\(\int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx\) [123]

Optimal result

Integrand size = 24, antiderivative size = 31 \[ \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {665} \[ \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \]

[In]

Int[1/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]

[Out]

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

Rule 665

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \]

[In]

Integrate[1/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]

[Out]

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {-e x +d}{d e \sqrt {-e^{2} x^{2}+d^{2}}}\)	\(29\)
trager	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d e \left (e x +d \right )}\)	\(30\)
default	\(-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{2} d \left (x +\frac {d}{e}\right )}\)	\(46\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx=-\frac {e x + d + \sqrt {-e^{2} x^{2} + d^{2}}}{d e^{2} x + d^{2} e} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{d e^{2} x + d^{2} e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx=\frac {2}{d {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.55 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^2}}{d\,e\,\left (d+e\,x\right )} \]

[In]

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

[Out]

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