3.279 \(\int \frac {\sqrt {-c+d x^2}}{\sqrt {a-b x^2}} \, dx\)

Optimal. Leaf size=89 \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2-c} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}}} \]

[Out]

EllipticE(x*b^(1/2)/a^(1/2),(a*d/b/c)^(1/2))*a^(1/2)*(1-b*x^2/a)^(1/2)*(d*x^2-c)^(1/2)/b^(1/2)/(-b*x^2+a)^(1/2
)/(1-d*x^2/c)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {427, 426, 424} \[ \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2-c} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*Sqrt[1 - (b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], (a*d)/(b*c)])/(Sqrt[b]*Sq
rt[a - b*x^2]*Sqrt[1 - (d*x^2)/c])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 427

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {-c+d x^2}}{\sqrt {a-b x^2}} \, dx &=\frac {\sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {-c+d x^2}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{\sqrt {a-b x^2}}\\ &=\frac {\left (\sqrt {1-\frac {b x^2}{a}} \sqrt {-c+d x^2}\right ) \int \frac {\sqrt {1-\frac {d x^2}{c}}}{\sqrt {1-\frac {b x^2}{a}}} \, dx}{\sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}}}\\ &=\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c+d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 89, normalized size = 1.00 \[ \frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {d x^2-c} E\left (\sin ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} \sqrt {a-b x^2} \sqrt {\frac {c-d x^2}{c}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a - b*x^2)/a]*Sqrt[-c + d*x^2]*EllipticE[ArcSin[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*Sqrt[a - b*x^2]*
Sqrt[(c - d*x^2)/c])

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{2} + a} \sqrt {d x^{2} - c}}{b x^{2} - a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-b*x^2 + a)*sqrt(d*x^2 - c)/(b*x^2 - a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 110, normalized size = 1.24 \[ \frac {\sqrt {d \,x^{2}-c}\, \sqrt {-b \,x^{2}+a}\, \sqrt {-\frac {b \,x^{2}-a}{a}}\, \sqrt {-\frac {d \,x^{2}-c}{c}}\, c \EllipticE \left (\sqrt {\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )}{\left (b d \,x^{4}-a d \,x^{2}-b c \,x^{2}+a c \right ) \sqrt {\frac {b}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x^2-c)^(1/2)*(-b*x^2+a)^(1/2)*c*(-(b*x^2-a)/a)^(1/2)*(-(d*x^2-c)/c)^(1/2)*EllipticE((1/a*b)^(1/2)*x,(a/b/c*
d)^(1/2))/(b*d*x^4-a*d*x^2-b*c*x^2+a*c)/(1/a*b)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {d\,x^2-c}}{\sqrt {a-b\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2 - c)^(1/2)/(a - b*x^2)^(1/2),x)

[Out]

int((d*x^2 - c)^(1/2)/(a - b*x^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c + d x^{2}}}{\sqrt {a - b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-c + d*x**2)/sqrt(a - b*x**2), x)

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