Optimal. Leaf size=238 \[ -\frac{3 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^2 \sqrt{d+e x}}+\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{4 \sqrt{c} \sqrt{d} g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}} \]
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Rubi [A] time = 0.345087, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.104, Rules used = {864, 891, 63, 217, 206} \[ -\frac{3 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^2 \sqrt{d+e x}}+\frac{3 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^2 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{4 \sqrt{c} \sqrt{d} g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 864
Rule 891
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} \sqrt{f+g x}} \, dx &=\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac{(3 (c d f-a e g)) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x} \sqrt{f+g x}} \, dx}{4 g}\\ &=-\frac{3 (c d f-a e g) \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^2 \sqrt{d+e x}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}+\frac{\left (3 (c d f-a e g)^2\right ) \int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g^2}\\ &=-\frac{3 (c d f-a e g) \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^2 \sqrt{d+e x}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}+\frac{\left (3 (c d f-a e g)^2 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \int \frac{1}{\sqrt{a e+c d x} \sqrt{f+g x}} \, dx}{8 g^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g) \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^2 \sqrt{d+e x}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}+\frac{\left (3 (c d f-a e g)^2 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{a e g}{c d}+\frac{g x^2}{c d}}} \, dx,x,\sqrt{a e+c d x}\right )}{4 c d g^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g) \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^2 \sqrt{d+e x}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}+\frac{\left (3 (c d f-a e g)^2 \sqrt{a e+c d x} \sqrt{d+e x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{c d}} \, dx,x,\frac{\sqrt{a e+c d x}}{\sqrt{f+g x}}\right )}{4 c d g^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{3 (c d f-a e g) \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^2 \sqrt{d+e x}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}+\frac{3 (c d f-a e g)^2 \sqrt{a e+c d x} \sqrt{d+e x} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{4 \sqrt{c} \sqrt{d} g^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 0.802621, size = 193, normalized size = 0.81 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\sqrt{g} (f+g x) (5 a e g+c d (2 g x-3 f))+\frac{3 \sqrt{c} \sqrt{d} (c d f-a e g)^{5/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right )}{(c d)^{3/2} \sqrt{a e+c d x}}\right )}{4 g^{5/2} \sqrt{d+e x} \sqrt{f+g x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.345, size = 325, normalized size = 1.4 \begin{align*}{\frac{1}{8\,{g}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}\sqrt{gx+f} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg}}{\sqrt{cdg}}} \right ){a}^{2}{e}^{2}{g}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg}}{\sqrt{cdg}}} \right ) acdefg+3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg}}{\sqrt{cdg}}} \right ){c}^{2}{d}^{2}{f}^{2}+4\,\sqrt{cdg}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }xcdg+10\,\sqrt{cdg}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }aeg-6\,\sqrt{cdg}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }cdf \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }}}{\frac{1}{\sqrt{cdg}}}} \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 (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}} \sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 6.25638, size = 1427, normalized size = 6. \begin{align*} \left [\frac{4 \,{\left (2 \, c^{2} d^{2} g^{2} x - 3 \, c^{2} d^{2} f g + 5 \, a c d e g^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} + 3 \,{\left (c^{2} d^{3} f^{2} - 2 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} +{\left (c^{2} d^{2} e f^{2} - 2 \, a c d e^{2} f g + a^{2} e^{3} g^{2}\right )} x\right )} \sqrt{c d g} \log \left (-\frac{8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d g x + c d f + a e g\right )} \sqrt{c d g} \sqrt{e x + d} \sqrt{g x + f} + 8 \,{\left (c^{2} d^{2} e f g +{\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} +{\left (c^{2} d^{2} e f^{2} + 2 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g +{\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{16 \,{\left (c d e g^{3} x + c d^{2} g^{3}\right )}}, \frac{2 \,{\left (2 \, c^{2} d^{2} g^{2} x - 3 \, c^{2} d^{2} f g + 5 \, a c d e g^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (c^{2} d^{3} f^{2} - 2 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} +{\left (c^{2} d^{2} e f^{2} - 2 \, a c d e^{2} f g + a^{2} e^{3} g^{2}\right )} x\right )} \sqrt{-c d g} \arctan \left (\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d g} \sqrt{e x + d} \sqrt{g x + f}}{2 \, c d e g x^{2} + c d^{2} f + a d e g +{\left (c d e f +{\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right )}{8 \,{\left (c d e g^{3} x + c d^{2} g^{3}\right )}}\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*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}} \sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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