Optimal. Leaf size=310 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 c^{3/2} d^{3/2} g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c d g^2 \sqrt{d+e x}}-\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^2 \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}} \]
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Rubi [A] time = 0.533453, antiderivative size = 310, 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} \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} (c d f-a e g)^3 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c} \sqrt{d} \sqrt{f+g x}}\right )}{8 c^{3/2} d^{3/2} g^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c d g^2 \sqrt{d+e x}}-\frac{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^2 \sqrt{d+e x}}+\frac{(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}} \]
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{\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}}{(d+e x)^{3/2}} \, dx &=\frac{(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}}{3 g (d+e x)^{3/2}}-\frac{(c d f-a e g) \int \frac{\sqrt{f+g x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{2 g}\\ &=-\frac{(c d f-a e g) (f+g x)^{3/2} \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{(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}}{3 g (d+e x)^{3/2}}+\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 g^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 d g^2 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{3/2} \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{(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}}{3 g (d+e x)^{3/2}}+\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 d g^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 d g^2 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{3/2} \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{(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}}{3 g (d+e x)^{3/2}}+\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 d g^2 \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 d g^2 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{3/2} \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{(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}}{3 g (d+e x)^{3/2}}+\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^2 d^2 g^2 \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 d g^2 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{3/2} \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{(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}}{3 g (d+e x)^{3/2}}+\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^2 d^2 g^2 \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 d g^2 \sqrt{d+e x}}-\frac{(c d f-a e g) (f+g x)^{3/2} \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{(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}}{3 g (d+e x)^{3/2}}+\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^{3/2} d^{3/2} 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 = 0.880542, size = 254, normalized size = 0.82 \[ \frac{\sqrt{c} \sqrt{d} \sqrt{d+e x} \left (\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{c d} (f+g x) (a e+c d x) \left (3 a^2 e^2 g^2+2 a c d e g (4 f+7 g x)+c^2 d^2 \left (-3 f^2+2 f g x+8 g^2 x^2\right )\right )+3 \sqrt{a e+c d x} (c d f-a e g)^{7/2} \sqrt{\frac{c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d f-a e g}}\right )\right )}{24 g^{5/2} (c d)^{5/2} \sqrt{f+g x} \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.327, size = 602, normalized size = 1.9 \begin{align*} -{\frac{1}{48\,cd{g}^{2}}\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}-28\,\sqrt{cdg}\sqrt{cdg{x}^{2}+aegx+cdfx+aef}xacde{g}^{2}-4\,\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{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}} \sqrt{g x + f}}{{\left (e x + d\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 = 9.61693, size = 1790, normalized size = 5.77 \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{3}{2}} \sqrt{g x + f}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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