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*(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)

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Rubi [A]  time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [C]  time = 0.06, size = 70, normalized size = 0.80 \[ -\frac {i c \sqrt {\frac {d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\frac {x}{2}\right )|\frac {4 d}{c}\right )-F\left (i \sinh ^{-1}\left (\frac {x}{2}\right )|\frac {4 d}{c}\right )\right )}{d \sqrt {c+d x^2}} \]

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])

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fricas [F]  time = 0.70, 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 = 76, normalized size = 0.86 \[ -\frac {2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \left (-\EllipticE \left (\sqrt {-\frac {d}{c}}\, x , \frac {\sqrt {\frac {c}{d}}}{2}\right )+\EllipticF \left (\sqrt {-\frac {d}{c}}\, x , \frac {\sqrt {\frac {c}{d}}}{2}\right )\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {-\frac {d}{c}}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

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(c + d*x**2)*sqrt(x**2 + 4)), x)

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