3.2.50 \(\int \frac {(2+3 x^2) \sqrt {3+5 x^2+x^4}}{x^{11}} \, dx\) [150]

Optimal. Leaf size=132 \[ -\frac {161 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{5184 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac {2093 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{10368 \sqrt {3}} \]

[Out]

-1/15*(x^4+5*x^2+3)^(3/2)/x^10-1/36*(x^4+5*x^2+3)^(3/2)/x^8+173/3240*(x^4+5*x^2+3)^(3/2)/x^6+2093/31104*arctan
h(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)-161/5184*(5*x^2+6)*(x^4+5*x^2+3)^(1/2)/x^4

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Rubi [A]
time = 0.07, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1265, 848, 820, 734, 738, 212} \begin {gather*} -\frac {161 \left (5 x^2+6\right ) \sqrt {x^4+5 x^2+3}}{5184 x^4}+\frac {2093 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{10368 \sqrt {3}}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}+\frac {173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^11,x]

[Out]

(-161*(6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/(5184*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(15*x^10) - (3 + 5*x^2 + x^4)^(3
/2)/(36*x^8) + (173*(3 + 5*x^2 + x^4)^(3/2))/(3240*x^6) + (2093*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2
+ x^4])])/(10368*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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 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 {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^{11}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(2+3 x) \sqrt {3+5 x+x^2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {1}{30} \text {Subst}\left (\int \frac {(-10+4 x) \sqrt {3+5 x+x^2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {1}{360} \text {Subst}\left (\int \frac {(-173-10 x) \sqrt {3+5 x+x^2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac {161}{432} \text {Subst}\left (\int \frac {\sqrt {3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {161 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{5184 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}-\frac {2093 \text {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )}{10368}\\ &=-\frac {161 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{5184 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac {2093 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {6+5 x^2}{\sqrt {3+5 x^2+x^4}}\right )}{5184}\\ &=-\frac {161 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{5184 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac {2093 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{10368 \sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 80, normalized size = 0.61 \begin {gather*} \frac {-\frac {3 \sqrt {3+5 x^2+x^4} \left (5184+10800 x^2+1176 x^4-1370 x^6+2641 x^8\right )}{x^{10}}-10465 \sqrt {3} \tanh ^{-1}\left (\frac {x^2-\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )}{77760} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^11,x]

[Out]

((-3*Sqrt[3 + 5*x^2 + x^4]*(5184 + 10800*x^2 + 1176*x^4 - 1370*x^6 + 2641*x^8))/x^10 - 10465*Sqrt[3]*ArcTanh[(
x^2 - Sqrt[3 + 5*x^2 + x^4])/Sqrt[3]])/77760

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Maple [A]
time = 0.22, size = 152, normalized size = 1.15

method	result	size
risch	\(-\frac {2641 x^{12}+11835 x^{10}+2249 x^{8}+12570 x^{6}+62712 x^{4}+58320 x^{2}+15552}{25920 x^{10} \sqrt {x^{4}+5 x^{2}+3}}+\frac {2093 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{31104}\)	\(81\)
trager	\(-\frac {\left (2641 x^{8}-1370 x^{6}+1176 x^{4}+10800 x^{2}+5184\right ) \sqrt {x^{4}+5 x^{2}+3}}{25920 x^{10}}-\frac {2093 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-5 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}-6 \RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{31104}\)	\(89\)
elliptic	\(\frac {137 \sqrt {x^{4}+5 x^{2}+3}}{2592 x^{4}}-\frac {2641 \sqrt {x^{4}+5 x^{2}+3}}{25920 x^{2}}+\frac {2093 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{31104}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{5 x^{10}}-\frac {5 \sqrt {x^{4}+5 x^{2}+3}}{12 x^{8}}-\frac {49 \sqrt {x^{4}+5 x^{2}+3}}{1080 x^{6}}\)	\(117\)
default	\(-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{36 x^{8}}+\frac {173 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{3240 x^{6}}-\frac {161 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{2592 x^{4}}+\frac {805 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{15552 x^{2}}-\frac {2093 \sqrt {x^{4}+5 x^{2}+3}}{31104}+\frac {2093 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{31104}-\frac {805 \left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{31104}-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{15 x^{10}}\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^11,x,method=_RETURNVERBOSE)

