Integrand size = 35, antiderivative size = 136 \[ \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\frac {3 \sqrt {b} B+\sqrt {3} (3 A+b C)}{27 \sqrt {b} \left (\sqrt {3} \sqrt {b}-3 x\right )}-\frac {\left (3 A-2 \sqrt {3} \sqrt {b} B-5 b C\right ) \log \left (\sqrt {b}-\sqrt {3} x\right )}{81 b}+\frac {\left (3 A-2 \sqrt {3} \sqrt {b} B+4 b C\right ) \log \left (2 \sqrt {b}+\sqrt {3} x\right )}{81 b} \] Output:
1/27*(3*b^(1/2)*B+3^(1/2)*(C*b+3*A))/b^(1/2)/(3^(1/2)*b^(1/2)-3*x)-1/81*(3 *A-2*3^(1/2)*b^(1/2)*B-5*C*b)*ln(b^(1/2)-x*3^(1/2))/b+1/81*(3*A-2*3^(1/2)* b^(1/2)*B+4*C*b)*ln(2*b^(1/2)+x*3^(1/2))/b
Time = 0.11 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.71 \[ \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\frac {\left (-\sqrt {3} \sqrt {b}+3 x\right ) \left (2 \sqrt {3} \sqrt {b}+3 x\right ) \left (-3 \left (3 A \sqrt {b}+\sqrt {3} b B+b^{3/2} C\right )+\left (-5 b^{3/2} C+6 \sqrt {b} B x+A \left (3 \sqrt {b}-3 \sqrt {3} x\right )+\sqrt {3} b (-2 B+5 C x)\right ) \log \left (-\sqrt {3} \sqrt {b}+3 x\right )+\left (-4 b^{3/2} C-6 \sqrt {b} B x+3 A \left (-\sqrt {b}+\sqrt {3} x\right )+2 \sqrt {3} b (B+2 C x)\right ) \log \left (2 \sqrt {3} \sqrt {b}+3 x\right )\right )}{81 \sqrt {3} b \left (2 \sqrt {3} b^{3/2}-9 b x+9 x^3\right )} \] Input:
Integrate[(A + B*x + C*x^2)/(2*Sqrt[3]*b^(3/2) - 9*b*x + 9*x^3),x]
Output:
((-(Sqrt[3]*Sqrt[b]) + 3*x)*(2*Sqrt[3]*Sqrt[b] + 3*x)*(-3*(3*A*Sqrt[b] + S qrt[3]*b*B + b^(3/2)*C) + (-5*b^(3/2)*C + 6*Sqrt[b]*B*x + A*(3*Sqrt[b] - 3 *Sqrt[3]*x) + Sqrt[3]*b*(-2*B + 5*C*x))*Log[-(Sqrt[3]*Sqrt[b]) + 3*x] + (- 4*b^(3/2)*C - 6*Sqrt[b]*B*x + 3*A*(-Sqrt[b] + Sqrt[3]*x) + 2*Sqrt[3]*b*(B + 2*C*x))*Log[2*Sqrt[3]*Sqrt[b] + 3*x]))/(81*Sqrt[3]*b*(2*Sqrt[3]*b^(3/2) - 9*b*x + 9*x^3))
Time = 0.49 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2525, 27, 2482, 27, 86, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx\) |
\(\Big \downarrow \) 2525 |
\(\displaystyle \frac {1}{27} \int \frac {9 (3 A+b C+3 B x)}{9 x^3-9 b x+2 \sqrt {3} b^{3/2}}dx+\frac {1}{27} C \log \left (2 \sqrt {3} b^{3/2}-9 b x+9 x^3\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {3 A+b C+3 B x}{9 x^3-9 b x+2 \sqrt {3} b^{3/2}}dx+\frac {1}{27} C \log \left (2 \sqrt {3} b^{3/2}-9 b x+9 x^3\right )\) |
\(\Big \downarrow \) 2482 |
\(\displaystyle 108 b^3 \int \frac {3 A+b C+3 B x}{108 \sqrt {3} b^3 \left (\sqrt {3} \sqrt {b}-3 x\right )^2 \left (\sqrt {3} x+2 \sqrt {b}\right )}dx+\frac {1}{27} C \log \left (2 \sqrt {3} b^{3/2}-9 b x+9 x^3\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {3 A+b C+3 B x}{\left (\sqrt {3} \sqrt {b}-3 x\right )^2 \left (\sqrt {3} x+2 \sqrt {b}\right )}dx}{\sqrt {3}}+\frac {1}{27} C \log \left (2 \sqrt {3} b^{3/2}-9 b x+9 x^3\right )\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {\int \left (\frac {3 A-2 \sqrt {3} \sqrt {b} B+b C}{27 b \left (\sqrt {3} x+2 \sqrt {b}\right )}+\frac {3 \sqrt {3} A-6 \sqrt {b} B+\sqrt {3} b C}{27 b \left (\sqrt {3} \sqrt {b}-3 x\right )}+\frac {3 