\(\int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx\) [11]

Optimal result

Integrand size = 36, antiderivative size = 208 \[ \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx=-\frac {(2 b B+3 a C) (3 a-b x) (3 a+2 b x)}{2 b^3 \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}+\frac {C (3 a-b x)^2 (3 a+2 b x)}{3 b^3 \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}-\frac {\left (4 A b^2-6 a b B+9 a^2 C\right ) (3 a+2 b x) \sqrt {3-\frac {b x}{a}} \text {arctanh}\left (\frac {1}{3} \sqrt {2} \sqrt {3-\frac {b x}{a}}\right )}{6 \sqrt {2} b^3 \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \] Output:

-1/2*(2*B*b+3*C*a)*(-b*x+3*a)*(2*b*x+3*a)/b^3/(-4*b^3*x^3+27*a^2*b*x+27*a^ 
3)^(1/2)+1/3*C*(-b*x+3*a)^2*(2*b*x+3*a)/b^3/(-4*b^3*x^3+27*a^2*b*x+27*a^3) 
^(1/2)-1/12*(4*A*b^2-6*B*a*b+9*C*a^2)*(2*b*x+3*a)*(3-b*x/a)^(1/2)*arctanh( 
1/3*2^(1/2)*(3-b*x/a)^(1/2))*2^(1/2)/b^3/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(1 
/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.65 \[ \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx=-\frac {\sqrt {3 a-b x} (3 a+2 b x) \left (2 \sqrt {a} \sqrt {3 a-b x} (6 b B+3 a C+2 b C x)+\sqrt {2} \left (4 A b^2-6 a b B+9 a^2 C\right ) \text {arctanh}\left (\frac {\sqrt {6 a-2 b x}}{3 \sqrt {a}}\right )\right )}{12 \sqrt {a} b^3 \sqrt {(3 a-b x) (3 a+2 b x)^2}} \] Input:

Integrate[(A + B*x + C*x^2)/Sqrt[27*a^3 + 27*a^2*b*x - 4*b^3*x^3],x]
 

Output:

-1/12*(Sqrt[3*a - b*x]*(3*a + 2*b*x)*(2*Sqrt[a]*Sqrt[3*a - b*x]*(6*b*B + 3 
*a*C + 2*b*C*x) + Sqrt[2]*(4*A*b^2 - 6*a*b*B + 9*a^2*C)*ArcTanh[Sqrt[6*a - 
 2*b*x]/(3*Sqrt[a])]))/(Sqrt[a]*b^3*Sqrt[(3*a - b*x)*(3*a + 2*b*x)^2])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2526, 27, 2483, 27, 90, 73, 221}

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.

\(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle -\frac {\int -\frac {3 b \left (9 C a^2+4 A b^2+4 b^2 B x\right )}{\sqrt {27 a^3+27 b x a^2-4 b^3 x^3}}dx}{12 b^3}-\frac {C \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}{6 b^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {9 C a^2+4 A b^2+4 b^2 B x}{\sqrt {27 a^3+27 b x a^2-4 b^3 x^3}}dx}{4 b^2}-\frac {C \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}{6 b^3}\)

\(\Big \downarrow \) 2483

\(\displaystyle \frac {81 \sqrt {3} a^2 (3 a+2 b x) \sqrt {3 a^3-a^2 b x} \int \frac {9 C a^2+4 A b^2+4 b^2 B x}{81 \sqrt {3} a^2 (3 a+2 b x) \sqrt {3 a^3-a^2 b x}}dx}{4 b^2 \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}-\frac {C \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}{6 b^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(3 a+2 b x) \sqrt {3 a^3-a^2 b x} \int \frac {9 C a^2+4 A b^2+4 b^2 B x}{(3 a+2 b x) \sqrt {3 a^3-a^2 b x}}dx}{4 b^2 \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}-\frac {C \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}{6 b^3}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {(3 a+2 b x) \sqrt {3 a^3-a^2 b x} \left (\left (9 a^2 C-6 a b B+4 A b^2\right ) \int \frac {1}{(3 a+2 b x) \sqrt {3 a^3-a^2 b x}}dx-\frac {4 B \sqrt {3 a^3-a^2 b x}}{a^2}\right )}{4 b^2 \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}-\frac {C \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}{6 b^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(3 a+2 b x) \sqrt {3 a^3-a^2 b x} \left (-\frac {2 \left (9 a^2 C-6 a b B+4 A b^2\right ) \int \frac {1}{9 a-\frac {2 \left (3 a^3-a^2 b x\right )}{a^2}}d\sqrt {3 a^3-a^2 b x}}{a^2 b}-\frac {4 B \sqrt {3 a^3-a^2 b x}}{a^2}\right )}{4 b^2 \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}-\frac {C \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}{6 b^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(3 a+2 b x) \sqrt {3 a^3-a^2 b x} \left (-\frac {4 B \sqrt {3 a^3-a^2 b x}}{a^2}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {3 a^3-a^2 b x}}{3 a^{3/2}}\right ) \left (9 a^2 C-6 a b B+4 A b^2\right )}{3 a^{3/2} b}\right )}{4 b^2 \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}-\frac {C \sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}}{6 b^3}\)

