\(\int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx\) [993]

Optimal result

Integrand size = 26, antiderivative size = 87 \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} E\left (\arcsin \left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {1+\frac {d x^2}{c}}}-\frac {c \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]

[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)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {507, 437, 435, 432, 430} \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} E\left (\arcsin \left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )}{d \sqrt {\frac {d x^2}{c}+1}}-\frac {c \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-\frac {4 d}{c}\right )}{d \sqrt {c+d x^2}} \]

[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 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]

Rule 435

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

Rule 437

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

Rule 507

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.68 \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx=\frac {c \sqrt {1+\frac {d x^2}{c}} \left (E\left (\arcsin \left (\frac {x}{2}\right )|-\frac {4 d}{c}\right )-\operatorname {EllipticF}\left (\arcsin \left (\frac {x}{2}\right ),-\frac {4 d}{c}\right )\right )}{d \sqrt {c+d x^2}} \]

[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])

Maple [A] (verified)

Time = 3.70 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.68

method	result	size
default	\(\frac {\left (-F\left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )+E\left (\frac {x}{2}, 2 \sqrt {-\frac {d}{c}}\right )\right ) c \sqrt {\frac {d \,x^{2}+c}{c}}}{\sqrt {d \,x^{2}+c}\, d}\)	\(59\)
elliptic	\(-\frac {\sqrt {-\left (d \,x^{2}+c \right ) \left (x^{2}-4\right )}\, c \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )-E\left (\frac {x}{2}, \sqrt {-1-\frac {-c +4 d}{c}}\right )\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {-d \,x^{4}-c \,x^{2}+4 d \,x^{2}+4 c}\, d}\)	\(111\)

[In]

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.77 \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx=-\frac {8 \, {\left (x E(\arcsin \left (\frac {2}{x}\right )\,|\,-\frac {c}{4 \, d}) - x F(\arcsin \left (\frac {2}{x}\right )\,|\,-\frac {c}{4 \, d})\right )} \sqrt {-d} + \sqrt {d x^{2} + c} \sqrt {-x^{2} + 4}}{d x} \]

[In]

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

[Out]

-(8*(x*elliptic_e(arcsin(2/x), -1/4*c/d) - x*elliptic_f(arcsin(2/x), -1/4*c/d))*sqrt(-d) + sqrt(d*x^2 + c)*sqr
t(-x^2 + 4))/(d*x)

Sympy [F]

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

[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)

Maxima [F]

\[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {d x^{2} + c} \sqrt {-x^{2} + 4}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {d x^{2} + c} \sqrt {-x^{2} + 4}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {4-x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^2}{\sqrt {4-x^2}\,\sqrt {d\,x^2+c}} \,d x \]

[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)