3.3.15 \(\int \frac {1}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx\) [215]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {432, 430} \begin {gather*} \frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} F\left (\text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 430

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

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), 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 {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\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 {c+d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.73, size = 87, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\frac {a-b x^2}{a}} \sqrt {\frac {c+d x^2}{c}} F\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 {c+d x^2}} \end {gather*}

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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Maple [A]
time = 0.08, size = 103, normalized size = 1.18

method	result	size
default	\(\frac {\EllipticF \left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{\sqrt {\frac {b}{a}}\, \left (-b d \,x^{4}+a d \,x^{2}-c \,x^{2} b +a c \right )}\)	\(103\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-c \,x^{2} b +a c}}\)	\(127\)

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,method=_RETURNVERBOSE)

[Out]

EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*((d*x^2+c)/c)^(1/2)*((-b*x^2+a)/a)^(1/2)*(-b*x^2+a)^(1/2)*(d*x^2+c)^
(1/2)/(b/a)^(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.16, size = 39, normalized size = 0.45 \begin {gather*} \frac {\sqrt {a c} \sqrt {\frac {b}{a}} {\rm ellipticF}\left (x \sqrt {\frac {b}{a}}, -\frac {a d}{b c}\right )}{b c} \end {gather*}

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]

sqrt(a*c)*sqrt(b/a)*ellipticF(x*sqrt(b/a), -a*d/(b*c))/(b*c)

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}} \,d x \end {gather*}

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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