3.216 \(\int \frac {1}{\sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx\)

Optimal. Leaf size=87 \[ \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {421, 419} \[ \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 419

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

Rule 421

Int[1/(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[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 89, normalized size = 1.02 \[ \frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {\frac {c-d x^2}{c}} \operatorname {EllipticF}\left (\sin ^{-1}\left (x \sqrt {-\frac {b}{a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x^{2} + a} \sqrt {-d x^{2} + c}}{b d x^{4} - {\left (b c - a d\right )} x^{2} - a c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 106, normalized size = 1.22 \[ -\frac {\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {-\frac {d \,x^{2}-c}{c}}\, \sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}\, \EllipticF \left (\sqrt {\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \left (b d \,x^{4}+a d \,x^{2}-b c \,x^{2}-a c \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(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 \[ \int \frac {1}{\sqrt {b\,x^2+a}\,\sqrt {c-d\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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