Optimal. Leaf size=119 \[ -\frac{6 c^2 d^2 (d+e x)^{13/2} \left (c d^2-a e^2\right )}{13 e^4}+\frac{6 c d (d+e x)^{11/2} \left (c d^2-a e^2\right )^2}{11 e^4}-\frac{2 (d+e x)^{9/2} \left (c d^2-a e^2\right )^3}{9 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0816524, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {626, 43} \[ -\frac{6 c^2 d^2 (d+e x)^{13/2} \left (c d^2-a e^2\right )}{13 e^4}+\frac{6 c d (d+e x)^{11/2} \left (c d^2-a e^2\right )^2}{11 e^4}-\frac{2 (d+e x)^{9/2} \left (c d^2-a e^2\right )^3}{9 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 43
Rubi steps
\begin{align*} \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 \, dx &=\int (a e+c d x)^3 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3 (d+e x)^{7/2}}{e^3}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{9/2}}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{11/2}}{e^3}+\frac{c^3 d^3 (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac{2 \left (c d^2-a e^2\right )^3 (d+e x)^{9/2}}{9 e^4}+\frac{6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{11/2}}{11 e^4}-\frac{6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{13/2}}{13 e^4}+\frac{2 c^3 d^3 (d+e x)^{15/2}}{15 e^4}\\ \end{align*}
Mathematica [A] time = 0.0915999, size = 111, normalized size = 0.93 \[ \frac{2 (d+e x)^{9/2} \left (-195 a^2 c d e^4 (2 d-9 e x)+715 a^3 e^6+15 a c^2 d^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+c^3 d^3 \left (72 d^2 e x-16 d^3-198 d e^2 x^2+429 e^3 x^3\right )\right )}{6435 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 131, normalized size = 1.1 \begin{align*}{\frac{858\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+2970\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-396\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+3510\,{a}^{2}cd{e}^{5}x-1080\,a{c}^{2}{d}^{3}{e}^{3}x+144\,{c}^{3}{d}^{5}ex+1430\,{a}^{3}{e}^{6}-780\,{a}^{2}c{d}^{2}{e}^{4}+240\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03659, size = 185, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{3} d^{3} - 1485 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1755 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 715 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{6435 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.91741, size = 743, normalized size = 6.24 \begin{align*} \frac{2 \,{\left (429 \, c^{3} d^{3} e^{7} x^{7} - 16 \, c^{3} d^{10} + 120 \, a c^{2} d^{8} e^{2} - 390 \, a^{2} c d^{6} e^{4} + 715 \, a^{3} d^{4} e^{6} + 33 \,{\left (46 \, c^{3} d^{4} e^{6} + 45 \, a c^{2} d^{2} e^{8}\right )} x^{6} + 9 \,{\left (206 \, c^{3} d^{5} e^{5} + 600 \, a c^{2} d^{3} e^{7} + 195 \, a^{2} c d e^{9}\right )} x^{5} + 5 \,{\left (160 \, c^{3} d^{6} e^{4} + 1374 \, a c^{2} d^{4} e^{6} + 1326 \, a^{2} c d^{2} e^{8} + 143 \, a^{3} e^{10}\right )} x^{4} + 5 \,{\left (c^{3} d^{7} e^{3} + 636 \, a c^{2} d^{5} e^{5} + 1794 \, a^{2} c d^{3} e^{7} + 572 \, a^{3} d e^{9}\right )} x^{3} - 3 \,{\left (2 \, c^{3} d^{8} e^{2} - 15 \, a c^{2} d^{6} e^{4} - 1560 \, a^{2} c d^{4} e^{6} - 1430 \, a^{3} d^{2} e^{8}\right )} x^{2} +{\left (8 \, c^{3} d^{9} e - 60 \, a c^{2} d^{7} e^{3} + 195 \, a^{2} c d^{5} e^{5} + 2860 \, a^{3} d^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{6435 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.85638, size = 165, normalized size = 1.39 \begin{align*} \frac{2 \left (\frac{c^{3} d^{3} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{3}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (3 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right )}{13 e^{3}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (3 a^{2} c d e^{4} - 6 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}\right )}{11 e^{3}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}\right )}{9 e^{3}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26956, size = 1258, normalized size = 10.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]