Optimal. Leaf size=41 \[ -\frac {c^2}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {656, 621}
\begin {gather*} -\frac {c^2}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 621
Rule 656
Rubi steps
\begin {align*} \int \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{(d+e x)^6} \, dx &=c^3 \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {c^2}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 27, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {c (d+e x)^2}}{4 e (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.54, size = 35, normalized size = 0.85
method | result | size |
risch | \(-\frac {\sqrt {\left (e x +d \right )^{2} c}}{4 \left (e x +d \right )^{5} e}\) | \(24\) |
gosper | \(-\frac {\sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{4 \left (e x +d \right )^{5} e}\) | \(35\) |
default | \(-\frac {\sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{4 \left (e x +d \right )^{5} e}\) | \(35\) |
trager | \(\frac {\left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{4 d^{4} \left (e x +d \right )^{5}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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. 75 vs.
\(2 (37) = 74\).
time = 4.16, size = 75, normalized size = 1.83 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{4 \, {\left (x^{5} e^{6} + 5 \, d x^{4} e^{5} + 10 \, d^{2} x^{3} e^{4} + 10 \, d^{3} x^{2} e^{3} + 5 \, d^{4} x e^{2} + d^{5} e\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 \frac {\sqrt {c \left (d + e x\right )^{2}}}{\left (d + e x\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.95, size = 22, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {c} e^{\left (-1\right )} \mathrm {sgn}\left (x e + d\right )}{4 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.49, size = 34, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{4\,e\,{\left (d+e\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________