Optimal. Leaf size=207 \[ \frac{c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt{d+e x} (f+g x) (c d f-a e g)}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt{d+e x} (f+g x)^2} \]
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Rubi [A] time = 0.273324, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {862, 872, 874, 205} \[ \frac{c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}}+\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt{d+e x} (f+g x) (c d f-a e g)}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt{d+e x} (f+g x)^2} \]
Antiderivative was successfully verified.
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Rule 862
Rule 872
Rule 874
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} (f+g x)^3} \, dx &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt{d+e x} (f+g x)^2}+\frac{(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 g}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt{d+e x} (f+g x)^2}+\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt{d+e x} (f+g x)}+\frac{\left (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 g (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 g \sqrt{d+e x} (f+g x)^2}+\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt{d+e x} (f+g x)}+\frac{\left (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 g (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 g \sqrt{d+e x} (f+g x)^2}+\frac{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt{d+e x} (f+g x)}+\frac{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 g^{3/2} (c d f-a e g)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0418726, size = 79, normalized size = 0.38 \[ \frac{2 c^2 d^2 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{3 (d+e x)^{3/2} (c d f-a e g)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.329, size = 285, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( 4\,aeg-4\,cdf \right ) g \left ( gx+f \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ({\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ){x}^{2}{c}^{2}{d}^{2}{g}^{2}+2\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{2}{d}^{2}fg+{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ){c}^{2}{d}^{2}{f}^{2}-\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}xcdg-2\,\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}aeg+\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{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{\sqrt{e x + d}{\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 = 1.75227, size = 2148, normalized size = 10.38 \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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