3.993 \(\int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx\)

Optimal. Leaf size=87 \[ \frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}}-\frac {c \sqrt {\frac {d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {493, 426, 424, 421, 419} \[ \frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}}-\frac {c \sqrt {\frac {d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[c + d*x^2]*EllipticE[ArcSin[x/2], (-4*d)/c])/(d*Sqrt[1 + (d*x^2)/c]) - (c*Sqrt[1 + (d*x^2)/c]*EllipticF[
ArcSin[x/2], (-4*d)/c])/(d*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 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 493

Int[(x_)^(n_)/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/b, Int[Sqrt[a +
b*x^n]/Sqrt[c + d*x^n], x], x] - Dist[a/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c,
d}, x] && NeQ[b*c - a*d, 0] && (EqQ[n, 2] || EqQ[n, 4]) &&  !(EqQ[n, 2] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx &=\frac {\int \frac {\sqrt {c+d x^2}}{\sqrt {4-x^2}} \, dx}{d}-\frac {c \int \frac {1}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx}{d}\\ &=\frac {\sqrt {c+d x^2} \int \frac {\sqrt {1+\frac {d x^2}{c}}}{\sqrt {4-x^2}} \, dx}{d \sqrt {1+\frac {d x^2}{c}}}-\frac {\left (c \sqrt {1+\frac {d x^2}{c}}\right ) \int \frac {1}{\sqrt {4-x^2} \sqrt {1+\frac {d x^2}{c}}} \, dx}{d \sqrt {c+d x^2}}\\ &=\frac {\sqrt {c+d x^2} E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {1+\frac {d x^2}{c}}}-\frac {c \sqrt {1+\frac {d x^2}{c}} F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 59, normalized size = 0.68 \[ \frac {c \sqrt {\frac {d x^2}{c}+1} \left (E\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )-F\left (\sin ^{-1}\left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )\right )}{d \sqrt {c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(c*Sqrt[1 + (d*x^2)/c]*(EllipticE[ArcSin[x/2], (-4*d)/c] - EllipticF[ArcSin[x/2], (-4*d)/c]))/(d*Sqrt[c + d*x^
2])

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {d x^{2} + c} \sqrt {-x^{2} + 4} x^{2}}{d x^{4} + {\left (c - 4 \, d\right )} x^{2} - 4 \, c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(d*x^2 + c)*sqrt(-x^2 + 4)*x^2/(d*x^4 + (c - 4*d)*x^2 - 4*c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {d x^{2} + c} \sqrt {-x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(x^2/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)

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maple [A]  time = 0.05, size = 59, normalized size = 0.68 \[ \frac {\left (\EllipticE \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )-\EllipticF \left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )\right ) \sqrt {\frac {d \,x^{2}+c}{c}}\, c}{\sqrt {d \,x^{2}+c}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {d x^{2} + c} \sqrt {-x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{\sqrt {4-x^2}\,\sqrt {d\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((4 - x^2)^(1/2)*(c + d*x^2)^(1/2)),x)

[Out]

int(x^2/((4 - 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 \[ \int \frac {x^{2}}{\sqrt {- \left (x - 2\right ) \left (x + 2\right )} \sqrt {c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

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

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