Optimal. Leaf size=313 \[ -\frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 c^{5/2} d^{5/2} g^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c^2 d^2 g \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac{a e}{c d}-\frac{f}{g}\right )}{12 \sqrt{d+e x}} \]
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Rubi [A] time = 0.51782, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 c^{5/2} d^{5/2} g^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c^2 d^2 g \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac{a e}{c d}-\frac{f}{g}\right )}{12 \sqrt{d+e x}} \]
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)^{3/2} \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)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt{d+e x}}-\frac{(c d f-a e g) \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}{6 g}\\ &=\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt{d+e x}}-\frac{(c d f-a e g)^2 \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 c d g}\\ &=-\frac{(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 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt{d+e x}}-\frac{(c d f-a e g)^3 \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 c^2 d^2 g}\\ &=-\frac{(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 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt{d+e x}}-\frac{\left ((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 c^2 d^2 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{(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 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt{d+e x}}-\frac{\left ((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^3 d^3 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{(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 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt{d+e x}}-\frac{\left ((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^3 d^3 g \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac{(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 c^2 d^2 g \sqrt{d+e x}}+\frac{\left (\frac{a e}{c d}-\frac{f}{g}\right ) (f+g x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 \sqrt{d+e x}}+\frac{(f+g x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt{d+e x}}-\frac{(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 c^{5/2} d^{5/2} 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 = 0.874469, size = 255, normalized size = 0.81 \[ \frac{\sqrt{c d} \sqrt{d+e x} \left (-\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) \left (3 a^2 e^2 g^2-2 a c d e g (4 f+g x)-c^2 d^2 \left (3 f^2+14 f g x+8 g^2 x^2\right )\right )-3 \sqrt{a e+c d x} (c d f-a e g)^{7/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right )\right )}{24 c^{7/2} d^{7/2} g^{3/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.328, size = 602, normalized size = 1.9 \begin{align*}{\frac{1}{48\,{c}^{2}{d}^{2}g}\sqrt{gx+f}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ){a}^{3}{e}^{3}{g}^{3}-9\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ){a}^{2}cd{e}^{2}f{g}^{2}+9\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ) a{c}^{2}{d}^{2}e{f}^{2}g-3\,\ln \left ( 1/2\,{\frac{2\,xcdg+aeg+cdf+2\,\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}}{\sqrt{cdg}}} \right ){c}^{3}{d}^{3}{f}^{3}+16\,{x}^{2}{c}^{2}{d}^{2}{g}^{2}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}+4\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}xacde{g}^{2}+28\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}x{c}^{2}{d}^{2}fg-6\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{a}^{2}{e}^{2}{g}^{2}+16\,acdefg\sqrt{cdg{x}^{2}+aegx+cdfx+aef}\sqrt{cdg}+6\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}{c}^{2}{d}^{2}{f}^{2} \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{3}{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 = 9.43747, size = 1790, normalized size = 5.72 \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{3}{2}}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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