A+\sqrt {3} \sqrt {b} B+b C}{3 \sqrt {b} \left (\sqrt {3} \sqrt {b}-3 x\right )^2}\right )dx}{\sqrt {3}}+\frac {1}{27} C \log \left (2 \sqrt {3} b^{3/2}-9 b x+9 x^3\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3 A+\sqrt {3} \sqrt {b} B+b C}{9 \sqrt {b} \left (\sqrt {3} \sqrt {b}-3 x\right )}+\frac {\log \left (\sqrt {b}-\sqrt {3} x\right ) \left (6 \sqrt {b} B-\sqrt {3} (3 A+b C)\right )}{81 b}-\frac {\log \left (2 \sqrt {b}+\sqrt {3} x\right ) \left (6 \sqrt {b} B-\sqrt {3} (3 A+b C)\right )}{81 b}}{\sqrt {3}}+\frac {1}{27} C \log \left (2 \sqrt {3} b^{3/2}-9 b x+9 x^3\right )\) |
Input:
Int[(A + B*x + C*x^2)/(2*Sqrt[3]*b^(3/2) - 9*b*x + 9*x^3),x]
Output:
((3*A + Sqrt[3]*Sqrt[b]*B + b*C)/(9*Sqrt[b]*(Sqrt[3]*Sqrt[b] - 3*x)) + ((6 *Sqrt[b]*B - Sqrt[3]*(3*A + b*C))*Log[Sqrt[b] - Sqrt[3]*x])/(81*b) - ((6*S qrt[b]*B - Sqrt[3]*(3*A + b*C))*Log[2*Sqrt[b] + Sqrt[3]*x])/(81*b))/Sqrt[3 ] + (C*Log[2*Sqrt[3]*b^(3/2) - 9*b*x + 9*x^3])/27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_.), x_ Symbol] :> Simp[1/(3^(3*p)*a^(2*p)) Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[4*b^3 + 27*a^2* d, 0] && IntegerQ[p]
Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Si mp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coeff[Qn, x, n])), x] + Simp[1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x ]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.39
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \sqrt {3}\, b^{\frac {3}{2}}-9 b \textit {\_Z} +9 \textit {\_Z}^{3}\right )}{\sum }\frac {\left (C \,\textit {\_R}^{2}+B \textit {\_R} +A \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-b}\right )}{9}\) | \(53\) |
Input:
int((C*x^2+B*x+A)/(2*3^(1/2)*b^(3/2)-9*b*x+9*x^3),x,method=_RETURNVERBOSE)
Output:
1/9*sum((C*_R^2+B*_R+A)/(3*_R^2-b)*ln(x-_R),_R=RootOf(2*3^(1/2)*b^(3/2)-9* b*_Z+9*_Z^3))
Time = 0.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=-\frac {3 \, C b^{2} + 9 \, B b x + 3 \, \sqrt {3} {\left (B b + {\left (C b + 3 \, A\right )} x\right )} \sqrt {b} + 9 \, A b + {\left (4 \, C b^{2} - 3 \, {\left (4 \, C b + 3 \, A\right )} x^{2} + 2 \, \sqrt {3} {\left (3 \, B x^{2} - B b\right )} \sqrt {b} + 3 \, A b\right )} \log \left (2 \, \sqrt {3} \sqrt {b} + 3 \, x\right ) + {\left (5 \, C b^{2} - 3 \, {\left (5 \, C b - 3 \, A\right )} x^{2} - 2 \, \sqrt {3} {\left (3 \, B x^{2} - B b\right )} \sqrt {b} - 3 \, A b\right )} \log \left (-\sqrt {3} \sqrt {b} + 3 \, x\right )}{81 \, {\left (3 \, b x^{2} - b^{2}\right )}} \] Input:
integrate((C*x^2+B*x+A)/(2*3^(1/2)*b^(3/2)-9*b*x+9*x^3),x, algorithm="fric as")
Output:
-1/81*(3*C*b^2 + 9*B*b*x + 3*sqrt(3)*(B*b + (C*b + 3*A)*x)*sqrt(b) + 9*A*b + (4*C*b^2 - 3*(4*C*b + 3*A)*x^2 + 2*sqrt(3)*(3*B*x^2 - B*b)*sqrt(b) + 3* A*b)*log(2*sqrt(3)*sqrt(b) + 3*x) + (5*C*b^2 - 3*(5*C*b - 3*A)*x^2 - 2*sqr t(3)*(3*B*x^2 - B*b)*sqrt(b) - 3*A*b)*log(-sqrt(3)*sqrt(b) + 3*x))/(3*b*x^ 2 - b^2)
Leaf count of result is larger than twice the leaf count of optimal. 7218 vs. \(2 (122) = 244\).
