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

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

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Rubi [A]  time = 0.01, 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 {\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*}

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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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-a}+e (c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.45, size = 21, normalized size = 0.91 \begin {gather*} \frac {\log \left (d e x + c e + \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, algorithm="fricas")

[Out]

log(d*e*x + c*e + sqrt(-a))/(d*e)

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giac [A]  time = 1.07, 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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maple [A]  time = 0.00, size = 22, normalized size = 0.96 \begin {gather*} \frac {\ln \left (d e x +c e +\sqrt {-a}\right )}{d e} \end {gather*}

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 = 0.67, size = 21, normalized size = 0.91 \begin {gather*} \frac {\log \left ({\left (d x + c\right )} e + \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, algorithm="maxima")

[Out]

log((d*x + c)*e + sqrt(-a))/(d*e)

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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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sympy [A]  time = 0.09, 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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