\(\int \frac {(1+2 x)^2 (1+3 x+4 x^2)}{(2-x+3 x^2)^{3/2}} \, dx\) [84]

Optimal result

Integrand size = 32, antiderivative size = 82 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {2 (1249-2273 x)}{621 \sqrt {2-x+3 x^2}}+\frac {112}{27} \sqrt {2-x+3 x^2}+\frac {8}{9} x \sqrt {2-x+3 x^2}-\frac {64 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{9 \sqrt {3}} \] Output:

2/621*(1249-2273*x)/(3*x^2-x+2)^(1/2)+112/27*(3*x^2-x+2)^(1/2)+8/9*x*(3*x^ 
2-x+2)^(1/2)-64/27*arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.79 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {2 \left (1275-1003 x+1196 x^2+276 x^3\right )}{207 \sqrt {2-x+3 x^2}}-\frac {64 \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{9 \sqrt {3}} \] Input:

Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(3/2),x]
 

Output:

(2*(1275 - 1003*x + 1196*x^2 + 276*x^3))/(207*Sqrt[2 - x + 3*x^2]) - (64*L 
og[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]])/(9*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2191, 27, 2192, 27, 1160, 1090, 222}

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.

\(\displaystyle \int \frac {(2 x+1)^2 \left (4 x^2+3 x+1\right )}{\left (3 x^2-x+2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2191

\(\displaystyle \frac {2}{23} \int \frac {46 \left (36 x^2+75 x+46\right )}{27 \sqrt {3 x^2-x+2}}dx+\frac {2 (1249-2273 x)}{621 \sqrt {3 x^2-x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{27} \int \frac {36 x^2+75 x+46}{\sqrt {3 x^2-x+2}}dx+\frac {2 (1249-2273 x)}{621 \sqrt {3 x^2-x+2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {4}{27} \left (\frac {1}{6} \int \frac {12 (42 x+17)}{\sqrt {3 x^2-x+2}}dx+6 \sqrt {3 x^2-x+2} x\right )+\frac {2 (1249-2273 x)}{621 \sqrt {3 x^2-x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{27} \left (2 \int \frac {42 x+17}{\sqrt {3 x^2-x+2}}dx+6 \sqrt {3 x^2-x+2} x\right )+\frac {2 (1249-2273 x)}{621 \sqrt {3 x^2-x+2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {4}{27} \left (2 \left (24 \int \frac {1}{\sqrt {3 x^2-x+2}}dx+14 \sqrt {3 x^2-x+2}\right )+6 \sqrt {3 x^2-x+2} x\right )+\frac {2 (1249-2273 x)}{621 \sqrt {3 x^2-x+2}}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {4}{27} \left (2 \left (8 \sqrt {\frac {3}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)+14 \sqrt {3 x^2-x+2}\right )+6 \sqrt {3 x^2-x+2} x\right )+\frac {2 (1249-2273 x)}{621 \sqrt {3 x^2-x+2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {4}{27} \left (2 \left (8 \sqrt {3} \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )+14 \sqrt {3 x^2-x+2}\right )+6 \sqrt {3 x^2-x+2} x\right )+\frac {2 (1249-2273 x)}{621 \sqrt {3 x^2-x+2}}\)

Input:

Int[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(3/2),x]
 

Output:

(2*(1249 - 2273*x))/(621*Sqrt[2 - x + 3*x^2]) + (4*(6*x*Sqrt[2 - x + 3*x^2 
] + 2*(14*Sqrt[2 - x + 3*x^2] + 8*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])))/ 
27
 

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, 
b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
 

rule 1160

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

rule 2191

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = 
PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P 
q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + 
c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ 
(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int 
[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* 
(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 
2 - 4*a*c, 0] && LtQ[p, -1]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.55

method	result	size
risch	\(\frac {\frac {8}{3} x^{3}+\frac {104}{9} x^{2}-\frac {2006}{207} x +\frac {850}{69}}{\sqrt {3 x^{2}-x +2}}+\frac {64 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{27}\)	\(45\)
trager	\(\frac {\frac {8}{3} x^{3}+\frac {104}{9} x^{2}-\frac {2006}{207} x +\frac {850}{69}}{\sqrt {3 x^{2}-x +2}}-\frac {64 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{27}\)	\(70\)
default	\(-\frac {89 \left (6 x -1\right )}{207 \sqrt {3 x^{2}-x +2}}+\frac {107}{9 \sqrt {3 x^{2}-x +2}}-\frac {64 x}{9 \sqrt {3 x^{2}-x +2}}+\frac {64 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{27}+\frac {104 x^{2}}{9 \sqrt {3 x^{2}-x +2}}+\frac {8 x^{3}}{3 \sqrt {3 x^{2}-x +2}}\)	\(98\)

