Optimal. Leaf size=113 \[ \frac{6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt{d+e x}}-\frac{2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac{2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}+\frac{2 c^3 d^3 \sqrt{d+e x}}{e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0530466, antiderivative size = 113, 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} \[ \frac{6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt{d+e x}}-\frac{2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac{2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}+\frac{2 c^3 d^3 \sqrt{d+e x}}{e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 43
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 )^3}{(d+e x)^{13/2}} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{7/2}}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^{5/2}}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^{3/2}}+\frac{c^3 d^3}{e^3 \sqrt{d+e x}}\right ) \, dx\\ &=\frac{2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}-\frac{2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac{6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 c^3 d^3 \sqrt{d+e x}}{e^4}\\ \end{align*}
Mathematica [A] time = 0.0561081, size = 109, normalized size = 0.96 \[ -\frac{2 \left (a^2 c d e^4 (2 d+5 e x)+a^3 e^6+a c^2 d^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-c^3 d^3 \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )\right )}{5 e^4 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 130, normalized size = 1.2 \begin{align*} -{\frac{-10\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+30\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-60\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+10\,{a}^{2}cd{e}^{5}x+40\,a{c}^{2}{d}^{3}{e}^{3}x-80\,{c}^{3}{d}^{5}ex+2\,{a}^{3}{e}^{6}+4\,{a}^{2}c{d}^{2}{e}^{4}+16\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{5\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04851, size = 189, normalized size = 1.67 \begin{align*} \frac{2 \,{\left (\frac{5 \, \sqrt{e x + d} c^{3} d^{3}}{e^{3}} + \frac{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 15 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{2} - 5 \,{\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 )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{3}}\right )}}{5 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.93331, size = 323, normalized size = 2.86 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e^{3} x^{3} + 16 \, c^{3} d^{6} - 8 \, a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 15 \,{\left (2 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (8 \, c^{3} d^{5} e - 4 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{5 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 47.2321, size = 654, normalized size = 5.79 \begin{align*} \begin{cases} - \frac{2 a^{3} e^{6}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{4 a^{2} c d^{2} e^{4}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{10 a^{2} c d e^{5} x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{16 a c^{2} d^{4} e^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{40 a c^{2} d^{3} e^{3} x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} - \frac{30 a c^{2} d^{2} e^{4} x^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{32 c^{3} d^{6}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{80 c^{3} d^{5} e x}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{60 c^{3} d^{4} e^{2} x^{2}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} + \frac{10 c^{3} d^{3} e^{3} x^{3}}{5 d^{2} e^{4} \sqrt{d + e x} + 10 d e^{5} x \sqrt{d + e x} + 5 e^{6} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{3} x^{4}}{4 \sqrt{d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23598, size = 252, normalized size = 2.23 \begin{align*} 2 \, \sqrt{x e + d} c^{3} d^{3} e^{\left (-4\right )} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{5} c^{3} d^{4} - 5 \,{\left (x e + d\right )}^{4} c^{3} d^{5} +{\left (x e + d\right )}^{3} c^{3} d^{6} - 15 \,{\left (x e + d\right )}^{5} a c^{2} d^{2} e^{2} + 10 \,{\left (x e + d\right )}^{4} a c^{2} d^{3} e^{2} - 3 \,{\left (x e + d\right )}^{3} a c^{2} d^{4} e^{2} - 5 \,{\left (x e + d\right )}^{4} a^{2} c d e^{4} + 3 \,{\left (x e + d\right )}^{3} a^{2} c d^{2} e^{4} -{\left (x e + d\right )}^{3} a^{3} e^{6}\right )} e^{\left (-4\right )}}{5 \,{\left (x e + d\right )}^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]