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

Optimal. Leaf size=36 \[ -\frac{2 \sqrt{c d^2-c e^2 x^2}}{c e \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.0127333, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {649} \[ -\frac{2 \sqrt{c d^2-c e^2 x^2}}{c e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

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

Rule 649

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x}}{\sqrt{c d^2-c e^2 x^2}} \, dx &=-\frac{2 \sqrt{c d^2-c e^2 x^2}}{c e \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0439942, size = 35, normalized size = 0.97 \[ -\frac{2 \sqrt{c \left (d^2-e^2 x^2\right )}}{c e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

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

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Maple [A]  time = 0.042, size = 36, normalized size = 1. \begin{align*} -2\,{\frac{ \left ( -ex+d \right ) \sqrt{ex+d}}{e\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13023, size = 39, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (\sqrt{c} e x - \sqrt{c} d\right )}}{\sqrt{-e x + d} c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.11583, size = 82, normalized size = 2.28 \begin{align*} -\frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{c e^{2} x + c d e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d + e x}}{\sqrt{- c \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((e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)/sqrt(-c*(-d + e*x)*(d + e*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{\sqrt{-c e^{2} x^{2} + c d^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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