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

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

[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 = 0.02, 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 = {506, 422} \begin {gather*} \frac {x \sqrt {c+d x^2}}{d \sqrt {x^2+4}}-\frac {\sqrt {c+d x^2} E\left (\text {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 )}}} \end {gather*}

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 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*} \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] Result contains complex when optimal does not.
time = 0.53, size = 70, normalized size = 0.80 \begin {gather*} -\frac {i c \sqrt {1+\frac {d x^2}{c}} \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}} \end {gather*}

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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Maple [A]
time = 0.12, size = 76, normalized size = 0.86

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

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,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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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