Optimal. Leaf size=170 \[ -\frac{\left (2 a e^2 g-c d (d g+e f)\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{3/2} (c d f-a e g)^{3/2}}-\frac{(e f-d g) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt{d+e x} (f+g x) (c d f-a e g)} \]
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Rubi [A] time = 0.233627, antiderivative size = 170, 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 = {878, 874, 205} \[ -\frac{\left (2 a e^2 g-c d (d g+e f)\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{3/2} (c d f-a e g)^{3/2}}-\frac{(e f-d g) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt{d+e x} (f+g x) (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 878
Rule 874
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt{d+e x} (f+g x)}+\frac{\left (e \left (\frac{1}{2} c d e^2 f+\frac{3}{2} c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\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}{g \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt{d+e x} (f+g x)}-\frac{\left (e^2 \left (2 a e^2 g-c d (e f+d g)\right )\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 )}{g (c d f-a e g)}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt{d+e x} (f+g x)}-\frac{\left (2 a e^2 g-c d (e f+d g)\right ) \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 )}{g^{3/2} (c d f-a e g)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.157788, size = 154, normalized size = 0.91 \[ \frac{\sqrt{d+e x} \left (\frac{\sqrt{g} (e f-d g) (a e+c d x)}{f+g x}-\frac{\sqrt{a e+c d x} \left (c d (d g+e f)-2 a e^2 g\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right )}{\sqrt{c d f-a e g}}\right )}{g^{3/2} \sqrt{(d+e x) (a e+c d x)} (a e g-c d f)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.331, size = 347, normalized size = 2. \begin{align*}{\frac{1}{ \left ( aeg-cdf \right ) g \left ( gx+f \right ) } \left ( -2\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) xa{e}^{2}{g}^{2}+{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) xc{d}^{2}{g}^{2}+{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) xcdefg-2\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) a{e}^{2}fg+{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) c{d}^{2}fg+{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ) cde{f}^{2}-\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}dg+\sqrt{ \left ( aeg-cdf \right ) g}\sqrt{cdx+ae}ef \right ) \sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\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{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58205, size = 1863, normalized size = 10.96 \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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