Optimal. Leaf size=113 \[ \frac {2 \left (c d^2-a e^2\right )^3}{e^4 \sqrt {d+e x}}+\frac {6 c d \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{e^4}-\frac {2 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{e^4}+\frac {2 c^3 d^3 (d+e x)^{5/2}}{5 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, 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 = {640, 45}
\begin {gather*} -\frac {2 c^2 d^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{e^4}+\frac {6 c d \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{e^4}+\frac {2 \left (c d^2-a e^2\right )^3}{e^4 \sqrt {d+e x}}+\frac {2 c^3 d^3 (d+e x)^{5/2}}{5 e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 640
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)^{9/2}} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{3/2}}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 \sqrt {d+e x}}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{e^3}+\frac {c^3 d^3 (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac {2 \left (c d^2-a e^2\right )^3}{e^4 \sqrt {d+e x}}+\frac {6 c d \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{e^4}-\frac {2 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{e^4}+\frac {2 c^3 d^3 (d+e x)^{5/2}}{5 e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 109, normalized size = 0.96 \begin {gather*} \frac {2 \left (-5 a^3 e^6+15 a^2 c d e^4 (2 d+e x)-5 a c^2 d^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+c^3 d^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{5 e^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.70, size = 155, normalized size = 1.37
method | result | size |
risch | \(\frac {2 c d \left (e^{2} x^{2} c^{2} d^{2}+5 a c d \,e^{3} x -3 c^{2} d^{3} e x +15 a^{2} e^{4}-25 a c \,d^{2} e^{2}+11 c^{2} d^{4}\right ) \sqrt {e x +d}}{5 e^{4}}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{e^{4} \sqrt {e x +d}}\) | \(127\) |
gosper | \(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-5 a \,c^{2} d^{2} e^{4} x^{2}+2 c^{3} d^{4} e^{2} x^{2}-15 a^{2} c d \,e^{5} x +20 a \,c^{2} d^{3} e^{3} x -8 c^{3} d^{5} e x +5 e^{6} a^{3}-30 e^{4} d^{2} a^{2} c +40 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{5 \sqrt {e x +d}\, e^{4}}\) | \(131\) |
trager | \(-\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-5 a \,c^{2} d^{2} e^{4} x^{2}+2 c^{3} d^{4} e^{2} x^{2}-15 a^{2} c d \,e^{5} x +20 a \,c^{2} d^{3} e^{3} x -8 c^{3} d^{5} e x +5 e^{6} a^{3}-30 e^{4} d^{2} a^{2} c +40 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{5 \sqrt {e x +d}\, e^{4}}\) | \(131\) |
derivativedivides | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}+2 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-2 c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}+6 a^{2} c d \,e^{4} \sqrt {e x +d}-12 a \,c^{2} d^{3} e^{2} \sqrt {e x +d}+6 c^{3} d^{5} \sqrt {e x +d}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{\sqrt {e x +d}}}{e^{4}}\) | \(155\) |
default | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}+2 a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-2 c^{3} d^{4} \left (e x +d \right )^{\frac {3}{2}}+6 a^{2} c d \,e^{4} \sqrt {e x +d}-12 a \,c^{2} d^{3} e^{2} \sqrt {e x +d}+6 c^{3} d^{5} \sqrt {e x +d}-\frac {2 \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{\sqrt {e x +d}}}{e^{4}}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 139, normalized size = 1.23 \begin {gather*} \frac {2}{5} \, {\left ({\left ({\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} - 5 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 15 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \sqrt {x e + d}\right )} e^{\left (-3\right )} + \frac {5 \, {\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 )} e^{\left (-3\right )}}{\sqrt {x e + d}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.31, size = 131, normalized size = 1.16 \begin {gather*} \frac {2 \, {\left (8 \, c^{3} d^{5} x e + 16 \, c^{3} d^{6} + 15 \, a^{2} c d x e^{5} - 5 \, a^{3} e^{6} + 5 \, {\left (a c^{2} d^{2} x^{2} + 6 \, a^{2} c d^{2}\right )} e^{4} + {\left (c^{3} d^{3} x^{3} - 20 \, a c^{2} d^{3} x\right )} e^{3} - 2 \, {\left (c^{3} d^{4} x^{2} + 20 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {x e + d}}{5 \, {\left (x e^{5} + d e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 230 vs.
\(2 (105) = 210\).
time = 1.12, size = 230, normalized size = 2.04 \begin {gather*} \begin {cases} - \frac {2 a^{3} e^{2}}{\sqrt {d + e x}} + \frac {12 a^{2} c d^{2}}{\sqrt {d + e x}} + \frac {6 a^{2} c d e x}{\sqrt {d + e x}} - \frac {16 a c^{2} d^{4}}{e^{2} \sqrt {d + e x}} - \frac {8 a c^{2} d^{3} x}{e \sqrt {d + e x}} + \frac {2 a c^{2} d^{2} x^{2}}{\sqrt {d + e x}} + \frac {32 c^{3} d^{6}}{5 e^{4} \sqrt {d + e x}} + \frac {16 c^{3} d^{5} x}{5 e^{3} \sqrt {d + e x}} - \frac {4 c^{3} d^{4} x^{2}}{5 e^{2} \sqrt {d + e x}} + \frac {2 c^{3} d^{3} x^{3}}{5 e \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {3}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.66, size = 163, normalized size = 1.44 \begin {gather*} \frac {2}{5} \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} e^{16} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{4} e^{16} + 15 \, \sqrt {x e + d} c^{3} d^{5} e^{16} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{2} e^{18} - 30 \, \sqrt {x e + d} a c^{2} d^{3} e^{18} + 15 \, \sqrt {x e + d} a^{2} c d e^{20}\right )} e^{\left (-20\right )} + \frac {2 \, {\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 )} e^{\left (-4\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 133, normalized size = 1.18 \begin {gather*} \frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4}-\frac {2\,a^3\,e^6-6\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2-2\,c^3\,d^6}{e^4\,\sqrt {d+e\,x}}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,\sqrt {d+e\,x}}{e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________