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

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

[Out]

Log[Sqrt[-a] + c*e + d*e*x]/(d*e)

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Rubi [A]  time = 0.0140041, antiderivative size = 23, normalized size of antiderivative = 1., 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} \[ \frac{\log \left (\sqrt{-a}+c e+d e x\right )}{d e} \]

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 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]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-a}+e (c+d x)} \, dx &=\frac{\operatorname{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]  time = 0.0088897, size = 23, normalized size = 1. \[ \frac{\log \left (\sqrt{-a}+c e+d e x\right )}{d e} \]

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 = 0.001, size = 22, normalized size = 1. \begin{align*}{\frac{1}{ed}\ln \left ( ce+dex+\sqrt{-a} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02679, size = 28, normalized size = 1.22 \begin{align*} \frac{\log \left ({\left (d x + c\right )} e + \sqrt{-a}\right )}{d e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((d*x + c)*e + sqrt(-a))/(d*e)

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Fricas [A]  time = 1.76498, size = 47, normalized size = 2.04 \begin{align*} \frac{\log \left (d e x + c e + \sqrt{-a}\right )}{d e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(d*e*x + c*e + sqrt(-a))/(d*e)

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Sympy [A]  time = 0.069685, size = 19, normalized size = 0.83 \begin{align*} \frac{\log{\left (c e + d e x + \sqrt{- a} \right )}}{d e} \end{align*}

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 = 1.18278, size = 30, normalized size = 1.3 \begin{align*} \frac{e^{\left (-1\right )} \log \left ({\left |{\left (d x + c\right )} e + \sqrt{-a} \right |}\right )}{d} \end{align*}

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