[Out]

-1/36*(x^4+5*x^2+3)^(3/2)/x^8+173/3240*(x^4+5*x^2+3)^(3/2)/x^6-161/2592/x^4*(x^4+5*x^2+3)^(3/2)+805/15552/x^2*
(x^4+5*x^2+3)^(3/2)-2093/31104*(x^4+5*x^2+3)^(1/2)+2093/31104*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2
))*3^(1/2)-805/31104*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)-1/15*(x^4+5*x^2+3)^(3/2)/x^10

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Maxima [A]
time = 0.48, size = 133, normalized size = 1.01 \begin {gather*} \frac {2093}{31104} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {161}{2592} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {805 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{5184 \, x^{2}} - \frac {161 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{2592 \, x^{4}} + \frac {173 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{3240 \, x^{6}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{36 \, x^{8}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{15 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^11,x, algorithm="maxima")

[Out]

2093/31104*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 161/2592*sqrt(x^4 + 5*x^2 + 3) + 805
/5184*sqrt(x^4 + 5*x^2 + 3)/x^2 - 161/2592*(x^4 + 5*x^2 + 3)^(3/2)/x^4 + 173/3240*(x^4 + 5*x^2 + 3)^(3/2)/x^6
- 1/36*(x^4 + 5*x^2 + 3)^(3/2)/x^8 - 1/15*(x^4 + 5*x^2 + 3)^(3/2)/x^10

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Fricas [A]
time = 0.37, size = 100, normalized size = 0.76 \begin {gather*} \frac {10465 \, \sqrt {3} x^{10} \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 ) - 15846 \, x^{10} - 6 \, {\left (2641 \, x^{8} - 1370 \, x^{6} + 1176 \, x^{4} + 10800 \, x^{2} + 5184\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{155520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^11,x, algorithm="fricas")

[Out]

1/155520*(10465*sqrt(3)*x^10*log((25*x^2 + 2*sqrt(3)*(5*x^2 + 6) + 2*sqrt(x^4 + 5*x^2 + 3)*(5*sqrt(3) + 6) + 3
0)/x^2) - 15846*x^10 - 6*(2641*x^8 - 1370*x^6 + 1176*x^4 + 10800*x^2 + 5184)*sqrt(x^4 + 5*x^2 + 3))/x^10

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}}{x^{11}}\, 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)**(1/2)/x**11,x)

[Out]

Integral((3*x**2 + 2)*sqrt(x**4 + 5*x**2 + 3)/x**11, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (106) = 212\).
time = 3.19, size = 255, normalized size = 1.93 \begin {gather*} -\frac {2093}{31104} \, \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 {10465 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{9} - 42830 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{7} + 1270080 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{6} + 7060800 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} + 15310080 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{4} + 16095870 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} + 7568640 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} + 1096335 \, x^{2} - 1096335 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 202176}{25920 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^11,x, algorithm="giac")

[Out]

-2093/31104*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))) + 1/2
5920*(10465*(x^2 - sqrt(x^4 + 5*x^2 + 3))^9 - 42830*(x^2 - sqrt(x^4 + 5*x^2 + 3))^7 + 1270080*(x^2 - sqrt(x^4
+ 5*x^2 + 3))^6 + 7060800*(x^2 - sqrt(x^4 + 5*x^2 + 3))^5 + 15310080*(x^2 - sqrt(x^4 + 5*x^2 + 3))^4 + 1609587
0*(x^2 - sqrt(x^4 + 5*x^2 + 3))^3 + 7568640*(x^2 - sqrt(x^4 + 5*x^2 + 3))^2 + 1096335*x^2 - 1096335*sqrt(x^4 +
5*x^2 + 3) + 202176)/((x^2 - sqrt(x^4 + 5*x^2 + 3))^2 - 3)^5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3}}{x^{11}} \,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)^(1/2))/x^11,x)

[Out]

int(((3*x^2 + 2)*(5*x^2 + x^4 + 3)^(1/2))/x^11, x)

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