Input:

Int[(A + B*x + C*x^2)/Sqrt[27*a^3 + 27*a^2*b*x - 4*b^3*x^3],x]
 

Output:

-1/6*(C*Sqrt[27*a^3 + 27*a^2*b*x - 4*b^3*x^3])/b^3 + ((3*a + 2*b*x)*Sqrt[3 
*a^3 - a^2*b*x]*((-4*B*Sqrt[3*a^3 - a^2*b*x])/a^2 - (Sqrt[2]*(4*A*b^2 - 6* 
a*b*B + 9*a^2*C)*ArcTanh[(Sqrt[2]*Sqrt[3*a^3 - a^2*b*x])/(3*a^(3/2))])/(3* 
a^(3/2)*b)))/(4*b^2*Sqrt[27*a^3 + 27*a^2*b*x - 4*b^3*x^3])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 221

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

rule 2483

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> Simp[(a + b*x + d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p))   In 
t[(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, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*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^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.70

method	result	size
risch	\(-\frac {\left (2 C x b +6 B b +3 C a \right ) \left (-b x +3 a \right ) \left (2 b x +3 a \right )}{6 b^{3} \sqrt {\left (-b x +3 a \right ) \left (2 b x +3 a \right )^{2}}}-\frac {\left (4 A \,b^{2}-6 a b B +9 C \,a^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) \sqrt {-b x +3 a}\, \left (2 b x +3 a \right )}{12 b^{3} \sqrt {a}\, \sqrt {\left (-b x +3 a \right ) \left (2 b x +3 a \right )^{2}}}\)	\(145\)
default	\(-\frac {\left (2 b x +3 a \right ) \sqrt {-b x +3 a}\, \left (4 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) b^{2}-6 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) a b +9 C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b x +3 a}\, \sqrt {2}}{3 \sqrt {a}}\right ) a^{2}-4 C \left (-b x +3 a \right )^{\frac {3}{2}} \sqrt {a}+12 \sqrt {-b x +3 a}\, B b \sqrt {a}+18 \sqrt {-b x +3 a}\, C \,a^{\frac {3}{2}}\right )}{12 \sqrt {-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}}\, b^{3} \sqrt {a}}\)	\(183\)

Input:

int((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/6*(2*C*b*x+6*B*b+3*C*a)/b^3*(-b*x+3*a)/((-b*x+3*a)*(2*b*x+3*a)^2)^(1/2) 
*(2*b*x+3*a)-1/12*(4*A*b^2-6*B*a*b+9*C*a^2)/b^3/a^(1/2)*2^(1/2)*arctanh(1/ 
3*(-b*x+3*a)^(1/2)*2^(1/2)/a^(1/2))/((-b*x+3*a)*(2*b*x+3*a)^2)^(1/2)*(-b*x 
+3*a)^(1/2)*(2*b*x+3*a)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.75 \[ \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx=\left [\frac {\sqrt {2} {\left (27 \, C a^{3} - 18 \, B a^{2} b + 12 \, A a b^{2} + 2 \, {\left (9 \, C a^{2} b - 6 \, B a b^{2} + 4 \, A b^{3}\right )} x\right )} \sqrt {a} \log \left (\frac {4 \, b^{2} x^{2} - 24 \, a b x - 45 \, a^{2} + 6 \, \sqrt {2} \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}} \sqrt {a}}{4 \, b^{2} x^{2} + 12 \, a b x + 9 \, a^{2}}\right ) - 4 \, \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}} {\left (2 \, C a b x + 3 \, C a^{2} + 6 \, B a b\right )}}{24 \, {\left (2 \, a b^{4} x + 3 \, a^{2} b^{3}\right )}}, -\frac {\sqrt {2} {\left (27 \, C a^{3} - 18 \, B a^{2} b + 12 \, A a b^{2} + 2 \, {\left (9 \, C a^{2} b - 6 \, B a b^{2} + 4 \, A b^{3}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {3 \, \sqrt {2} \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}} \sqrt {-a}}{2 \, {\left (2 \, b^{2} x^{2} - 3 \, a b x - 9 \, a^{2}\right )}}\right ) + 2 \, \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}} {\left (2 \, C a b x + 3 \, C a^{2} + 6 \, B a b\right )}}{12 \, {\left (2 \, a b^{4} x + 3 \, a^{2} b^{3}\right )}}\right ] \] Input:

integrate((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(1/2),x, algorithm= 
"fricas")
 

Output:

