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

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

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 c e (d+e x)^{3/2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 1.05 \begin {gather*} -\frac {2 (d-e x) \sqrt {c \left (d^2-e^2 x^2\right )}}{3 e \sqrt {d+e x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.09, size = 47, normalized size = 1.24 \begin {gather*} \frac {2 (e x-d) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{3 e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 44, normalized size = 1.16 \begin {gather*} \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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-c e^{2} x^{2} + c d^{2}}}{\sqrt {e x + d}}\,{d x} \end {gather*}

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)

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maple [A]  time = 0.05, size = 36, normalized size = 0.95 \begin {gather*} -\frac {2 \left (-e x +d \right ) \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{3 \sqrt {e x +d}\, e} \end {gather*}

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 = 1.48, size = 26, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (\sqrt {c} e x - \sqrt {c} d\right )} \sqrt {-e x + d}}{3 \, e} \end {gather*}

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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mupad [B]  time = 0.51, size = 35, normalized size = 0.92 \begin {gather*} \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x}{3}-\frac {2\,d}{3\,e}\right )}{\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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