Optimal. Leaf size=233 \[ \frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.207986, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 656
Rule 648
Rubi steps
\begin{align*} \int (d+e 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 &=\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d}+\frac{\left (6 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{11 d}\\ &=\frac{4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d}+\frac{\left (8 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{33 d^2}\\ &=\frac{16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d}+\frac{\left (16 \left (d^2-\frac{a e^2}{c}\right )^3\right ) \int \frac{\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}{231 d^3}\\ &=\frac{32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 c^4 d^4 (d+e x)^{5/2}}+\frac{16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 c^3 d^3 (d+e x)^{3/2}}+\frac{4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 c d}\\ \end{align*}
Mathematica [A] time = 0.125224, size = 132, normalized size = 0.57 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (8 a^2 c d e^4 (11 d+5 e x)-16 a^3 e^6-2 a c^2 d^2 e^2 \left (99 d^2+110 d e x+35 e^2 x^2\right )+c^3 d^3 \left (495 d^2 e x+231 d^3+385 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 c^4 d^4 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 168, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -105\,{e}^{3}{x}^{3}{c}^{3}{d}^{3}+70\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-385\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-40\,{a}^{2}cd{e}^{5}x+220\,a{c}^{2}{d}^{3}{e}^{3}x-495\,{c}^{3}{d}^{5}ex+16\,{a}^{3}{e}^{6}-88\,{a}^{2}c{d}^{2}{e}^{4}+198\,a{c}^{2}{d}^{4}{e}^{2}-231\,{c}^{3}{d}^{6} \right ) }{1155\,{c}^{4}{d}^{4}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08584, size = 369, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (105 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 198 \, a^{3} c^{2} d^{4} e^{4} + 88 \, a^{4} c d^{2} e^{6} - 16 \, a^{5} e^{8} + 35 \,{\left (11 \, c^{5} d^{6} e^{2} + 4 \, a c^{4} d^{4} e^{4}\right )} x^{4} + 5 \,{\left (99 \, c^{5} d^{7} e + 110 \, a c^{4} d^{5} e^{3} + a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (77 \, c^{5} d^{8} + 264 \, a c^{4} d^{6} e^{2} + 11 \, a^{2} c^{3} d^{4} e^{4} - 2 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} +{\left (462 \, a c^{4} d^{7} e + 99 \, a^{2} c^{3} d^{5} e^{3} - 44 \, a^{3} c^{2} d^{3} e^{5} + 8 \, a^{4} c d e^{7}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{1155 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.9506, size = 614, normalized size = 2.64 \begin{align*} \frac{2 \,{\left (105 \, c^{5} d^{5} e^{3} x^{5} + 231 \, a^{2} c^{3} d^{6} e^{2} - 198 \, a^{3} c^{2} d^{4} e^{4} + 88 \, a^{4} c d^{2} e^{6} - 16 \, a^{5} e^{8} + 35 \,{\left (11 \, c^{5} d^{6} e^{2} + 4 \, a c^{4} d^{4} e^{4}\right )} x^{4} + 5 \,{\left (99 \, c^{5} d^{7} e + 110 \, a c^{4} d^{5} e^{3} + a^{2} c^{3} d^{3} e^{5}\right )} x^{3} + 3 \,{\left (77 \, c^{5} d^{8} + 264 \, a c^{4} d^{6} e^{2} + 11 \, a^{2} c^{3} d^{4} e^{4} - 2 \, a^{3} c^{2} d^{2} e^{6}\right )} x^{2} +{\left (462 \, a c^{4} d^{7} e + 99 \, a^{2} c^{3} d^{5} e^{3} - 44 \, a^{3} c^{2} d^{3} e^{5} + 8 \, a^{4} c d e^{7}\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}}{1155 \,{\left (c^{4} d^{4} e x + c^{4} d^{5}\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(-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]