3.883 \(\int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} e} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {661, 208} \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} e} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2]),x]

[Out]

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 661

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(2*c*d + e^2*x^2
), x], x, Sqrt[a + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx &=(2 e) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} e}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 86, normalized size = 1.32 \[ -\frac {\sqrt {2} \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {d} e \sqrt {c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2]),x]

[Out]

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

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fricas [A]  time = 0.91, size = 169, normalized size = 2.60 \[ \left [\frac {\sqrt {2} \sqrt {\frac {1}{c d}} \log \left (-\frac {e^{2} x^{2} - 2 \, d e x + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {\frac {1}{c d}} - 3 \, d^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, e}, -\frac {\sqrt {2} \sqrt {-\frac {1}{c d}} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {-\frac {1}{c d}}}{e^{2} x^{2} - d^{2}}\right )}{e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(2)*sqrt(1/(c*d))*log(-(e^2*x^2 - 2*d*e*x + 2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*d*sqrt(1
/(c*d)) - 3*d^2)/(e^2*x^2 + 2*d*e*x + d^2))/e, -sqrt(2)*sqrt(-1/(c*d))*arctan(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)
*sqrt(e*x + d)*d*sqrt(-1/(c*d))/(e^2*x^2 - d^2))/e]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 74, normalized size = 1.14 \[ -\frac {\sqrt {-\left (e^{2} x^{2}-d^{2}\right ) c}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )}{\sqrt {e x +d}\, \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {c\,d^2-c\,e^2\,x^2}\,\sqrt {d+e\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )} \sqrt {d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-c*(-d + e*x)*(d + e*x))*sqrt(d + e*x)), x)

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