Optimal. Leaf size=36 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {623}
\begin {gather*} \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 623
Rubi steps
\begin {align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 25, normalized size = 0.69 \begin {gather*} \frac {(d+e x) \left (c (d+e x)^2\right )^{5/2}}{6 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.52, size = 33, normalized size = 0.92
method | result | size |
risch | \(\frac {c^{2} \left (e x +d \right )^{5} \sqrt {\left (e x +d \right )^{2} c}}{6 e}\) | \(27\) |
default | \(\frac {\left (e x +d \right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{6 e}\) | \(33\) |
gosper | \(\frac {x \left (e^{5} x^{5}+6 d \,e^{4} x^{4}+15 d^{2} e^{3} x^{3}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{6 \left (e x +d \right )^{5}}\) | \(84\) |
trager | \(\frac {c^{2} x \left (e^{5} x^{5}+6 d \,e^{4} x^{4}+15 d^{2} e^{3} x^{3}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{6 e x +6 d}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 53, normalized size = 1.47 \begin {gather*} \frac {1}{6} \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {5}{2}} d e^{\left (-1\right )} + \frac {1}{6} \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {5}{2}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs.
\(2 (32) = 64\).
time = 2.30, size = 101, normalized size = 2.81 \begin {gather*} \frac {{\left (c^{2} x^{6} e^{5} + 6 \, c^{2} d x^{5} e^{4} + 15 \, c^{2} d^{2} x^{4} e^{3} + 20 \, c^{2} d^{3} x^{3} e^{2} + 15 \, c^{2} d^{4} x^{2} e + 6 \, c^{2} d^{5} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{6 \, {\left (x e + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs.
\(2 (32) = 64\).
time = 1.12, size = 88, normalized size = 2.44 \begin {gather*} \frac {1}{6} \, {\left (3 \, {\left (x^{2} e + 2 \, d x\right )} c^{2} d^{4} \mathrm {sgn}\left (x e + d\right ) + 3 \, {\left (x^{2} e + 2 \, d x\right )}^{2} c^{2} d^{2} e \mathrm {sgn}\left (x e + d\right ) + {\left (x^{2} e + 2 \, d x\right )}^{3} c^{2} e^{2} \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 36, normalized size = 1.00 \begin {gather*} \frac {\left (x\,e^2+d\,e\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{6\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________