[1/24*(sqrt(2)*(27*C*a^3 - 18*B*a^2*b + 12*A*a*b^2 + 2*(9*C*a^2*b - 6*B*a* 
b^2 + 4*A*b^3)*x)*sqrt(a)*log((4*b^2*x^2 - 24*a*b*x - 45*a^2 + 6*sqrt(2)*s 
qrt(-4*b^3*x^3 + 27*a^2*b*x + 27*a^3)*sqrt(a))/(4*b^2*x^2 + 12*a*b*x + 9*a 
^2)) - 4*sqrt(-4*b^3*x^3 + 27*a^2*b*x + 27*a^3)*(2*C*a*b*x + 3*C*a^2 + 6*B 
*a*b))/(2*a*b^4*x + 3*a^2*b^3), -1/12*(sqrt(2)*(27*C*a^3 - 18*B*a^2*b + 12 
*A*a*b^2 + 2*(9*C*a^2*b - 6*B*a*b^2 + 4*A*b^3)*x)*sqrt(-a)*arctan(3/2*sqrt 
(2)*sqrt(-4*b^3*x^3 + 27*a^2*b*x + 27*a^3)*sqrt(-a)/(2*b^2*x^2 - 3*a*b*x - 
 9*a^2)) + 2*sqrt(-4*b^3*x^3 + 27*a^2*b*x + 27*a^3)*(2*C*a*b*x + 3*C*a^2 + 
 6*B*a*b))/(2*a*b^4*x + 3*a^2*b^3)]
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx=\int \frac {A + B x + C x^{2}}{\sqrt {- \left (- 3 a + b x\right ) \left (3 a + 2 b x\right )^{2}}}\, dx \] Input:

integrate((C*x**2+B*x+A)/(-4*b**3*x**3+27*a**2*b*x+27*a**3)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2)/sqrt(-(-3*a + b*x)*(3*a + 2*b*x)**2), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(1/2),x, algorithm= 
"maxima")
 

Output:

integrate((C*x^2 + B*x + A)/sqrt(-4*b^3*x^3 + 27*a^2*b*x + 27*a^3), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx=-\frac {\sqrt {2} {\left (9 \, C a^{2} - 6 \, B a b + 4 \, A b^{2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-b x + 3 \, a}}{3 \, \sqrt {-a}}\right )}{12 \, \sqrt {-a} b^{3} \mathrm {sgn}\left (-2 \, b x - 3 \, a\right )} - \frac {2 \, {\left (-b x + 3 \, a\right )}^{\frac {3}{2}} C b^{6} - 9 \, \sqrt {-b x + 3 \, a} C a b^{6} - 6 \, \sqrt {-b x + 3 \, a} B b^{7}}{6 \, b^{9} \mathrm {sgn}\left (-2 \, b x - 3 \, a\right )} \] Input:

integrate((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(1/2),x, algorithm= 
"giac")
 

Output:

-1/12*sqrt(2)*(9*C*a^2 - 6*B*a*b + 4*A*b^2)*arctan(1/3*sqrt(2)*sqrt(-b*x + 
 3*a)/sqrt(-a))/(sqrt(-a)*b^3*sgn(-2*b*x - 3*a)) - 1/6*(2*(-b*x + 3*a)^(3/ 
2)*C*b^6 - 9*sqrt(-b*x + 3*a)*C*a*b^6 - 6*sqrt(-b*x + 3*a)*B*b^7)/(b^9*sgn 
(-2*b*x - 3*a))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx=\int \frac {C\,x^2+B\,x+A}{\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}} \,d x \] Input:

int((A + B*x + C*x^2)/(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2),x)
 

Output:

int((A + B*x + C*x^2)/(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x+C x^2}{\sqrt {27 a^3+27 a^2 b x-4 b^3 x^3}} \, dx=\frac {-12 \sqrt {-b x +3 a}\, a c -24 \sqrt {-b x +3 a}\, b^{2}-8 \sqrt {-b x +3 a}\, b c x +9 \sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}-3 \sqrt {a}\, \sqrt {2}\right ) a c -2 \sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}-3 \sqrt {a}\, \sqrt {2}\right ) b^{2}-9 \sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}+3 \sqrt {a}\, \sqrt {2}\right ) a c +2 \sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {-b x +3 a}+3 \sqrt {a}\, \sqrt {2}\right ) b^{2}}{24 b^{3}} \] Input:

int((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(1/2),x)
 

Output:

( - 12*sqrt(3*a - b*x)*a*c - 24*sqrt(3*a - b*x)*b**2 - 8*sqrt(3*a - b*x)*b 
*c*x + 9*sqrt(a)*sqrt(2)*log(2*sqrt(3*a - b*x) - 3*sqrt(a)*sqrt(2))*a*c - 
2*sqrt(a)*sqrt(2)*log(2*sqrt(3*a - b*x) - 3*sqrt(a)*sqrt(2))*b**2 - 9*sqrt 
(a)*sqrt(2)*log(2*sqrt(3*a - b*x) + 3*sqrt(a)*sqrt(2))*a*c + 2*sqrt(a)*sqr 
t(2)*log(2*sqrt(3*a - b*x) + 3*sqrt(a)*sqrt(2))*b**2)/(24*b**3)