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

Optimal. Leaf size=38 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 c e (d+e x)^{3/2}} \]

[Out]

(-2*(c*d^2 - c*e^2*x^2)^(3/2))/(3*c*e*(d + e*x)^(3/2))

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Rubi [A]  time = 0.012885, antiderivative size = 38, 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} \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 c e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(c*d^2 - c*e^2*x^2)^(3/2))/(3*c*e*(d + e*x)^(3/2))

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{c d^2-c e^2 x^2}}{\sqrt{d+e x}} \, dx &=-\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 c e (d+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0374206, size = 40, normalized size = 1.05 \[ -\frac{2 (d-e x) \sqrt{c \left (d^2-e^2 x^2\right )}}{3 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d - e*x)*Sqrt[c*(d^2 - e^2*x^2)])/(3*e*Sqrt[d + e*x])

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Maple [A]  time = 0.04, size = 36, normalized size = 1. \begin{align*} -{\frac{-2\,ex+2\,d}{3\,e}\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-e*x+d)*(-c*e^2*x^2+c*d^2)^(1/2)/e/(e*x+d)^(1/2)

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Maxima [A]  time = 1.06111, size = 35, normalized size = 0.92 \begin{align*} \frac{2 \,{\left (\sqrt{c} e x - \sqrt{c} d\right )} \sqrt{-e x + d}}{3 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(c)*e*x - sqrt(c)*d)*sqrt(-e*x + d)/e

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Fricas [A]  time = 2.01604, size = 92, normalized size = 2.42 \begin{align*} \frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}{\left (e x - d\right )}}{3 \,{\left (e^{2} x + d e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*(e*x - d)/(e^2*x + d*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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