3.2.88 \(\int \frac {2+3 x^2}{x^7 \sqrt {3+5 x^2+x^4}} \, dx\) [188]

Optimal. Leaf size=104 \[ -\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {13 \sqrt {3+5 x^2+x^4}}{108 x^2}-\frac {61 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{216 \sqrt {3}} \]

[Out]

-61/648*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)-1/9*(x^4+5*x^2+3)^(1/2)/x^6-1/54*(x^4+5*x^2
+3)^(1/2)/x^4+13/108*(x^4+5*x^2+3)^(1/2)/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1265, 848, 820, 738, 212} \begin {gather*} \frac {13 \sqrt {x^4+5 x^2+3}}{108 x^2}-\frac {\sqrt {x^4+5 x^2+3}}{54 x^4}-\frac {61 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{216 \sqrt {3}}-\frac {\sqrt {x^4+5 x^2+3}}{9 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^2)/(x^7*Sqrt[3 + 5*x^2 + x^4]),x]

[Out]

-1/9*Sqrt[3 + 5*x^2 + x^4]/x^6 - Sqrt[3 + 5*x^2 + x^4]/(54*x^4) + (13*Sqrt[3 + 5*x^2 + x^4])/(108*x^2) - (61*A
rcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(216*Sqrt[3])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 848

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

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {2+3 x^2}{x^7 \sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {2+3 x}{x^4 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {1}{18} \text {Subst}\left (\int \frac {-2+4 x}{x^3 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {1}{108} \text {Subst}\left (\int \frac {-39-2 x}{x^2 \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {13 \sqrt {3+5 x^2+x^4}}{108 x^2}+\frac {61}{216} \text {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {13 \sqrt {3+5 x^2+x^4}}{108 x^2}-\frac {61}{108} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {6+5 x^2}{\sqrt {3+5 x^2+x^4}}\right )\\ &=-\frac {\sqrt {3+5 x^2+x^4}}{9 x^6}-\frac {\sqrt {3+5 x^2+x^4}}{54 x^4}+\frac {13 \sqrt {3+5 x^2+x^4}}{108 x^2}-\frac {61 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{216 \sqrt {3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.20, size = 75, normalized size = 0.72 \begin {gather*} \frac {\sqrt {3+5 x^2+x^4} \left (-12-2 x^2+13 x^4\right )}{108 x^6}+\frac {61 \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}-\frac {\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )}{108 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x^2)/(x^7*Sqrt[3 + 5*x^2 + x^4]),x]

[Out]

(Sqrt[3 + 5*x^2 + x^4]*(-12 - 2*x^2 + 13*x^4))/(108*x^6) + (61*ArcTanh[x^2/Sqrt[3] - Sqrt[3 + 5*x^2 + x^4]/Sqr
t[3]])/(108*Sqrt[3])

________________________________________________________________________________________

Maple [A]
time = 0.19, size = 83, normalized size = 0.80

method	result	size
risch	\(\frac {13 x^{8}+63 x^{6}+17 x^{4}-66 x^{2}-36}{108 x^{6} \sqrt {x^{4}+5 x^{2}+3}}-\frac {61 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{648}\)	\(71\)
trager	\(\frac {\left (13 x^{4}-2 x^{2}-12\right ) \sqrt {x^{4}+5 x^{2}+3}}{108 x^{6}}+\frac {61 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-5 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}-6 \RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{648}\)	\(79\)
default	\(-\frac {61 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{648}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{9 x^{6}}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{54 x^{4}}+\frac {13 \sqrt {x^{4}+5 x^{2}+3}}{108 x^{2}}\)	\(83\)
elliptic	\(-\frac {61 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{648}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{9 x^{6}}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{54 x^{4}}+\frac {13 \sqrt {x^{4}+5 x^{2}+3}}{108 x^{2}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+2)/x^7/(x^4+5*x^2+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-61/648*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)-1/9*(x^4+5*x^2+3)^(1/2)/x^6-1/54*(x^4+5*x^2
+3)^(1/2)/x^4+13/108*(x^4+5*x^2+3)^(1/2)/x^2

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 85, normalized size = 0.82 \begin {gather*} -\frac {61}{648} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {13 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{108 \, x^{2}} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{54 \, x^{4}} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{9 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^7/(x^4+5*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

-61/648*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 13/108*sqrt(x^4 + 5*x^2 + 3)/x^2 - 1/54
*sqrt(x^4 + 5*x^2 + 3)/x^4 - 1/9*sqrt(x^4 + 5*x^2 + 3)/x^6

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 90, normalized size = 0.87 \begin {gather*} \frac {61 \, \sqrt {3} x^{6} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) + 78 \, x^{6} + 6 \, {\left (13 \, x^{4} - 2 \, x^{2} - 12\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{648 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^7/(x^4+5*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

1/648*(61*sqrt(3)*x^6*log((25*x^2 - 2*sqrt(3)*(5*x^2 + 6) - 2*sqrt(x^4 + 5*x^2 + 3)*(5*sqrt(3) - 6) + 30)/x^2)
+ 78*x^6 + 6*(13*x^4 - 2*x^2 - 12)*sqrt(x^4 + 5*x^2 + 3))/x^6

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{2} + 2}{x^{7} \sqrt {x^{4} + 5 x^{2} + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+2)/x**7/(x**4+5*x**2+3)**(1/2),x)

[Out]

Integral((3*x**2 + 2)/(x**7*sqrt(x**4 + 5*x**2 + 3)), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (82) = 164\).
time = 3.63, size = 167, normalized size = 1.61 \begin {gather*} \frac {61}{648} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) - \frac {61 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} - 920 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} - 2052 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 1449 \, x^{2} + 1449 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 108}{108 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^7/(x^4+5*x^2+3)^(1/2),x, algorithm="giac")

[Out]

61/648*sqrt(3)*log((x^2 + sqrt(3) - sqrt(x^4 + 5*x^2 + 3))/(x^2 - sqrt(3) - sqrt(x^4 + 5*x^2 + 3))) - 1/108*(6
1*(x^2 - sqrt(x^4 + 5*x^2 + 3))^5 - 920*(x^2 - sqrt(x^4 + 5*x^2 + 3))^3 - 2052*(x^2 - sqrt(x^4 + 5*x^2 + 3))^2
- 1449*x^2 + 1449*sqrt(x^4 + 5*x^2 + 3) - 108)/((x^2 - sqrt(x^4 + 5*x^2 + 3))^2 - 3)^3

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x^2+2}{x^7\,\sqrt {x^4+5\,x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 + 2)/(x^7*(5*x^2 + x^4 + 3)^(1/2)),x)

[Out]

int((3*x^2 + 2)/(x^7*(5*x^2 + x^4 + 3)^(1/2)), x)

________________________________________________________________________________________