Time = 1.13 (sec) , antiderivative size = 7218, normalized size of antiderivative = 53.07 \[ \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\text {Too large to display} \] Input:
integrate((C*x**2+B*x+A)/(2*3**(1/2)*b**(3/2)-9*b*x+9*x**3),x)
Output:
(C/18 - sqrt(36*A**2*b**8 - 48*sqrt(3)*A*B*b**(17/2) - 12*A*C*b**9 + 48*B* *2*b**9 + 8*sqrt(3)*B*C*b**(19/2) + C**2*b**10)/(162*b**5))*log(648*sqrt(3 )*A**4*b**(25/2)/(3888*A**4*b**12 - 10368*sqrt(3)*A**3*B*b**(25/2) - 2592* A**3*C*b**13 + 31104*A**2*B**2*b**13 + 5184*sqrt(3)*A**2*B*C*b**(27/2) + 6 48*A**2*C**2*b**14 - 13824*sqrt(3)*A*B**3*b**(27/2) - 10368*A*B**2*C*b**14 - 864*sqrt(3)*A*B*C**2*b**(29/2) - 72*A*C**3*b**15 + 6912*B**4*b**14 + 23 04*sqrt(3)*B**3*C*b**(29/2) + 864*B**2*C**2*b**15 + 48*sqrt(3)*B*C**3*b**( 31/2) + 3*C**4*b**16) - 5184*A**3*B*b**13/(3888*A**4*b**12 - 10368*sqrt(3) *A**3*B*b**(25/2) - 2592*A**3*C*b**13 + 31104*A**2*B**2*b**13 + 5184*sqrt( 3)*A**2*B*C*b**(27/2) + 648*A**2*C**2*b**14 - 13824*sqrt(3)*A*B**3*b**(27/ 2) - 10368*A*B**2*C*b**14 - 864*sqrt(3)*A*B*C**2*b**(29/2) - 72*A*C**3*b** 15 + 6912*B**4*b**14 + 2304*sqrt(3)*B**3*C*b**(29/2) + 864*B**2*C**2*b**15 + 48*sqrt(3)*B*C**3*b**(31/2) + 3*C**4*b**16) - 3348*sqrt(3)*A**3*C*b**(2 7/2)/(3888*A**4*b**12 - 10368*sqrt(3)*A**3*B*b**(25/2) - 2592*A**3*C*b**13 + 31104*A**2*B**2*b**13 + 5184*sqrt(3)*A**2*B*C*b**(27/2) + 648*A**2*C**2 *b**14 - 13824*sqrt(3)*A*B**3*b**(27/2) - 10368*A*B**2*C*b**14 - 864*sqrt( 3)*A*B*C**2*b**(29/2) - 72*A*C**3*b**15 + 6912*B**4*b**14 + 2304*sqrt(3)*B **3*C*b**(29/2) + 864*B**2*C**2*b**15 + 48*sqrt(3)*B*C**3*b**(31/2) + 3*C* *4*b**16) + 5184*sqrt(3)*A**2*B**2*b**(27/2)/(3888*A**4*b**12 - 10368*sqrt (3)*A**3*B*b**(25/2) - 2592*A**3*C*b**13 + 31104*A**2*B**2*b**13 + 5184...
