Optimal. Leaf size=194 \[ \frac{16 g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}+\frac{8 g \sqrt{d+e x}}{3 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 \sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
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Rubi [A] time = 0.219604, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {868, 860} \[ \frac{16 g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^3}+\frac{8 g \sqrt{d+e x}}{3 \sqrt{f+g x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 \sqrt{f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 868
Rule 860
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^{3/2}}{3 (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}}-\frac{(4 g) \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}{3 (c d f-a e g)}\\ &=-\frac{2 (d+e x)^{3/2}}{3 (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}}+\frac{8 g \sqrt{d+e x}}{3 (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}}+\frac{\left (8 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}{3 (c d f-a e g)^2}\\ &=-\frac{2 (d+e x)^{3/2}}{3 (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}}+\frac{8 g \sqrt{d+e x}}{3 (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}}+\frac{16 g^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt{d+e x} \sqrt{f+g x}}\\ \end{align*}
Mathematica [A] time = 0.0780404, size = 103, normalized size = 0.53 \[ \frac{2 (d+e x)^{3/2} \left (3 a^2 e^2 g^2+6 a c d e g (f+2 g x)+c^2 d^2 \left (-f^2+4 f g x+8 g^2 x^2\right )\right )}{3 \sqrt{f+g x} ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 169, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{g}^{2}{x}^{2}+12\,acde{g}^{2}x+4\,{c}^{2}{d}^{2}fgx+3\,{a}^{2}{e}^{2}{g}^{2}+6\,acdefg-{c}^{2}{d}^{2}{f}^{2} \right ) }{3\,{a}^{3}{e}^{3}{g}^{3}-9\,{a}^{2}cd{e}^{2}f{g}^{2}+9\,a{c}^{2}{d}^{2}e{f}^{2}g-3\,{c}^{3}{d}^{3}{f}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{gx+f}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{5}{2}}}} \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{5}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91612, size = 1281, normalized size = 6.6 \begin{align*} \frac{2 \,{\left (8 \, c^{2} d^{2} g^{2} x^{2} - c^{2} d^{2} f^{2} + 6 \, a c d e f g + 3 \, a^{2} e^{2} g^{2} + 4 \,{\left (c^{2} d^{2} f g + 3 \, a c d e g^{2}\right )} x\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}}{3 \,{\left (a^{2} c^{3} d^{4} e^{2} f^{4} - 3 \, a^{3} c^{2} d^{3} e^{3} f^{3} g + 3 \, a^{4} c d^{2} e^{4} f^{2} g^{2} - a^{5} d e^{5} f g^{3} +{\left (c^{5} d^{5} e f^{3} g - 3 \, a c^{4} d^{4} e^{2} f^{2} g^{2} + 3 \, a^{2} c^{3} d^{3} e^{3} f g^{3} - a^{3} c^{2} d^{2} e^{4} g^{4}\right )} x^{4} +{\left (c^{5} d^{5} e f^{4} +{\left (c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} f^{3} g - 3 \,{\left (a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} f^{2} g^{2} +{\left (3 \, a^{2} c^{3} d^{4} e^{2} + 5 \, a^{3} c^{2} d^{2} e^{4}\right )} f g^{3} -{\left (a^{3} c^{2} d^{3} e^{3} + 2 \, a^{4} c d e^{5}\right )} g^{4}\right )} x^{3} +{\left ({\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} f^{4} -{\left (a c^{4} d^{5} e + 5 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{3} g - 3 \,{\left (a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g^{2} +{\left (5 \, a^{3} c^{2} d^{3} e^{3} + a^{4} c d e^{5}\right )} f g^{3} -{\left (2 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} g^{4}\right )} x^{2} -{\left (a^{5} d e^{5} g^{4} -{\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} f^{4} +{\left (5 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{3} g - 3 \,{\left (a^{3} c^{2} d^{3} e^{3} + a^{4} c d e^{5}\right )} f^{2} g^{2} -{\left (a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} f g^{3}\right )} x\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*} \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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