3.3.1 \(\int \frac {2+3 x^2}{(3+5 x^2+x^4)^{3/2}} \, dx\) [201]

Optimal. Leaf size=282 \[ \frac {4 x \left (5+\sqrt {13}+2 x^2\right )}{39 \sqrt {3+5 x^2+x^4}}-\frac {x \left (7+8 x^2\right )}{39 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{39 \sqrt {3+5 x^2+x^4}}+\frac {11 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}} \]

[Out]

-1/39*x*(8*x^2+7)/(x^4+5*x^2+3)^(1/2)+4/39*x*(5+2*x^2+13^(1/2))/(x^4+5*x^2+3)^(1/2)-2/117*(1/(36+x^2*(30+6*13^
(1/2))))^(1/2)*(36+x^2*(30+6*13^(1/2)))^(1/2)*EllipticE(x*(30+6*13^(1/2))^(1/2)/(36+x^2*(30+6*13^(1/2)))^(1/2)
,1/6*(-78+30*13^(1/2))^(1/2))*(6+x^2*(5+13^(1/2)))*(30+6*13^(1/2))^(1/2)*((6+x^2*(5-13^(1/2)))/(6+x^2*(5+13^(1
/2))))^(1/2)/(x^4+5*x^2+3)^(1/2)+11/13*(1/(36+x^2*(30+6*13^(1/2))))^(1/2)*(36+x^2*(30+6*13^(1/2)))^(1/2)*Ellip
ticF(x*(30+6*13^(1/2))^(1/2)/(36+x^2*(30+6*13^(1/2)))^(1/2),1/6*(-78+30*13^(1/2))^(1/2))*(6+x^2*(5+13^(1/2)))*
((6+x^2*(5-13^(1/2)))/(6+x^2*(5+13^(1/2))))^(1/2)/(x^4+5*x^2+3)^(1/2)/(30+6*13^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1192, 1203, 1113, 1149} \begin {gather*} \frac {11 \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\text {ArcTan}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {x^4+5 x^2+3}}-\frac {2 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\text {ArcTan}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{39 \sqrt {x^4+5 x^2+3}}+\frac {4 x \left (2 x^2+\sqrt {13}+5\right )}{39 \sqrt {x^4+5 x^2+3}}-\frac {x \left (8 x^2+7\right )}{39 \sqrt {x^4+5 x^2+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2),x]

[Out]

(4*x*(5 + Sqrt[13] + 2*x^2))/(39*Sqrt[3 + 5*x^2 + x^4]) - (x*(7 + 8*x^2))/(39*Sqrt[3 + 5*x^2 + x^4]) - (2*Sqrt
[(2*(5 + Sqrt[13]))/3]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*Ellipt
icE[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(39*Sqrt[3 + 5*x^2 + x^4]) + (11*Sqrt[(6 + (5 - S
qrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticF[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (
-13 + 5*Sqrt[13])/6])/(13*Sqrt[6*(5 + Sqrt[13])]*Sqrt[3 + 5*x^2 + x^4])

Rule 1113

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(2*a + (b + q
)*x^2)*(Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*Elli
pticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSq
rtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1149

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b +
q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4])), x] - Simp[Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*(Sqrt[(2*a + (
b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q
/(b + q))], x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; Fre
eQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1203

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[d, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e, Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b +
q)/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {2+3 x^2}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx &=-\frac {x \left (7+8 x^2\right )}{39 \sqrt {3+5 x^2+x^4}}-\frac {1}{39} \int \frac {-33-8 x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=-\frac {x \left (7+8 x^2\right )}{39 \sqrt {3+5 x^2+x^4}}+\frac {8}{39} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx+\frac {11}{13} \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {4 x \left (5+\sqrt {13}+2 x^2\right )}{39 \sqrt {3+5 x^2+x^4}}-\frac {x \left (7+8 x^2\right )}{39 \sqrt {3+5 x^2+x^4}}-\frac {2 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{39 \sqrt {3+5 x^2+x^4}}+\frac {11 \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{13 \sqrt {6 \left (5+\sqrt {13}\right )} \sqrt {3+5 x^2+x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 10.20, size = 219, normalized size = 0.78 \begin {gather*} \frac {-2 x \left (7+8 x^2\right )+4 i \sqrt {2} \left (-5+\sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-i \sqrt {2} \left (13+4 \sqrt {13}\right ) \sqrt {\frac {-5+\sqrt {13}-2 x^2}{-5+\sqrt {13}}} \sqrt {5+\sqrt {13}+2 x^2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )}{78 \sqrt {3+5 x^2+x^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(2 + 3*x^2)/(3 + 5*x^2 + x^4)^(3/2),x]

[Out]

(-2*x*(7 + 8*x^2) + (4*I)*Sqrt[2]*(-5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[
13] + 2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(13 + 4*Sqrt[13
])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sq
rt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(78*Sqrt[3 + 5*x^2 + x^4])

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 240, normalized size = 0.85

method	result	size
risch	\(-\frac {x \left (8 x^{2}+7\right )}{39 \sqrt {x^{4}+5 x^{2}+3}}-\frac {96 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+\frac {66 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\)	\(216\)
elliptic	\(-\frac {2 \left (\frac {4}{39} x^{3}+\frac {7}{78} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}-\frac {96 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}+\frac {66 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}\)	\(217\)
default	\(-\frac {6 \left (\frac {1}{13} x^{3}+\frac {5}{26} x \right )}{\sqrt {x^{4}+5 x^{2}+3}}+\frac {66 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {96 \sqrt {1-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-30+6 \sqrt {13}}}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{13 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (5+\sqrt {13}\right )}-\frac {4 \left (-\frac {19}{78} x -\frac {5}{78} x^{3}\right )}{\sqrt {x^{4}+5 x^{2}+3}}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+2)/(x^4+5*x^2+3)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-6*(1/13*x^3+5/26*x)/(x^4+5*x^2+3)^(1/2)+66/13/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-5
/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)*EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2
))-96/13/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2
+3)^(1/2)/(5+13^(1/2))*(EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))-EllipticE(1/6*x*(-30+
6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2)))-4*(-19/78*x-5/78*x^3)/(x^4+5*x^2+3)^(1/2)

________________________________________________________________________________________

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((3*x^2+2)/(x^4+5*x^2+3)^(3/2),x, algorithm="maxima")

[Out]

integrate((3*x^2 + 2)/(x^4 + 5*x^2 + 3)^(3/2), x)

________________________________________________________________________________________

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((3*x^2+2)/(x^4+5*x^2+3)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{2} + 2}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+2)/(x**4+5*x**2+3)**(3/2),x)

[Out]

Integral((3*x**2 + 2)/(x**4 + 5*x**2 + 3)**(3/2), x)

________________________________________________________________________________________

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((3*x^2+2)/(x^4+5*x^2+3)^(3/2),x, algorithm="giac")

[Out]

integrate((3*x^2 + 2)/(x^4 + 5*x^2 + 3)^(3/2), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {3\,x^2+2}{{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 + 2)/(5*x^2 + x^4 + 3)^(3/2),x)

[Out]

int((3*x^2 + 2)/(5*x^2 + x^4 + 3)^(3/2), x)

________________________________________________________________________________________