\(\int (1+2 x) (2-x+3 x^2)^{5/2} (1+3 x+4 x^2) \, dx\) [59]

Optimal result

Integrand size = 30, antiderivative size = 139 \[ \int (1+2 x) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=-\frac {1177025 (1-6 x) \sqrt {2-x+3 x^2}}{5971968}-\frac {51175 (1-6 x) \left (2-x+3 x^2\right )^{3/2}}{746496}-\frac {445 (1-6 x) \left (2-x+3 x^2\right )^{5/2}}{15552}+\frac {2}{27} (1+2 x)^2 \left (2-x+3 x^2\right )^{7/2}+\frac {1}{648} (137+122 x) \left (2-x+3 x^2\right )^{7/2}-\frac {27071575 \text {arcsinh}\left (\frac {1-6 x}{\sqrt {23}}\right )}{11943936 \sqrt {3}} \] Output:

-1177025/5971968*(1-6*x)*(3*x^2-x+2)^(1/2)-51175/746496*(1-6*x)*(3*x^2-x+2 
)^(3/2)-445/15552*(1-6*x)*(3*x^2-x+2)^(5/2)+2/27*(1+2*x)^2*(3*x^2-x+2)^(7/ 
2)+1/648*(137+122*x)*(3*x^2-x+2)^(7/2)-27071575/35831808*arcsinh(1/23*(1-6 
*x)*23^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.65 \[ \int (1+2 x) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {6 \sqrt {2-x+3 x^2} \left (10960335+19860062 x+41031048 x^2+58946544 x^3+66969216 x^4+80034048 x^5+79377408 x^6+30357504 x^7+47775744 x^8\right )-27071575 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{35831808} \] Input:

Integrate[(1 + 2*x)*(2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2),x]
 

Output:

(6*Sqrt[2 - x + 3*x^2]*(10960335 + 19860062*x + 41031048*x^2 + 58946544*x^ 
3 + 66969216*x^4 + 80034048*x^5 + 79377408*x^6 + 30357504*x^7 + 47775744*x 
^8) - 27071575*Sqrt[3]*Log[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]])/35831808
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2184, 27, 1225, 1087, 1087, 1087, 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.

\(\displaystyle \int (2 x+1) \left (3 x^2-x+2\right )^{5/2} \left (4 x^2+3 x+1\right ) \, dx\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {1}{108} \int 4 (2 x+1) (61 x+18) \left (3 x^2-x+2\right )^{5/2}dx+\frac {2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{27} \int (2 x+1) (61 x+18) \left (3 x^2-x+2\right )^{5/2}dx+\frac {2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{27} \left (\frac {445}{16} \int \left (3 x^2-x+2\right )^{5/2}dx+\frac {1}{24} (122 x+137) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{27} \left (\frac {445}{16} \left (\frac {115}{72} \int \left (3 x^2-x+2\right )^{3/2}dx-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{24} (122 x+137) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{27} \left (\frac {445}{16} \left (\frac {115}{72} \left (\frac {23}{16} \int \sqrt {3 x^2-x+2}dx-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{24} (122 x+137) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{27} \left (\frac {445}{16} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {23}{24} \int \frac {1}{\sqrt {3 x^2-x+2}}dx-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{24} (122 x+137) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{27} \left (\frac {445}{16} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {1}{24} \sqrt {\frac {23}{3}} \int \frac {1}{\sqrt {\frac {1}{23} (6 x-1)^2+1}}d(6 x-1)-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{24} (122 x+137) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{27} \left (\frac {445}{16} \left (\frac {115}{72} \left (\frac {23}{16} \left (\frac {23 \text {arcsinh}\left (\frac {6 x-1}{\sqrt {23}}\right )}{24 \sqrt {3}}-\frac {1}{12} (1-6 x) \sqrt {3 x^2-x+2}\right )-\frac {1}{24} (1-6 x) \left (3 x^2-x+2\right )^{3/2}\right )-\frac {1}{36} (1-6 x) \left (3 x^2-x+2\right )^{5/2}\right )+\frac {1}{24} (122 x+137) \left (3 x^2-x+2\right )^{7/2}\right )+\frac {2}{27} (2 x+1)^2 \left (3 x^2-x+2\right )^{7/2}\)

