3.9.31 \(\int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx\) [831]

Optimal. Leaf size=31 \[ -\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {665} \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[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*} \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)}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 31, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[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)))

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Maple [A]
time = 0.48, size = 46, normalized size = 1.48

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	\(-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{2} d \left (x +\frac {d}{e}\right )}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 29, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {-x^{2} e^{2} + d^{2}}}{d x e^{2} + d^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.69, size = 35, normalized size = 1.13 \begin {gather*} -\frac {x e + d + \sqrt {-x^{2} e^{2} + d^{2}}}{d x e^{2} + d^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]
time = 2.25, size = 38, normalized size = 1.23 \begin {gather*} \frac {2 \, e^{\left (-1\right )}}{d {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.44, size = 29, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}}{d\,e\,\left (d+e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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