\(\int (3-x+2 x^2)^{3/2} (2+3 x+5 x^2) \, dx\) [102]

Optimal result

Integrand size = 25, antiderivative size = 105 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=-\frac {4117 (1-4 x) \sqrt {3-x+2 x^2}}{8192}-\frac {179 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {107}{240} \left (3-x+2 x^2\right )^{5/2}+\frac {5}{12} x \left (3-x+2 x^2\right )^{5/2}-\frac {94691 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16384 \sqrt {2}} \] Output:

-4117/8192*(1-4*x)*(2*x^2-x+3)^(1/2)-179/1536*(1-4*x)*(2*x^2-x+3)^(3/2)+10 
7/240*(2*x^2-x+3)^(5/2)+5/12*x*(2*x^2-x+3)^(5/2)-94691/32768*arcsinh(1/23* 
(1-4*x)*23^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (388341+565276 x+319072 x^2+561024 x^3+14336 x^4+204800 x^5\right )-1420365 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{491520} \] Input:

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

Output:

(4*Sqrt[3 - x + 2*x^2]*(388341 + 565276*x + 319072*x^2 + 561024*x^3 + 1433 
6*x^4 + 204800*x^5) - 1420365*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2 
]])/491520
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2192, 27, 1160, 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.

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{12} \int \frac {1}{2} (107 x+18) \left (2 x^2-x+3\right )^{3/2}dx+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{24} \int (107 x+18) \left (2 x^2-x+3\right )^{3/2}dx+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \int \left (2 x^2-x+3\right )^{3/2}dx+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{24} \left (\frac {179}{4} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {107}{10} \left (2 x^2-x+3\right )^{5/2}\right )+\frac {5}{12} x \left (2 x^2-x+3\right )^{5/2}\)

Input:

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

Output:

(5*x*(3 - x + 2*x^2)^(5/2))/12 + ((107*(3 - x + 2*x^2)^(5/2))/10 + (179*(- 
1/16*((1 - 4*x)*(3 - x + 2*x^2)^(3/2)) + (69*(-1/8*((1 - 4*x)*Sqrt[3 - x + 
 2*x^2]) + (23*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/32))/4)/24
 

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 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 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 = 1.88 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {\left (204800 x^{5}+14336 x^{4}+561024 x^{3}+319072 x^{2}+565276 x +388341\right ) \sqrt {2 x^{2}-x +3}}{122880}+\frac {94691 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{32768}\)	\(55\)
trager	\(\left (\frac {5}{3} x^{5}+\frac {7}{60} x^{4}+\frac {1461}{320} x^{3}+\frac {9971}{3840} x^{2}+\frac {141319}{30720} x +\frac {129447}{40960}\right ) \sqrt {2 x^{2}-x +3}-\frac {94691 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{32768}\)	\(79\)
default	\(\frac {179 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{1536}+\frac {4117 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{8192}+\frac {94691 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{32768}+\frac {107 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{240}+\frac {5 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{12}\)	\(83\)

Input:

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

Output:

1/122880*(204800*x^5+14336*x^4+561024*x^3+319072*x^2+565276*x+388341)*(2*x 
^2-x+3)^(1/2)+94691/32768*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {1}{122880} \, {\left (204800 \, x^{5} + 14336 \, x^{4} + 561024 \, x^{3} + 319072 \, x^{2} + 565276 \, x + 388341\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {94691}{65536} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:

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

Output:

1/122880*(204800*x^5 + 14336*x^4 + 561024*x^3 + 319072*x^2 + 565276*x + 38 
8341)*sqrt(2*x^2 - x + 3) + 94691/65536*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 
- x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
 

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {5 x^{5}}{3} + \frac {7 x^{4}}{60} + \frac {1461 x^{3}}{320} + \frac {9971 x^{2}}{3840} + \frac {141319 x}{30720} + \frac {129447}{40960}\right ) + \frac {94691 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{32768} \] Input:

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

Output:

sqrt(2*x**2 - x + 3)*(5*x**5/3 + 7*x**4/60 + 1461*x**3/320 + 9971*x**2/384 
0 + 141319*x/30720 + 129447/40960) + 94691*sqrt(2)*asinh(4*sqrt(23)*(x - 1 
/4)/23)/32768
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {5}{12} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {107}{240} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {179}{384} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {179}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {4117}{2048} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {94691}{32768} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {4117}{8192} \, \sqrt {2 \, x^{2} - x + 3} \] Input:

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

Output:

5/12*(2*x^2 - x + 3)^(5/2)*x + 107/240*(2*x^2 - x + 3)^(5/2) + 179/384*(2* 
x^2 - x + 3)^(3/2)*x - 179/1536*(2*x^2 - x + 3)^(3/2) + 4117/2048*sqrt(2*x 
^2 - x + 3)*x + 94691/32768*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 411 
7/8192*sqrt(2*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {1}{122880} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x + 7\right )} x + 4383\right )} x + 9971\right )} x + 141319\right )} x + 388341\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {94691}{32768} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:

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

Output:

1/122880*(4*(8*(4*(16*(100*x + 7)*x + 4383)*x + 9971)*x + 141319)*x + 3883 
41)*sqrt(2*x^2 - x + 3) - 94691/32768*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - 
sqrt(2*x^2 - x + 3)) + 1)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {5 \sqrt {2 x^{2}-x +3}\, x^{5}}{3}+\frac {7 \sqrt {2 x^{2}-x +3}\, x^{4}}{60}+\frac {1461 \sqrt {2 x^{2}-x +3}\, x^{3}}{320}+\frac {9971 \sqrt {2 x^{2}-x +3}\, x^{2}}{3840}+\frac {141319 \sqrt {2 x^{2}-x +3}\, x}{30720}+\frac {129447 \sqrt {2 x^{2}-x +3}}{40960}+\frac {94691 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{32768} \] Input:

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

Output:

(819200*sqrt(2*x**2 - x + 3)*x**5 + 57344*sqrt(2*x**2 - x + 3)*x**4 + 2244 
096*sqrt(2*x**2 - x + 3)*x**3 + 1276288*sqrt(2*x**2 - x + 3)*x**2 + 226110 
4*sqrt(2*x**2 - x + 3)*x + 1553364*sqrt(2*x**2 - x + 3) + 1420365*sqrt(2)* 
log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23)))/491520