Optimal. Leaf size=304 \[ \frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 g^3 \sqrt{d+e x}}-\frac{5 \sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{12 g^2 (d+e x)^{3/2}}-\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 \sqrt{c} \sqrt{d} g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}} \]
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Rubi [A] time = 0.487025, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 g^3 \sqrt{d+e x}}-\frac{5 \sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{12 g^2 (d+e x)^{3/2}}-\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 \sqrt{c} \sqrt{d} g^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}} \]
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 )^{5/2}}{(d+e x)^{5/2} \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 )^{5/2}}{3 g (d+e x)^{5/2}}-\frac{(5 (c d f-a e g)) \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}{6 g}\\ &=-\frac{5 (c d f-a e g) \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}}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}}+\frac{\left (5 (c d f-a e g)^2\right ) \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}{8 g^2}\\ &=\frac{5 (c d f-a e g)^2 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt{d+e x}}-\frac{5 (c d f-a e g) \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}}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}}-\frac{\left (5 (c d f-a e g)^3\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}{16 g^3}\\ &=\frac{5 (c d f-a e g)^2 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt{d+e x}}-\frac{5 (c d f-a e g) \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}}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}}-\frac{\left (5 (c d f-a e g)^3 \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}{16 g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{5 (c d f-a e g)^2 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt{d+e x}}-\frac{5 (c d f-a e g) \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}}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}}-\frac{\left (5 (c d f-a e g)^3 \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 )}{8 c d g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{5 (c d f-a e g)^2 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt{d+e x}}-\frac{5 (c d f-a e g) \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}}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}}-\frac{\left (5 (c d f-a e g)^3 \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 )}{8 c d g^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac{5 (c d f-a e g)^2 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt{d+e x}}-\frac{5 (c d f-a e g) \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}}{12 g^2 (d+e x)^{3/2}}+\frac{\sqrt{f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2}}-\frac{5 (c d f-a e g)^3 \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 )}{8 \sqrt{c} \sqrt{d} g^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 1.04351, size = 229, normalized size = 0.75 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\sqrt{g} (f+g x) \left (33 a^2 e^2 g^2+2 a c d e g (13 g x-20 f)+c^2 d^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )-\frac{15 \sqrt{c} \sqrt{d} (c d f-a e g)^{7/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right )}{(c d)^{3/2} \sqrt{a e+c d x}}\right )}{24 g^{7/2} \sqrt{d+e x} \sqrt{f+g x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.368, size = 508, normalized size = 1.7 \begin{align*}{\frac{1}{48\,{g}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}\sqrt{gx+f} \left ( 15\,\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}^{3}{e}^{3}{g}^{3}-45\,\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}cd{e}^{2}f{g}^{2}+45\,\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{c}^{2}{d}^{2}e{f}^{2}g-15\,\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}^{3}{d}^{3}{f}^{3}+16\,{x}^{2}{c}^{2}{d}^{2}{g}^{2}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg}+52\,\sqrt{cdg}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }xacde{g}^{2}-20\,\sqrt{cdg}\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }x{c}^{2}{d}^{2}fg+66\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg}{a}^{2}{e}^{2}{g}^{2}-80\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg}acdefg+30\,\sqrt{ \left ( cdx+ae \right ) \left ( gx+f \right ) }\sqrt{cdg}{c}^{2}{d}^{2}{f}^{2} \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{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{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 = 8.05044, size = 1793, normalized size = 5.9 \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] 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{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}} \sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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