Optimal. Leaf size=385 \[ -\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac{a e}{c d}-\frac{f}{g}\right )}{24 \sqrt{d+e x}} \]
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Rubi [A] time = 0.718058, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {864, 870, 891, 63, 217, 206} \[ -\frac{5 \sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{5 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac{a e}{c d}-\frac{f}{g}\right )}{24 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 864
Rule 870
Rule 891
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx &=\frac{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt{d+e x}}-\frac{(c d f-a e g) \int \frac{\sqrt{d+e x} (f+g x)^{5/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g}\\ &=\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt{d+e x}}-\frac{\left (5 (c d f-a e g)^2\right ) \int \frac{\sqrt{d+e x} (f+g x)^{3/2}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{48 c d g}\\ &=-\frac{5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt{d+e x}}-\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}{64 c^2 d^2 g}\\ &=-\frac{5 (c d f-a e g)^3 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt{d+e x}}-\frac{\left (5 (c d f-a e g)^4\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}{128 c^3 d^3 g}\\ &=-\frac{5 (c d f-a e g)^3 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt{d+e x}}-\frac{\left (5 (c d f-a e g)^4 \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}{128 c^3 d^3 g \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)^3 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt{d+e x}}-\frac{\left (5 (c d f-a e g)^4 \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 )}{64 c^4 d^4 g \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)^3 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt{d+e x}}-\frac{\left (5 (c d f-a e g)^4 \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 )}{64 c^4 d^4 g \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)^3 \sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^3 d^3 g \sqrt{d+e x}}-\frac{5 (c d f-a e g)^2 (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt{d+e x}}+\frac{(f+g x)^{7/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt{d+e x}}-\frac{5 (c d f-a e g)^4 \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 )}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 1.20928, size = 300, normalized size = 0.78 \[ \frac{\sqrt{c d} \sqrt{d+e x} \left (\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) \left (-5 a^2 c d e^2 g^2 (11 f+2 g x)+15 a^3 e^3 g^3+a c^2 d^2 e g \left (73 f^2+36 f g x+8 g^2 x^2\right )+c^3 d^3 \left (118 f^2 g x+15 f^3+136 f g^2 x^2+48 g^3 x^3\right )\right )-15 \sqrt{a e+c d x} (c d f-a e g)^{9/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right )\right )}{192 c^{9/2} d^{9/2} g^{3/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.338, size = 870, normalized size = 2.3 \begin{align*} -{\frac{1}{384\,{c}^{3}{d}^{3}g}\sqrt{gx+f}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( -96\,{x}^{3}{c}^{3}{d}^{3}{g}^{3}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}+15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ){a}^{4}{e}^{4}{g}^{4}-60\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ){a}^{3}cd{e}^{3}f{g}^{3}+90\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ){a}^{2}{c}^{2}{d}^{2}{e}^{2}{f}^{2}{g}^{2}-60\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ) a{c}^{3}{d}^{3}e{f}^{3}g+15\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ){c}^{4}{d}^{4}{f}^{4}-16\,{x}^{2}a{c}^{2}{d}^{2}e{g}^{3}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}-272\,{x}^{2}{c}^{3}{d}^{3}f{g}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}+20\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}x{a}^{2}cd{e}^{2}{g}^{3}-72\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}xa{c}^{2}{d}^{2}ef{g}^{2}-236\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}x{c}^{3}{d}^{3}{f}^{2}g-30\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{a}^{3}{e}^{3}{g}^{3}+110\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{a}^{2}cd{e}^{2}f{g}^{2}-146\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}a{c}^{2}{d}^{2}e{f}^{2}g-30\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{c}^{3}{d}^{3}{f}^{3} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{cdg{x}^{2}+aegx+cdfx+aef}}}{\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{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{5}{2}}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 15.9162, size = 2244, normalized size = 5.83 \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{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{5}{2}}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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