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

Optimal. Leaf size=88 \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]

[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))])

________________________________________________________________________________________

Rubi [A]  time = 0.0361522, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {492, 411} \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]

Antiderivative was successfully verified.

[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 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[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]

Rule 411

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

Rubi steps

\begin{align*} \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{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]  time = 0.0489789, size = 70, normalized size = 0.8 \[ -\frac{i c \sqrt{\frac{d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\frac{x}{2}\right )|\frac{4 d}{c}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{2}\right ),\frac{4 d}{c}\right )\right )}{d \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-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]  time = 0.016, size = 76, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{\sqrt{d{x}^{2}+c}}\sqrt{{\frac{d{x}^{2}+c}{c}}} \left ({\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},1/2\,\sqrt{{\frac{c}{d}}} \right ) -{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},1/2\,\sqrt{{\frac{c}{d}}} \right ) \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \end{align*}

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]

-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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{d x^{2} + c} \sqrt{x^{2} + 4}}\,{d x} \end{align*}

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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{c + d x^{2}} \sqrt{x^{2} + 4}}\, dx \end{align*}

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(c + d*x**2)*sqrt(x**2 + 4)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{d x^{2} + c} \sqrt{x^{2} + 4}}\,{d x} \end{align*}

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)