\(\int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx\) [36]

Optimal result

Integrand size = 17, antiderivative size = 23 \[ \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx=\frac {\log \left (\sqrt {-a}+c e+d e x\right )}{d e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {33, 31} \[ \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx=\frac {\log \left (\sqrt {-a}+c e+d e x\right )}{d e} \]

[In]

Int[(Sqrt[-a] + e*(c + d*x))^(-1),x]

[Out]

Log[Sqrt[-a] + c*e + d*e*x]/(d*e)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {-a}+e x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\log \left (\sqrt {-a}+c e+d e x\right )}{d e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx=\frac {\log \left (\sqrt {-a}+c e+d e x\right )}{d e} \]

[In]

Integrate[(Sqrt[-a] + e*(c + d*x))^(-1),x]

[Out]

Log[Sqrt[-a] + c*e + d*e*x]/(d*e)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\ln \left (c e +d e x +\sqrt {-a}\right )}{d e}\)	\(22\)
norman	\(\frac {\ln \left (c e +d e x +\sqrt {-a}\right )}{d e}\)	\(22\)
parallelrisch	\(\frac {\ln \left (c e +d e x +\sqrt {-a}\right )}{d e}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx=\frac {\log \left (d e x + c e + \sqrt {-a}\right )}{d e} \]

[In]

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

[Out]

log(d*e*x + c*e + sqrt(-a))/(d*e)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx=\frac {\log {\left (c e + d e x + \sqrt {- a} \right )}}{d e} \]

[In]

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

[Out]

log(c*e + d*e*x + sqrt(-a))/(d*e)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx=\frac {\log \left ({\left (d x + c\right )} e + \sqrt {-a}\right )}{d e} \]

[In]

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

[Out]

log((d*x + c)*e + sqrt(-a))/(d*e)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx=\frac {\log \left ({\left | {\left (d x + c\right )} e + \sqrt {-a} \right |}\right )}{d e} \]

[In]

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

[Out]

log(abs((d*x + c)*e + sqrt(-a)))/(d*e)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx=\frac {\ln \left (\sqrt {-a}+c\,e+d\,e\,x\right )}{d\,e} \]

[In]

int(1/((-a)^(1/2) + e*(c + d*x)),x)

[Out]

log((-a)^(1/2) + c*e + d*e*x)/(d*e)