-c f)^2 \sqrt{e+f x}}+\frac{2 b^3 \sqrt{e+f x}}{d f^3}-\frac{\left (2 (b c-a d)^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 f (d e-c f)^2}\\ &=-\frac{2 (b e-a f)^3}{3 f^3 (d e-c f) (e+f x)^{3/2}}+\frac{2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{f^3 (d e-c f)^2 \sqrt{e+f x}}+\frac{2 b^3 \sqrt{e+f x}}{d f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.135197, size = 165, normalized size = 1.01 \[ \frac{2 \left (-\frac{b \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{f^3}+\frac{3 b^2 d (e+f x) (-3 a d f+b c f+2 b d e)}{f^3}+\frac{(b c-a d)^3 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{d (e+f x)}{d e-c f}\right )}{c f-d e}+\frac{3 b^3 d^2 (e+f x)^2}{f^3}\right )}{3 d^3 (e+f x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 501, normalized size = 3.1 \begin{align*} 2\,{\frac{{b}^{3}\sqrt{fx+e}}{d{f}^{3}}}-{\frac{2\,{a}^{3}}{3\,cf-3\,de} \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{{a}^{2}be}{ \left ( cf-de \right ) f \left ( fx+e \right ) ^{3/2}}}-2\,{\frac{a{b}^{2}{e}^{2}}{{f}^{2} \left ( cf-de \right ) \left ( fx+e \right ) ^{3/2}}}+{\frac{2\,{b}^{3}{e}^{3}}{3\,{f}^{3} \left ( cf-de \right ) } \left ( fx+e \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{{a}^{3}d}{ \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-6\,{\frac{{a}^{2}bc}{ \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+12\,{\frac{a{b}^{2}ce}{f \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-6\,{\frac{a{b}^{2}d{e}^{2}}{{f}^{2} \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}-6\,{\frac{{b}^{3}c{e}^{2}}{{f}^{2} \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+4\,{\frac{{b}^{3}d{e}^{3}}{{f}^{3} \left ( cf-de \right ) ^{2}\sqrt{fx+e}}}+2\,{\frac{{d}^{2}{a}^{3}}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{d{a}^{2}cb}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{a{b}^{2}{c}^{2}}{ \left ( cf-de \right ) ^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{b}^{3}{c}^{3}}{d \left ( cf-de \right ) ^{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 [B] time = 1.76337, size = 2836, normalized size = 17.4 \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 [A] time = 78.1227, size = 153, normalized size = 0.94 \begin{align*} \frac{2 b^{3} \sqrt{e + f x}}{d f^{3}} + \frac{2 \left (a f - b e\right )^{2} \left (a d f - 3 b c f + 2 b d e\right )}{f^{3} \sqrt{e + f x} \left (c f - d e\right )^{2}} - \frac{2 \left (a f - b e\right )^{3}}{3 f^{3} \left (e + f x\right )^{\frac{3}{2}} \left (c f - d e\right )} + \frac{2 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d^{2} \sqrt{\frac{c f - d e}{d}} \left (c f - d e\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.74893, size = 455, normalized size = 2.79 \begin{align*} -\frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c^{2} d f^{2} - 2 \, c d^{2} f e + d^{3} e^{2}\right )} \sqrt{c d f - d^{2} e}} + \frac{2 \, \sqrt{f x + e} b^{3}}{d f^{3}} - \frac{2 \,{\left (9 \,{\left (f x + e\right )} a^{2} b c f^{3} - 3 \,{\left (f x + e\right )} a^{3} d f^{3} + a^{3} c f^{4} - 18 \,{\left (f x + e\right )} a b^{2} c f^{2} e - 3 \, a^{2} b c f^{3} e - a^{3} d f^{3} e + 9 \,{\left (f x + e\right )} b^{3} c f e^{2} + 9 \,{\left (f x + e\right )} a b^{2} d f e^{2} + 3 \, a b^{2} c f^{2} e^{2} + 3 \, a^{2} b d f^{2} e^{2} - 6 \,{\left (f x + e\right )} b^{3} d e^{3} - b^{3} c f e^{3} - 3 \, a b^{2} d f e^{3} + b^{3} d e^{4}\right )}}{3 \,{\left (c^{2} f^{5} - 2 \, c d f^{4} e + d^{2} f^{3} e^{2}\right )}{\left (f x + e\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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