\[ \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\int { \frac {C x^{2} + B x + A}{9 \, x^{3} + 2 \, \sqrt {3} b^{\frac {3}{2}} - 9 \, b x} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(2*3^(1/2)*b^(3/2)-9*b*x+9*x^3),x, algorithm="maxi ma")
Output:
integrate((C*x^2 + B*x + A)/(9*x^3 + 2*sqrt(3)*b^(3/2) - 9*b*x), x)
Exception generated. \[ \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((C*x^2+B*x+A)/(2*3^(1/2)*b^(3/2)-9*b*x+9*x^3),x, algorithm="giac ")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 12.62 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\ln \left (x-\frac {\sqrt {3}\,\sqrt {b}}{3}\right )\,\left (\frac {C}{18}+\frac {b\,\left (12\,B\,\sqrt {27\,b}+\sqrt {3}\,C\,\sqrt {b}\,\sqrt {27\,b}\right )-6\,\sqrt {3}\,A\,\sqrt {b}\,\sqrt {27\,b}}{1458\,b^2}\right )+\ln \left (x+\frac {2\,\sqrt {3}\,\sqrt {b}}{3}\right )\,\left (\frac {C}{18}-\frac {b\,\left (12\,B\,\sqrt {27\,b}+\sqrt {3}\,C\,\sqrt {b}\,\sqrt {27\,b}\right )-6\,\sqrt {3}\,A\,\sqrt {b}\,\sqrt {27\,b}}{1458\,b^2}\right )-\frac {\frac {B}{27}+\frac {\sqrt {3}\,\left (\frac {A}{27}+\frac {C\,b}{81}\right )}{\sqrt {b}}}{x-\frac {\sqrt {3}\,\sqrt {b}}{3}} \] Input:
int((A + B*x + C*x^2)/(2*3^(1/2)*b^(3/2) - 9*b*x + 9*x^3),x)
Output:
log(x - (3^(1/2)*b^(1/2))/3)*(C/18 + (b*(12*B*(27*b)^(1/2) + 3^(1/2)*C*b^( 1/2)*(27*b)^(1/2)) - 6*3^(1/2)*A*b^(1/2)*(27*b)^(1/2))/(1458*b^2)) + log(x + (2*3^(1/2)*b^(1/2))/3)*(C/18 - (b*(12*B*(27*b)^(1/2) + 3^(1/2)*C*b^(1/2 )*(27*b)^(1/2)) - 6*3^(1/2)*A*b^(1/2)*(27*b)^(1/2))/(1458*b^2)) - (B/27 + (3^(1/2)*(A/27 + (C*b)/81))/b^(1/2))/(x - (3^(1/2)*b^(1/2))/3)
Time = 0.19 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.99 \[ \int \frac {A+B x+C x^2}{2 \sqrt {3} b^{3/2}-9 b x+9 x^3} \, dx=\frac {-2 \sqrt {b}\, \sqrt {3}\, \mathrm {log}\left (-2 \sqrt {b}\, \sqrt {3}-3 x \right ) b^{2}+6 \sqrt {b}\, \sqrt {3}\, \mathrm {log}\left (-2 \sqrt {b}\, \sqrt {3}-3 x \right ) b \,x^{2}+2 \sqrt {b}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {3}-3 x \right ) b^{2}-6 \sqrt {b}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {3}-3 x \right ) b \,x^{2}+9 \sqrt {b}\, \sqrt {3}\, a x +3 \sqrt {b}\, \sqrt {3}\, b c x +9 \sqrt {b}\, \sqrt {3}\, b \,x^{2}+3 \,\mathrm {log}\left (-2 \sqrt {b}\, \sqrt {3}-3 x \right ) a b -9 \,\mathrm {log}\left (-2 \sqrt {b}\, \sqrt {3}-3 x \right ) a \,x^{2}+4 \,\mathrm {log}\left (-2 \sqrt {b}\, \sqrt {3}-3 x \right ) b^{2} c -12 \,\mathrm {log}\left (-2 \sqrt {b}\, \sqrt {3}-3 x \right ) b c \,x^{2}-3 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {3}-3 x \right ) a b +9 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {3}-3 x \right ) a \,x^{2}+5 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {3}-3 x \right ) b^{2} c -15 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {3}-3 x \right ) b c \,x^{2}+27 a \,x^{2}+9 b^{2} x +9 b c \,x^{2}}{81 b \left (-3 x^{2}+b \right )} \] Input:
int((C*x^2+B*x+A)/(2*3^(1/2)*b^(3/2)-9*b*x+9*x^3),x)
Output:
( - 2*sqrt(b)*sqrt(3)*log( - 2*sqrt(b)*sqrt(3) - 3*x)*b**2 + 6*sqrt(b)*sqr t(3)*log( - 2*sqrt(b)*sqrt(3) - 3*x)*b*x**2 + 2*sqrt(b)*sqrt(3)*log(sqrt(b )*sqrt(3) - 3*x)*b**2 - 6*sqrt(b)*sqrt(3)*log(sqrt(b)*sqrt(3) - 3*x)*b*x** 2 + 9*sqrt(b)*sqrt(3)*a*x + 3*sqrt(b)*sqrt(3)*b*c*x + 9*sqrt(b)*sqrt(3)*b* x**2 + 3*log( - 2*sqrt(b)*sqrt(3) - 3*x)*a*b - 9*log( - 2*sqrt(b)*sqrt(3) - 3*x)*a*x**2 + 4*log( - 2*sqrt(b)*sqrt(3) - 3*x)*b**2*c - 12*log( - 2*sqr t(b)*sqrt(3) - 3*x)*b*c*x**2 - 3*log(sqrt(b)*sqrt(3) - 3*x)*a*b + 9*log(sq rt(b)*sqrt(3) - 3*x)*a*x**2 + 5*log(sqrt(b)*sqrt(3) - 3*x)*b**2*c - 15*log (sqrt(b)*sqrt(3) - 3*x)*b*c*x**2 + 27*a*x**2 + 9*b**2*x + 9*b*c*x**2)/(81* b*(b - 3*x**2))