3.831 \(\int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx\)

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 = {651} \[ -\frac {\sqrt {d^2-e^2 x^2}}{d e (d+e x)} \]

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 651

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 = 0.01, size = 32, normalized size = 1.03 \[ -\frac {\sqrt {d^2-e^2 x^2}}{d^2 e+d e^2 x} \]

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^2*e + d*e^2*x))

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fricas [A]  time = 0.63, size = 35, normalized size = 1.13 \[ -\frac {e x + d + \sqrt {-e^{2} x^{2} + d^{2}}}{d e^{2} x + d^{2} e} \]

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]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError >> Unable to parse Giac output: -2*exp(2)*atan((-1/2*(-2*d*exp(1)-2*sqrt
(d^2-x^2*exp(2))*exp(1))/x+exp(2))/sqrt(-exp(1)^4+exp(2)^2))/sqrt(-exp(1)^4+exp(2)^2)/d/exp(1)

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maple [A]  time = 0.04, size = 29, normalized size = 0.94 \[ -\frac {-e x +d}{\sqrt {-e^{2} x^{2}+d^{2}}\, d e} \]

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)

[Out]

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

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maxima [A]  time = 2.92, size = 30, normalized size = 0.97 \[ -\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{d e^{2} x + d^{2} e} \]

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(-e^2*x^2 + d^2)/(d*e^2*x + d^2*e)

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

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

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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