Optimal. Leaf size=213 \[ \frac{3 c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 \sqrt{g} (c d f-a e g)^{5/2}}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)} \]
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Rubi [A] time = 0.313584, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {872, 874, 205} \[ \frac{3 c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 \sqrt{g} (c d f-a e g)^{5/2}}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt{d+e x} (f+g x) (c d f-a e g)^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt{d+e x} (f+g x)^2 (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 872
Rule 874
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{(f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt{d+e x} (f+g x)^2}+\frac{(3 c d) \int \frac{\sqrt{d+e x}}{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 (c d f-a e g)}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt{d+e x} (f+g x)^2}+\frac{3 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)}+\frac{\left (3 c^2 d^2\right ) \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}{8 (c d f-a e g)^2}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt{d+e x} (f+g x)^2}+\frac{3 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)}+\frac{\left (3 c^2 d^2 e^2\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 )}{4 (c d f-a e g)^2}\\ &=\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g) \sqrt{d+e x} (f+g x)^2}+\frac{3 c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^2 \sqrt{d+e x} (f+g x)}+\frac{3 c^2 d^2 \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 )}{4 \sqrt{g} (c d f-a e g)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0475734, size = 77, normalized size = 0.36 \[ \frac{2 c^2 d^2 \sqrt{(d+e x) (a e+c d x)} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{\sqrt{d+e x} (c d f-a e g)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.329, size = 285, normalized size = 1.3 \begin{align*} -{\frac{1}{4\, \left ( aeg-cdf \right ) ^{2} \left ( gx+f \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{2}{d}^{2}{g}^{2}+6\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{2}{d}^{2}fg+3\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{2}{d}^{2}{f}^{2}-3\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xcdg+2\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}aeg-5\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}cdf \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \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{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88753, size = 2587, normalized size = 12.15 \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 [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(-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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