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

Optimal. Leaf size=91 \[ \frac {\sqrt {c} \sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2-c}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {427, 426, 424} \[ \frac {\sqrt {c} \sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2-c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.05, size = 91, normalized size = 1.00 \[ \frac {\sqrt {-a-b x^2} \sqrt {\frac {c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt {\frac {d}{c}} x\right )|-\frac {b c}{a d}\right )}{\sqrt {\frac {d}{c}} \sqrt {\frac {a+b x^2}{a}} \sqrt {d x^2-c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-b x^{2} - a}}{\sqrt {d x^{2} - c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 110, normalized size = 1.21 \[ \frac {\sqrt {-b \,x^{2}-a}\, \sqrt {d \,x^{2}-c}\, \sqrt {-\frac {d \,x^{2}-c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, a \EllipticE \left (\sqrt {\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )}{\left (b d \,x^{4}+a d \,x^{2}-b c \,x^{2}-a c \right ) \sqrt {\frac {d}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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