\(\int \frac {x^2}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx\) [994]

Optimal result

Integrand size = 24, antiderivative size = 88 \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\frac {x \sqrt {c+d x^2}}{d \sqrt {4+x^2}}-\frac {\sqrt {c+d x^2} E\left (\arctan \left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{d \sqrt {4+x^2} \sqrt {\frac {c+d x^2}{c \left (4+x^2\right )}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {506, 422} \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\frac {x \sqrt {c+d x^2}}{d \sqrt {x^2+4}}-\frac {\sqrt {c+d x^2} E\left (\arctan \left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{d \sqrt {x^2+4} \sqrt {\frac {c+d x^2}{c \left (x^2+4\right )}}} \]

[In]

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

[Out]

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

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {c+d x^2}}{d \sqrt {4+x^2}}-\frac {4 \int \frac {\sqrt {c+d x^2}}{\left (4+x^2\right )^{3/2}} \, dx}{d} \\ & = \frac {x \sqrt {c+d x^2}}{d \sqrt {4+x^2}}-\frac {\sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {x}{2}\right )|1-\frac {4 d}{c}\right )}{d \sqrt {4+x^2} \sqrt {\frac {c+d x^2}{c \left (4+x^2\right )}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=-\frac {i c \sqrt {1+\frac {d x^2}{c}} \left (E\left (i \text {arcsinh}\left (\frac {x}{2}\right )|\frac {4 d}{c}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{2}\right ),\frac {4 d}{c}\right )\right )}{d \sqrt {c+d x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.81 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \frac {\sqrt {\frac {c}{d}}}{2}\right )-E\left (x \sqrt {-\frac {d}{c}}, \frac {\sqrt {\frac {c}{d}}}{2}\right )\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {-\frac {d}{c}}}\)	\(76\)
elliptic	\(-\frac {2 \sqrt {\left (d \,x^{2}+c \right ) \left (x^{2}+4\right )}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \frac {\sqrt {-4+\frac {c +4 d}{d}}}{2}\right )-E\left (x \sqrt {-\frac {d}{c}}, \frac {\sqrt {-4+\frac {c +4 d}{d}}}{2}\right )\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {-\frac {d}{c}}\, \sqrt {d \,x^{4}+c \,x^{2}+4 d \,x^{2}+4 c}}\)	\(124\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.10 \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=-\frac {c \sqrt {d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {4 \, d}{c}) - c \sqrt {d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {4 \, d}{c}) - \sqrt {d x^{2} + c} \sqrt {x^{2} + 4} d}{d^{2} x} \]

[In]

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

[Out]

-(c*sqrt(d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), 4*d/c) - c*sqrt(d)*x*sqrt(-c/d)*elliptic_f(arcsin(sq
rt(-c/d)/x), 4*d/c) - sqrt(d*x^2 + c)*sqrt(x^2 + 4)*d)/(d^2*x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/(sqrt(c + d*x**2)*sqrt(x**2 + 4)), 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. \[ \int \frac {x^2}{\sqrt {4+x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^2}{\sqrt {x^2+4}\,\sqrt {d\,x^2+c}} \,d x \]

[In]

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

[Out]

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