Input:

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

Output:

2/207*(276*x^3+1196*x^2-1003*x+1275)/(3*x^2-x+2)^(1/2)+64/27*3^(1/2)*arcsi 
nh(6/23*23^(1/2)*(x-1/6))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.12 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (368 \, \sqrt {3} {\left (3 \, x^{2} - x + 2\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 3 \, {\left (276 \, x^{3} + 1196 \, x^{2} - 1003 \, x + 1275\right )} \sqrt {3 \, x^{2} - x + 2}\right )}}{621 \, {\left (3 \, x^{2} - x + 2\right )}} \] Input:

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

Output:

2/621*(368*sqrt(3)*(3*x^2 - x + 2)*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x 
 - 1) - 72*x^2 + 24*x - 25) + 3*(276*x^3 + 1196*x^2 - 1003*x + 1275)*sqrt( 
3*x^2 - x + 2))/(3*x^2 - x + 2)
 

Sympy [F]

\[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx=\int \frac {\left (2 x + 1\right )^{2} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((1+2*x)**2*(4*x**2+3*x+1)/(3*x**2-x+2)**(3/2),x)
 

Output:

Integral((2*x + 1)**2*(4*x**2 + 3*x + 1)/(3*x**2 - x + 2)**(3/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {8 \, x^{3}}{3 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {104 \, x^{2}}{9 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {64}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {2006 \, x}{207 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {850}{69 \, \sqrt {3 \, x^{2} - x + 2}} \] Input:

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

Output:

8/3*x^3/sqrt(3*x^2 - x + 2) + 104/9*x^2/sqrt(3*x^2 - x + 2) + 64/27*sqrt(3 
)*arcsinh(1/23*sqrt(23)*(6*x - 1)) - 2006/207*x/sqrt(3*x^2 - x + 2) + 850/ 
69/sqrt(3*x^2 - x + 2)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx=-\frac {64}{27} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac {2 \, {\left ({\left (92 \, {\left (3 \, x + 13\right )} x - 1003\right )} x + 1275\right )}}{207 \, \sqrt {3 \, x^{2} - x + 2}} \] Input:

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

Output:

-64/27*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1) + 2/2 
07*((92*(3*x + 13)*x - 1003)*x + 1275)/sqrt(3*x^2 - x + 2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx=\int \frac {{\left (2\,x+1\right )}^2\,\left (4\,x^2+3\,x+1\right )}{{\left (3\,x^2-x+2\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.24 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx=\frac {4968 \sqrt {3 x^{2}-x +2}\, x^{3}+21528 \sqrt {3 x^{2}-x +2}\, x^{2}-18054 \sqrt {3 x^{2}-x +2}\, x +22950 \sqrt {3 x^{2}-x +2}+13248 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right ) x^{2}-4416 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right ) x +8832 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right )-13638 \sqrt {3}\, x^{2}+4546 \sqrt {3}\, x -9092 \sqrt {3}}{5589 x^{2}-1863 x +3726} \] Input:

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

Output:

(2*(2484*sqrt(3*x**2 - x + 2)*x**3 + 10764*sqrt(3*x**2 - x + 2)*x**2 - 902 
7*sqrt(3*x**2 - x + 2)*x + 11475*sqrt(3*x**2 - x + 2) + 6624*sqrt(3)*log(( 
2*sqrt(3*x**2 - x + 2)*sqrt(3) + 6*x - 1)/sqrt(23))*x**2 - 2208*sqrt(3)*lo 
g((2*sqrt(3*x**2 - x + 2)*sqrt(3) + 6*x - 1)/sqrt(23))*x + 4416*sqrt(3)*lo 
g((2*sqrt(3*x**2 - x + 2)*sqrt(3) + 6*x - 1)/sqrt(23)) - 6819*sqrt(3)*x**2 
 + 2273*sqrt(3)*x - 4546*sqrt(3)))/(1863*(3*x**2 - x + 2))