Optimal. Leaf size=133 \[ -\frac{2 \sqrt{d+e x}}{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac{2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{(c d f-a e g)^{3/2}} \]
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Rubi [A] time = 0.176276, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {868, 874, 205} \[ -\frac{2 \sqrt{d+e x}}{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}-\frac{2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{(c d f-a e g)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 868
Rule 874
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{g \int \frac{\sqrt{d+e x}}{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d f-a e g}\\ &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (2 e^2 g\right ) \operatorname{Subst}\left (\int \frac{1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )}{c d f-a e g}\\ &=-\frac{2 \sqrt{d+e x}}{(c d f-a e g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{2 \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d f-a e g} \sqrt{d+e x}}\right )}{(c d f-a e g)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0290364, size = 71, normalized size = 0.53 \[ -\frac{2 \sqrt{d+e x} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{\sqrt{(d+e x) (a e+c d x)} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.278, size = 128, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}{\sqrt{ex+d} \left ( cdx+ae \right ) \left ( aeg-cdf \right ) \sqrt{ \left ( aeg-cdf \right ) g}} \left ( g{\it Artanh} \left ({\frac{g\sqrt{cdx+ae}}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) \sqrt{cdx+ae}-\sqrt{ \left ( aeg-cdf \right ) g} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (g x + f\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74607, size = 1170, normalized size = 8.8 \begin{align*} \left [-\frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-\frac{g}{c d f - a e g}} \log \left (-\frac{c d e g x^{2} - c d^{2} f + 2 \, a d e g + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d f - a e g\right )} \sqrt{e x + d} \sqrt{-\frac{g}{c d f - a e g}} -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f +{\left (e f + d g\right )} x}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{a c d^{2} e f - a^{2} d e^{2} g +{\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} +{\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f -{\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x}, -\frac{2 \,{\left ({\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{\frac{g}{c d f - a e g}} \arctan \left (-\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d f - a e g\right )} \sqrt{e x + d} \sqrt{\frac{g}{c d f - a e g}}}{c d e g x^{2} + a d e g +{\left (c d^{2} + a e^{2}\right )} g x}\right ) + \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}\right )}}{a c d^{2} e f - a^{2} d e^{2} g +{\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} +{\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f -{\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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