Input:

Int[(1 + 2*x)*(2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2),x]
 

Output:

(2*(1 + 2*x)^2*(2 - x + 3*x^2)^(7/2))/27 + (((137 + 122*x)*(2 - x + 3*x^2) 
^(7/2))/24 + (445*(-1/36*((1 - 6*x)*(2 - x + 3*x^2)^(5/2)) + (115*(-1/24*( 
(1 - 6*x)*(2 - x + 3*x^2)^(3/2)) + (23*(-1/12*((1 - 6*x)*Sqrt[2 - x + 3*x^ 
2]) + (23*ArcSinh[(-1 + 6*x)/Sqrt[23]])/(24*Sqrt[3])))/16))/72))/16)/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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.50

method	result	size
risch	\(\frac {\left (47775744 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3 x^{2}-x +2}}{5971968}+\frac {27071575 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{35831808}\)	\(70\)
trager	\(\left (8 x^{8}+\frac {61}{12} x^{7}+\frac {319}{24} x^{6}+\frac {11579}{864} x^{5}+\frac {58133}{5184} x^{4}+\frac {409351}{41472} x^{3}+\frac {1709627}{248832} x^{2}+\frac {9930031}{2985984} x +\frac {1217815}{663552}\right ) \sqrt {3 x^{2}-x +2}+\frac {27071575 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}-x +2}\right )}{35831808}\)	\(96\)
default	\(\frac {445 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{15552}+\frac {51175 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{746496}+\frac {1177025 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{5971968}+\frac {27071575 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{35831808}+\frac {185 \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{648}+\frac {157 x \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{324}+\frac {8 x^{2} \left (3 x^{2}-x +2\right )^{\frac {7}{2}}}{27}\)	\(119\)

Input:

int((1+2*x)*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x,method=_RETURNVERBOSE)
 

Output:

1/5971968*(47775744*x^8+30357504*x^7+79377408*x^6+80034048*x^5+66969216*x^ 
4+58946544*x^3+41031048*x^2+19860062*x[phone])*(3*x^2-x+2)^(1/2)+2707157 
5/35831808*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.67 \[ \int (1+2 x) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{5971968} \, {\left (47775744 \, x^{8} + 30357504 \, x^{7} + 79377408 \, x^{6} + 80034048 \, x^{5} + 66969216 \, x^{4} + 58946544 \, x^{3} + 41031048 \, x^{2} + 19860062 \, x + 10960335\right )} \sqrt {3 \, x^{2} - x + 2} + \frac {27071575}{71663616} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \] Input:

integrate((1+2*x)*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="fricas")
 

Output:

1/5971968*(47775744*x^8 + 30357504*x^7 + 79377408*x^6 + 80034048*x^5 + 669 
69216*x^4 + 58946544*x^3 + 41031048*x^2 + 19860062*x + 10960335)*sqrt(3*x^ 
2 - x + 2) + 27071575/71663616*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)* 
(6*x - 1) - 72*x^2 + 24*x - 25)
 

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.63 \[ \int (1+2 x) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\sqrt {3 x^{2} - x + 2} \cdot \left (8 x^{8} + \frac {61 x^{7}}{12} + \frac {319 x^{6}}{24} + \frac {11579 x^{5}}{864} + \frac {58133 x^{4}}{5184} + \frac {409351 x^{3}}{41472} + \frac {1709627 x^{2}}{248832} + \frac {9930031 x}{2985984} + \frac {1217815}{663552}\right ) + \frac {27071575 \sqrt {3} \operatorname {asinh}{\left (\frac {6 \sqrt {23} \left (x - \frac {1}{6}\right )}{23} \right )}}{35831808} \] Input:

integrate((1+2*x)*(3*x**2-x+2)**(5/2)*(4*x**2+3*x+1),x)
 

