Optimal. Leaf size=129 \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.148876, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 872
Rule 860
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} (f+g x)^{11/2}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{9/2}}+\frac{(2 c d) \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} (f+g x)^{9/2}} \, dx}{9 (c d f-a e g)}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{9/2}}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0874178, size = 79, normalized size = 0.61 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} (c d (9 f+2 g x)-7 a e g)}{63 \sqrt{d+e x} (f+g x)^{9/2} (c d f-a e g)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.053, size = 99, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,xcdg+7\,aeg-9\,cdf \right ) }{63\,{a}^{2}{e}^{2}{g}^{2}-126\,acdefg+63\,{c}^{2}{d}^{2}{f}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( gx+f \right ) ^{-{\frac{9}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}}{\left (g x + f\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.70142, size = 1276, normalized size = 9.89 \begin{align*} \frac{2 \,{\left (2 \, c^{4} d^{4} g x^{4} + 9 \, a^{3} c d e^{3} f - 7 \, a^{4} e^{4} g +{\left (9 \, c^{4} d^{4} f - a c^{3} d^{3} e g\right )} x^{3} + 3 \,{\left (9 \, a c^{3} d^{3} e f - 5 \, a^{2} c^{2} d^{2} e^{2} g\right )} x^{2} +{\left (27 \, a^{2} c^{2} d^{2} e^{2} f - 19 \, a^{3} c d e^{3} g\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}}{63 \,{\left (c^{2} d^{3} f^{7} - 2 \, a c d^{2} e f^{6} g + a^{2} d e^{2} f^{5} g^{2} +{\left (c^{2} d^{2} e f^{2} g^{5} - 2 \, a c d e^{2} f g^{6} + a^{2} e^{3} g^{7}\right )} x^{6} +{\left (5 \, c^{2} d^{2} e f^{3} g^{4} + a^{2} d e^{2} g^{7} +{\left (c^{2} d^{3} - 10 \, a c d e^{2}\right )} f^{2} g^{5} -{\left (2 \, a c d^{2} e - 5 \, a^{2} e^{3}\right )} f g^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{2} d^{2} e f^{4} g^{3} + a^{2} d e^{2} f g^{6} +{\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{3} g^{4} - 2 \,{\left (a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{5}\right )} x^{4} + 10 \,{\left (c^{2} d^{2} e f^{5} g^{2} + a^{2} d e^{2} f^{2} g^{5} +{\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{4} g^{3} -{\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{4}\right )} x^{3} + 5 \,{\left (c^{2} d^{2} e f^{6} g + 2 \, a^{2} d e^{2} f^{3} g^{4} + 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} f^{5} g^{2} -{\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{4} g^{3}\right )} x^{2} +{\left (c^{2} d^{2} e f^{7} + 5 \, a^{2} d e^{2} f^{4} g^{3} +{\left (5 \, c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{6} g -{\left (10 \, a c d^{2} e - a^{2} e^{3}\right )} f^{5} g^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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}}{\left (g x + f\right )}^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]