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

Optimal. Leaf size=23 \[ \frac {\log \left (\sqrt {-a}+c e+d e x\right )}{d e} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {33, 31} \begin {gather*} \frac {\log \left (\sqrt {-a}+c e+d e x\right )}{d e} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx &=\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*}

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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sqrt {-a}+c e+d e x\right )}{d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Sqrt[-a] + c*e + d*e*x]/(d*e)

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Maple [A]
time = 0.10, size = 22, normalized size = 0.96

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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Maxima [A]
time = 0.28, size = 21, normalized size = 0.91 \begin {gather*} \frac {e^{\left (-1\right )} \log \left ({\left (d x + c\right )} e + \sqrt {-a}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1)*log((d*x + c)*e + sqrt(-a))/d

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Fricas [A]
time = 1.05, size = 21, normalized size = 0.91 \begin {gather*} \frac {e^{\left (-1\right )} \log \left ({\left (d x + c\right )} e + \sqrt {-a}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1)*log((d*x + c)*e + sqrt(-a))/d

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Sympy [A]
time = 0.01, size = 19, normalized size = 0.83 \begin {gather*} \frac {\log {\left (c e + d e x + \sqrt {- a} \right )}}{d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*e + d*e*x + sqrt(-a))/(d*e)

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Giac [A]
time = 1.33, size = 22, normalized size = 0.96 \begin {gather*} \frac {e^{\left (-1\right )} \log \left ({\left | {\left (d x + c\right )} e + \sqrt {-a} \right |}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-1)*log(abs((d*x + c)*e + sqrt(-a)))/d

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Mupad [B]
time = 0.14, size = 21, normalized size = 0.91 \begin {gather*} \frac {\ln \left (\sqrt {-a}+c\,e+d\,e\,x\right )}{d\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-a)^(1/2) + e*(c + d*x)),x)

[Out]

log((-a)^(1/2) + c*e + d*e*x)/(d*e)

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