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

Optimal. Leaf size=189 \[ \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {423, 427, 426, 424, 421, 419} \[ \frac {\sqrt {c} \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.05, size = 89, normalized size = 0.47 \[ \frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {c-d x^2} E\left (\sin ^{-1}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {a+b x^2} \sqrt {\frac {c-d x^2}{c}}} \]

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 = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-d x^{2} + c}}{\sqrt {b x^{2} + a}}, x\right ) \]

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(-d*x^2 + c)/sqrt(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 = 0.02, size = 164, normalized size = 0.87 \[ \frac {\left (a d \EllipticE \left (\sqrt {\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )-a d \EllipticF \left (\sqrt {\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )-b c \EllipticF \left (\sqrt {\frac {d}{c}}\, x , \sqrt {-\frac {b c}{a d}}\right )\right ) \sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, \sqrt {-\frac {d \,x^{2}-c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}}{\left (b d \,x^{4}+a d \,x^{2}-b c \,x^{2}-a c \right ) \sqrt {\frac {d}{c}}\, b} \]

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c - d*x^2)^(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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