Optimal. Leaf size=36 \[ -\frac {2 \sqrt {c d^2-c e^2 x^2}}{c e \sqrt {d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {663}
\begin {gather*} -\frac {2 \sqrt {c d^2-c e^2 x^2}}{c e \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 663
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx &=-\frac {2 \sqrt {c d^2-c e^2 x^2}}{c e \sqrt {d+e x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 35, normalized size = 0.97 \begin {gather*} -\frac {2 \sqrt {c \left (d^2-e^2 x^2\right )}}{c e \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.47, size = 32, normalized size = 0.89
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}}{\sqrt {e x +d}\, c e}\) | \(32\) |
gosper | \(-\frac {2 \left (-e x +d \right ) \sqrt {e x +d}}{e \sqrt {-x^{2} c \,e^{2}+c \,d^{2}}}\) | \(36\) |
risch | \(-\frac {2 \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, \left (-e x +d \right )}{\sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 30, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (\sqrt {c} x e - \sqrt {c} d\right )} e^{\left (-1\right )}}{\sqrt {-x e + d} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.03, size = 39, normalized size = 1.08 \begin {gather*} -\frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d}}{c x e^{2} + c d e} \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 {d + e x}}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.80, size = 38, normalized size = 1.06 \begin {gather*} 2 \, {\left (\frac {\sqrt {2} \sqrt {c d}}{c} - \frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.57, size = 32, normalized size = 0.89 \begin {gather*} -\frac {2\,\sqrt {c\,d^2-c\,e^2\,x^2}}{c\,e\,\sqrt {d+e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________