3.2.49 \(\int \frac {(2+3 x^2) \sqrt {3+5 x^2+x^4}}{x^9} \, dx\) [149]

Optimal. Leaf size=111 \[ \frac {67 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{1728 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac {871 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{3456 \sqrt {3}} \]

[Out]

-1/12*(x^4+5*x^2+3)^(3/2)/x^8-11/216*(x^4+5*x^2+3)^(3/2)/x^6-871/10368*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^
2+3)^(1/2))*3^(1/2)+67/1728*(5*x^2+6)*(x^4+5*x^2+3)^(1/2)/x^4

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Rubi [A]
time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {67 \left (5 x^2+6\right ) \sqrt {x^4+5 x^2+3}}{1728 x^4}-\frac {871 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3456 \sqrt {3}}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{12 x^8}-\frac {11 \left (x^4+5 x^2+3\right )^{3/2}}{216 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^9,x]

[Out]

(67*(6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/(1728*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(12*x^8) - (11*(3 + 5*x^2 + x^4)^(
3/2))/(216*x^6) - (871*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(3456*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^9} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(2+3 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}}{12 x^8}-\frac {1}{24} \text {Subst}\left (\int \frac {(-11+2 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}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac {67}{144} \text {Subst}\left (\int \frac {\sqrt {3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=\frac {67 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{1728 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}+\frac {871 \text {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )}{3456}\\ &=\frac {67 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{1728 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac {871 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {6+5 x^2}{\sqrt {3+5 x^2+x^4}}\right )}{1728}\\ &=\frac {67 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{1728 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac {871 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{3456 \sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 80, normalized size = 0.72 \begin {gather*} \frac {\sqrt {3+5 x^2+x^4} \left (-432-984 x^2-182 x^4+247 x^6\right )}{1728 x^8}+\frac {871 \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}-\frac {\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )}{1728 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^9,x]

[Out]

(Sqrt[3 + 5*x^2 + x^4]*(-432 - 984*x^2 - 182*x^4 + 247*x^6))/(1728*x^8) + (871*ArcTanh[x^2/Sqrt[3] - Sqrt[3 +
5*x^2 + x^4]/Sqrt[3]])/(1728*Sqrt[3])

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Maple [A]
time = 0.20, size = 135, normalized size = 1.22

method	result	size
risch	\(\frac {247 x^{10}+1053 x^{8}-1153 x^{6}-5898 x^{4}-5112 x^{2}-1296}{1728 x^{8} \sqrt {x^{4}+5 x^{2}+3}}-\frac {871 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{10368}\)	\(76\)
trager	\(\frac {\left (247 x^{6}-182 x^{4}-984 x^{2}-432\right ) \sqrt {x^{4}+5 x^{2}+3}}{1728 x^{8}}-\frac {871 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {x^{4}+5 x^{2}+3}}{x^{2}}\right )}{10368}\)	\(84\)
elliptic	\(-\frac {91 \sqrt {x^{4}+5 x^{2}+3}}{864 x^{4}}+\frac {247 \sqrt {x^{4}+5 x^{2}+3}}{1728 x^{2}}-\frac {871 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{10368}-\frac {\sqrt {x^{4}+5 x^{2}+3}}{4 x^{8}}-\frac {41 \sqrt {x^{4}+5 x^{2}+3}}{72 x^{6}}\)	\(100\)
default	\(-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{12 x^{8}}-\frac {11 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{216 x^{6}}+\frac {67 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{864 x^{4}}-\frac {335 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{5184 x^{2}}+\frac {871 \sqrt {x^{4}+5 x^{2}+3}}{10368}-\frac {871 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{10368}+\frac {335 \left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{10368}\)	\(135\)

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^9,x,method=_RETURNVERBOSE)

[Out]

-1/12*(x^4+5*x^2+3)^(3/2)/x^8-11/216*(x^4+5*x^2+3)^(3/2)/x^6+67/864/x^4*(x^4+5*x^2+3)^(3/2)-335/5184/x^2*(x^4+
5*x^2+3)^(3/2)+871/10368*(x^4+5*x^2+3)^(1/2)-871/10368*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1
/2)+335/10368*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)

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Maxima [A]
time = 0.50, size = 116, normalized size = 1.05 \begin {gather*} -\frac {871}{10368} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) - \frac {67}{864} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {335 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{1728 \, x^{2}} + \frac {67 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{864 \, x^{4}} - \frac {11 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{216 \, x^{6}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{12 \, x^{8}} \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^9,x, algorithm="maxima")

[Out]

-871/10368*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) - 67/864*sqrt(x^4 + 5*x^2 + 3) - 335/1
728*sqrt(x^4 + 5*x^2 + 3)/x^2 + 67/864*(x^4 + 5*x^2 + 3)^(3/2)/x^4 - 11/216*(x^4 + 5*x^2 + 3)^(3/2)/x^6 - 1/12
*(x^4 + 5*x^2 + 3)^(3/2)/x^8

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Fricas [A]
time = 0.36, size = 95, normalized size = 0.86 \begin {gather*} \frac {871 \, \sqrt {3} x^{8} \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 ) + 1482 \, x^{8} + 6 \, {\left (247 \, x^{6} - 182 \, x^{4} - 984 \, x^{2} - 432\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{10368 \, x^{8}} \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^9,x, algorithm="fricas")

[Out]

1/10368*(871*sqrt(3)*x^8*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) + 1482*x^8 + 6*(247*x^6 - 182*x^4 - 984*x^2 - 432)*sqrt(x^4 + 5*x^2 + 3))/x^8

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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^{9}}\, 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**9,x)

[Out]

Integral((3*x**2 + 2)*sqrt(x**4 + 5*x**2 + 3)/x**9, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (89) = 178\).
time = 3.54, size = 233, normalized size = 2.10 \begin {gather*} \frac {871}{10368} \, \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 {871 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{7} - 5184 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{6} - 57389 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} - 165888 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{4} - 204807 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} - 93312 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 2403 \, x^{2} + 2403 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5184}{1728 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{4}} \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^9,x, algorithm="giac")

[Out]

871/10368*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))) - 1/172
8*(871*(x^2 - sqrt(x^4 + 5*x^2 + 3))^7 - 5184*(x^2 - sqrt(x^4 + 5*x^2 + 3))^6 - 57389*(x^2 - sqrt(x^4 + 5*x^2
+ 3))^5 - 165888*(x^2 - sqrt(x^4 + 5*x^2 + 3))^4 - 204807*(x^2 - sqrt(x^4 + 5*x^2 + 3))^3 - 93312*(x^2 - sqrt(
x^4 + 5*x^2 + 3))^2 - 2403*x^2 + 2403*sqrt(x^4 + 5*x^2 + 3) - 5184)/((x^2 - sqrt(x^4 + 5*x^2 + 3))^2 - 3)^4

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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^9} \,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^9,x)

[Out]

int(((3*x^2 + 2)*(5*x^2 + x^4 + 3)^(1/2))/x^9, x)

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