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

Optimal. Leaf size=198 \[ -\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {b x^2-a} \sqrt {-c-d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{d \sqrt {b x^2-a} \sqrt {\frac {d x^2}{c}+1}} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {423, 427, 426, 424, 421, 419} \[ -\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) F\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {b x^2-a} \sqrt {-c-d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{d \sqrt {b x^2-a} \sqrt {\frac {d x^2}{c}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 423

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

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

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Mathematica [A]  time = 0.05, size = 93, normalized size = 0.47 \[ \frac {\sqrt {b x^2-a} \sqrt {\frac {c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {b c}{a d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {\frac {a-b x^2}{a}} \sqrt {-c-d x^2}} \]

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 = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b x^{2} - a} \sqrt {-d x^{2} - c}}{d x^{2} + c}, x\right ) \]

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)/(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 = 167, normalized size = 0.84 \[ \frac {\sqrt {b \,x^{2}-a}\, \sqrt {-d \,x^{2}-c}\, \sqrt {-\frac {b \,x^{2}-a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \left (a d \EllipticF \left (\sqrt {\frac {b}{a}}\, x , \sqrt {-\frac {a d}{b c}}\right )-b c \EllipticE \left (\sqrt {\frac {b}{a}}\, x , \sqrt {-\frac {a d}{b c}}\right )+b c \EllipticF \left (\sqrt {\frac {b}{a}}\, x , \sqrt {-\frac {a d}{b c}}\right )\right )}{\left (b d \,x^{4}-a d \,x^{2}+b c \,x^{2}-a c \right ) \sqrt {\frac {b}{a}}\, d} \]

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

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

[Out]

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

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

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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