Optimal. Leaf size=227 \[ \frac{3 g \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^2 d^2 \sqrt{d+e x}}+\frac{3 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{c^{5/2} d^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x} (f+g x)^{3/2}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.314217, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {866, 870, 891, 63, 217, 206} \[ \frac{3 g \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^2 d^2 \sqrt{d+e x}}+\frac{3 \sqrt{g} \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{c^{5/2} d^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 \sqrt{d+e x} (f+g x)^{3/2}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 866
Rule 870
Rule 891
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2} (f+g x)^{3/2}}{\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} (f+g x)^{3/2}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{(3 g) \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}{c d}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^{3/2}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3 g \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \sqrt{d+e x}}+\frac{(3 g (c d f-a e g)) \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}{2 c^2 d^2}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^{3/2}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3 g \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \sqrt{d+e x}}+\frac{\left (3 g (c d f-a e g) \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}{2 c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^{3/2}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3 g \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \sqrt{d+e x}}+\frac{\left (3 g (c d f-a e g) \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 )}{c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^{3/2}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3 g \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \sqrt{d+e x}}+\frac{\left (3 g (c d f-a e g) \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 )}{c^3 d^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{2 \sqrt{d+e x} (f+g x)^{3/2}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3 g \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2 \sqrt{d+e x}}+\frac{3 \sqrt{g} (c d f-a e g) \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 )}{c^{5/2} d^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [C] time = 0.087196, size = 100, normalized size = 0.44 \[ -\frac{2 \sqrt{d+e x} (f+g x)^{3/2} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{c d \sqrt{(d+e x) (a e+c d x)} \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.379, size = 396, normalized size = 1.7 \begin{align*} -{\frac{1}{ \left ( 2\,cdx+2\,ae \right ){c}^{2}{d}^{2}} \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 ) xacde{g}^{2}-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 ) x{c}^{2}{d}^{2}fg+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}-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 ) acdefg-2\,\sqrt{cdg}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }xcdg-6\,\sqrt{cdg}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }aeg+4\,\sqrt{cdg}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }cdf \right ) \sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }}}{\frac{1}{\sqrt{cdg}}}{\frac{1}{\sqrt{ex+d}}}} \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 (g x + f\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}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 7.00522, size = 1544, normalized size = 6.8 \begin{align*} \left [\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d g x - 2 \, c d f + 3 \, a e g\right )} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (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 )} \sqrt{\frac{g}{c d}} \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} + 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} - 4 \,{\left (2 \, c^{2} d^{2} g x + c^{2} d^{2} f + a c d e g\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} \sqrt{\frac{g}{c d}} +{\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 )}{4 \,{\left (c^{3} d^{3} e x^{2} + a c^{2} d^{3} e +{\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} x\right )}}, \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d g x - 2 \, c d f + 3 \, a e g\right )} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (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 )} \sqrt{-\frac{g}{c d}} \arctan \left (\frac{2 \, \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} c d \sqrt{-\frac{g}{c d}}}{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 )}{2 \,{\left (c^{3} d^{3} e x^{2} + a c^{2} d^{3} e +{\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} x\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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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