Output:

sqrt(3*x**2 - x + 2)*(8*x**8 + 61*x**7/12 + 319*x**6/24 + 11579*x**5/864 + 
 58133*x**4/5184 + 409351*x**3/41472 + 1709627*x**2/248832 + 9930031*x/298 
5984 + 1217815/663552) + 27071575*sqrt(3)*asinh(6*sqrt(23)*(x - 1/6)/23)/3 
5831808
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.08 \[ \int (1+2 x) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {8}{27} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {157}{324} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} x + \frac {185}{648} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {7}{2}} + \frac {445}{2592} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x - \frac {445}{15552} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} + \frac {51175}{124416} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x - \frac {51175}{746496} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} + \frac {1177025}{995328} \, \sqrt {3 \, x^{2} - x + 2} x + \frac {27071575}{35831808} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {1177025}{5971968} \, \sqrt {3 \, x^{2} - x + 2} \] Input:

integrate((1+2*x)*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="maxima")
 

Output:

8/27*(3*x^2 - x + 2)^(7/2)*x^2 + 157/324*(3*x^2 - x + 2)^(7/2)*x + 185/648 
*(3*x^2 - x + 2)^(7/2) + 445/2592*(3*x^2 - x + 2)^(5/2)*x - 445/15552*(3*x 
^2 - x + 2)^(5/2) + 51175/124416*(3*x^2 - x + 2)^(3/2)*x - 51175/746496*(3 
*x^2 - x + 2)^(3/2) + 1177025/995328*sqrt(3*x^2 - x + 2)*x + 27071575/3583 
1808*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) - 1177025/5971968*sqrt(3*x^2 
 - x + 2)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.63 \[ \int (1+2 x) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\frac {1}{5971968} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (96 \, x + 61\right )} x + 319\right )} x + 11579\right )} x + 58133\right )} x + 409351\right )} x + 1709627\right )} x + 9930031\right )} x + 10960335\right )} \sqrt {3 \, x^{2} - x + 2} - \frac {27071575}{35831808} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) \] Input:

integrate((1+2*x)*(3*x^2-x+2)^(5/2)*(4*x^2+3*x+1),x, algorithm="giac")
 

Output:

1/5971968*(2*(12*(6*(8*(6*(36*(2*(96*x + 61)*x + 319)*x + 11579)*x + 58133 
)*x + 409351)*x + 1709627)*x + 9930031)*x + 10960335)*sqrt(3*x^2 - x + 2) 
- 27071575/35831808*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2 
)) + 1)
 

Mupad [F(-1)]

Timed out. \[ \int (1+2 x) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=\int \left (2\,x+1\right )\,{\left (3\,x^2-x+2\right )}^{5/2}\,\left (4\,x^2+3\,x+1\right ) \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.22 \[ \int (1+2 x) \left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right ) \, dx=8 \sqrt {3 x^{2}-x +2}\, x^{8}+\frac {61 \sqrt {3 x^{2}-x +2}\, x^{7}}{12}+\frac {319 \sqrt {3 x^{2}-x +2}\, x^{6}}{24}+\frac {11579 \sqrt {3 x^{2}-x +2}\, x^{5}}{864}+\frac {58133 \sqrt {3 x^{2}-x +2}\, x^{4}}{5184}+\frac {409351 \sqrt {3 x^{2}-x +2}\, x^{3}}{41472}+\frac {1709627 \sqrt {3 x^{2}-x +2}\, x^{2}}{248832}+\frac {9930031 \sqrt {3 x^{2}-x +2}\, x}{2985984}+\frac {1217815 \sqrt {3 x^{2}-x +2}}{663552}+\frac {27071575 \sqrt {3}\, \mathrm {log}\left (\frac {2 \sqrt {3 x^{2}-x +2}\, \sqrt {3}+6 x -1}{\sqrt {23}}\right )}{35831808} \] Input:

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

Output:

(286654464*sqrt(3*x**2 - x + 2)*x**8 + 182145024*sqrt(3*x**2 - x + 2)*x**7 
 + 476264448*sqrt(3*x**2 - x + 2)*x**6 + 480204288*sqrt(3*x**2 - x + 2)*x* 
*5 + 401815296*sqrt(3*x**2 - x + 2)*x**4 + 353679264*sqrt(3*x**2 - x + 2)* 
x**3 + 246186288*sqrt(3*x**2 - x + 2)*x**2 + 119160372*sqrt(3*x**2 - x + 2 
)*x + 65762010*sqrt(3*x**2 - x + 2) + 27071575*sqrt(3)*log((2*sqrt(3*x**2 
- x + 2)*sqrt(3) + 6*x - 1)/sqrt(23)))/35831808