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]

-(Sqrt[d^2 - e^2*x^2]/(d*e*(d + e*x)))

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Rubi [A]  time = 0.0094422, antiderivative size = 31, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0057505, 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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Maple [A]  time = 0.045, size = 29, normalized size = 0.9 \begin{align*} -{\frac{-ex+d}{de}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.01641, size = 72, normalized size = 2.32 \begin{align*} -\frac{e x + d + \sqrt{-e^{2} x^{2} + d^{2}}}{d e^{2} x + d^{2} e} \end{align*}

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

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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