\(\int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx\) [882]

Optimal result

Integrand size = 29, antiderivative size = 36 \[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2 \sqrt {c d^2-c e^2 x^2}}{c e \sqrt {d+e x}} \]

[Out]

-2*(-c*e^2*x^2+c*d^2)^(1/2)/c/e/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {663} \[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2 \sqrt {c d^2-c e^2 x^2}}{c e \sqrt {d+e x}} \]

[In]

Int[Sqrt[d + e*x]/Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-2*Sqrt[c*d^2 - c*e^2*x^2])/(c*e*Sqrt[d + e*x])

Rule 663

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(p + 1))), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {c d^2-c e^2 x^2}}{c e \sqrt {d+e x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2 \sqrt {c \left (d^2-e^2 x^2\right )}}{c e \sqrt {d+e x}} \]

[In]

Integrate[Sqrt[d + e*x]/Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-2*Sqrt[c*(d^2 - e^2*x^2)])/(c*e*Sqrt[d + e*x])

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}}{\sqrt {e x +d}\, c e}\)	\(32\)
gosper	\(-\frac {2 \left (-e x +d \right ) \sqrt {e x +d}}{e \sqrt {-c \,x^{2} e^{2}+c \,d^{2}}}\)	\(36\)
risch	\(-\frac {2 \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, \left (-e x +d \right )}{\sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, e \sqrt {-c \left (e x -d \right )}}\)	\(74\)

[In]

int((e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/(e*x+d)^(1/2)*(c*(-e^2*x^2+d^2))^(1/2)/c/e

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{c e^{2} x + c d e} \]

[In]

integrate((e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(c*e^2*x + c*d*e)

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \]

[In]

integrate((e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)/sqrt(-c*(-d + e*x)*(d + e*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx=\frac {2 \, {\left (\sqrt {c} e x - \sqrt {c} d\right )}}{\sqrt {-e x + d} c e} \]

[In]

integrate((e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2*(sqrt(c)*e*x - sqrt(c)*d)/(sqrt(-e*x + d)*c*e)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx=\frac {2 \, {\left (\frac {\sqrt {2} \sqrt {c d}}{c} - \frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d}}{c}\right )}}{e} \]

[In]

integrate((e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="giac")

[Out]

2*(sqrt(2)*sqrt(c*d)/c - sqrt(-(e*x + d)*c + 2*c*d)/c)/e

Mupad [B] (verification not implemented)

Time = 9.92 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx=-\frac {2\,\sqrt {c\,d^2-c\,e^2\,x^2}}{c\,e\,\sqrt {d+e\,x}} \]

[In]

int((d + e*x)^(1/2)/(c*d^2 - c*e^2*x^2)^(1/2),x)

[Out]

-(2*(c*d^2 - c*e^2*x^2)^(1/2))/(c*e*(d + e*x)^(1/2))