3.2.46 \(\int \frac {(2+3 x^2) \sqrt {3+5 x^2+x^4}}{x^3} \, dx\) [146]

Optimal. Leaf size=97 \[ -\frac {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}+\frac {19}{4} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )-\frac {7 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{\sqrt {3}} \]

[Out]

19/4*arctanh(1/2*(2*x^2+5)/(x^4+5*x^2+3)^(1/2))-7/3*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)
-1/2*(-3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^2

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Rubi [A]
time = 0.05, antiderivative size = 97, 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, 826, 857, 635, 212, 738} \begin {gather*} -\frac {\sqrt {x^4+5 x^2+3} \left (2-3 x^2\right )}{2 x^2}+\frac {19}{4} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {7 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((2 - 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^2 + (19*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/4 - (7*ArcT
anh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 826

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

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

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^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(2+3 x) \sqrt {3+5 x+x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {-28-19 x}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}+\frac {19}{4} \text {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )+7 \text {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}+\frac {19}{2} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {5+2 x^2}{\sqrt {3+5 x^2+x^4}}\right )-14 \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 {\left (2-3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}+\frac {19}{4} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )-\frac {7 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 91, normalized size = 0.94 \begin {gather*} \frac {\left (-2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{2 x^2}+\frac {14 \tanh ^{-1}\left (\frac {x^2-\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {19}{4} \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/(2*x^2) + (14*ArcTanh[(x^2 - Sqrt[3 + 5*x^2 + x^4])/Sqrt[3]])/Sqrt[3] - (
19*Log[-5 - 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/4

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Maple [A]
time = 0.29, size = 104, normalized size = 1.07

method	result	size
risch	\(-\frac {\sqrt {x^{4}+5 x^{2}+3}}{x^{2}}+\frac {3 \sqrt {x^{4}+5 x^{2}+3}}{2}+\frac {19 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}-\frac {7 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{3}\)	\(83\)
elliptic	\(-\frac {\sqrt {x^{4}+5 x^{2}+3}}{x^{2}}+\frac {3 \sqrt {x^{4}+5 x^{2}+3}}{2}+\frac {19 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}-\frac {7 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{3}\)	\(83\)
trager	\(\frac {\left (3 x^{2}-2\right ) \sqrt {x^{4}+5 x^{2}+3}}{2 x^{2}}+\frac {7 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {-5 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}-6 \RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{3}+\frac {19 \ln \left (-2 x^{2}-2 \sqrt {x^{4}+5 x^{2}+3}-5\right )}{4}\)	\(98\)
default	\(-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{3 x^{2}}+\frac {7 \sqrt {x^{4}+5 x^{2}+3}}{3}+\frac {19 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}-\frac {7 \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{3}+\frac {\left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{6}\)	\(104\)

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^3,x,method=_RETURNVERBOSE)

[Out]

-1/3/x^2*(x^4+5*x^2+3)^(3/2)+7/3*(x^4+5*x^2+3)^(1/2)+19/4*ln(x^2+5/2+(x^4+5*x^2+3)^(1/2))-7/3*arctanh(1/6*(5*x
^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)+1/6*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)

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Maxima [A]
time = 0.49, size = 89, normalized size = 0.92 \begin {gather*} -\frac {7}{3} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {3}{2} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {\sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {19}{4} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \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^3,x, algorithm="maxima")

[Out]

-7/3*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 3/2*sqrt(x^4 + 5*x^2 + 3) - sqrt(x^4 + 5*x
^2 + 3)/x^2 + 19/4*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]
time = 0.35, size = 112, normalized size = 1.15 \begin {gather*} \frac {56 \, \sqrt {3} x^{2} \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 ) - 114 \, x^{2} \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) + 21 \, x^{2} + 12 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (3 \, x^{2} - 2\right )}}{24 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

1/24*(56*sqrt(3)*x^2*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)
- 114*x^2*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) + 21*x^2 + 12*sqrt(x^4 + 5*x^2 + 3)*(3*x^2 - 2))/x^2

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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^{3}}\, 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**3,x)

[Out]

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

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Giac [A]
time = 3.49, size = 138, normalized size = 1.42 \begin {gather*} \frac {7}{3} \, \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 {3}{2} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {5 \, x^{2} - 5 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 6}{{\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3} - \frac {19}{4} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \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^3,x, algorithm="giac")

[Out]

7/3*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))) + 3/2*sqrt(x^
4 + 5*x^2 + 3) + (5*x^2 - 5*sqrt(x^4 + 5*x^2 + 3) + 6)/((x^2 - sqrt(x^4 + 5*x^2 + 3))^2 - 3) - 19/4*log(2*x^2
- 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Mupad [B]
time = 0.88, size = 84, normalized size = 0.87 \begin {gather*} \frac {19\,\ln \left (\sqrt {x^4+5\,x^2+3}+x^2+\frac {5}{2}\right )}{4}-\frac {\sqrt {x^4+5\,x^2+3}}{x^2}-\frac {7\,\sqrt {3}\,\ln \left (\frac {3}{x^2}+\frac {\sqrt {3}\,\sqrt {x^4+5\,x^2+3}}{x^2}+\frac {5}{2}\right )}{3}+\frac {3\,\sqrt {x^4+5\,x^2+3}}{2} \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^3,x)

[Out]

(19*log((5*x^2 + x^4 + 3)^(1/2) + x^2 + 5/2))/4 - (5*x^2 + x^4 + 3)^(1/2)/x^2 - (7*3^(1/2)*log(3/x^2 + (3^(1/2
)*(5*x^2 + x^4 + 3)^(1/2))/x^2 + 5/2))/3 + (3*(5*x^2 + x^4 + 3)^(1/2))/2

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