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

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

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Rubi [A]  time = 0.01, antiderivative size = 36, 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 \sqrt {c d^2-c e^2 x^2}}{c e \sqrt {d+e x}} \end {gather*}

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*}

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Mathematica [A]  time = 0.05, size = 35, normalized size = 0.97 \begin {gather*} -\frac {2 \sqrt {c \left (d^2-e^2 x^2\right )}}{c e \sqrt {d+e x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.09, size = 41, normalized size = 1.14 \begin {gather*} -\frac {2 \sqrt {2 c d (d+e x)-c (d+e x)^2}}{c e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 39, normalized size = 1.08 \begin {gather*} -\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{c e^{2} x + c d e} \end {gather*}

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

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)

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maple [A]  time = 0.04, size = 36, normalized size = 1.00 \begin {gather*} -\frac {2 \left (-e x +d \right ) \sqrt {e x +d}}{\sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, e} \end {gather*}

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 = 1.49, size = 29, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (\sqrt {c} e x - \sqrt {c} d\right )}}{\sqrt {-e x + d} c e} \end {gather*}

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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mupad [B]  time = 0.57, size = 32, normalized size = 0.89 \begin {gather*} -\frac {2\,\sqrt {c\,d^2-c\,e^2\,x^2}}{c\,e\,\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(c*d^2 - c*e^2*x^2)^(1/2))/(c*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 {d + e x}}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \end {gather*}

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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