\(\int (3-x+2 x^2)^{5/2} (2+3 x+5 x^2) \, dx\) [109]

Optimal result

Integrand size = 25, antiderivative size = 128 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=-\frac {732665 (1-4 x) \sqrt {3-x+2 x^2}}{524288}-\frac {31855 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{98304}-\frac {277 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{3072}+\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}-\frac {16851295 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1048576 \sqrt {2}} \] Output:

-732665/524288*(1-4*x)*(2*x^2-x+3)^(1/2)-31855/98304*(1-4*x)*(2*x^2-x+3)^( 
3/2)-277/3072*(1-4*x)*(2*x^2-x+3)^(5/2)+141/448*(2*x^2-x+3)^(7/2)+5/16*x*( 
2*x^2-x+3)^(7/2)-16851295/2097152*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.66 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (58536675+148957444 x+67272352 x^2+172684416 x^3-1619968 x^4+118808576 x^5-13565952 x^6+27525120 x^7\right )-353877195 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{44040192} \] Input:

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

Output:

(4*Sqrt[3 - x + 2*x^2]*(58536675 + 148957444*x + 67272352*x^2 + 172684416* 
x^3 - 1619968*x^4 + 118808576*x^5 - 13565952*x^6 + 27525120*x^7) - 3538771 
95*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/44040192
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2192, 27, 1160, 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.

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{16} \int \frac {1}{2} (141 x+34) \left (2 x^2-x+3\right )^{5/2}dx+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \int (141 x+34) \left (2 x^2-x+3\right )^{5/2}dx+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \int \left (2 x^2-x+3\right )^{5/2}dx+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \int \left (2 x^2-x+3\right )^{3/2}dx-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{32} \left (\frac {277}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )+\frac {141}{14} \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}\)

Input:

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

Output:

(5*x*(3 - x + 2*x^2)^(7/2))/16 + ((141*(3 - x + 2*x^2)^(7/2))/14 + (277*(- 
1/24*((1 - 4*x)*(3 - x + 2*x^2)^(5/2)) + (115*(-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))/48))/4)/32
 

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 = 1.91 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.51

method	result	size
risch	\(\frac {\left (27525120 x^{7}-13565952 x^{6}[phone] x^{5}-1619968 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {2 x^{2}-x +3}}{11010048}+\frac {16851295 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2097152}\)	\(65\)
trager	\(\left (\frac {5}{2} x^{7}-\frac {69}{56} x^{6}+\frac {14503}{1344} x^{5}-\frac {113}{768} x^{4}+\frac {449699}{28672} x^{3}+\frac {300323}{49152} x^{2}+\frac {37239361}{2752512} x +\frac {19512225}{3670016}\right ) \sqrt {2 x^{2}-x +3}-\frac {16851295 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{2097152}\)	\(89\)
default	\(\frac {277 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{3072}+\frac {31855 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{98304}+\frac {732665 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{524288}+\frac {16851295 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2097152}+\frac {141 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{448}+\frac {5 x \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{16}\)	\(102\)

Input:

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

Output:

1/11010048*(27525120*x^7-13565952*x^6+118808576*x^5-1619968*x^4+172684416* 
x^3+67272352*x^2+148957444*x[phone])*(2*x^2-x+3)^(1/2)[phone]/2097152* 
2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.69 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {1}{11010048} \, {\left (27525120 \, x^{7} - 13565952 \, x^{6} + 118808576 \, x^{5} - 1619968 \, x^{4} + 172684416 \, x^{3} + 67272352 \, x^{2} + 148957444 \, x + 58536675\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {16851295}{4194304} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:

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

Output:

1/11010048*(27525120*x^7 - 13565952*x^6 + 118808576*x^5 - 1619968*x^4 + 17 
2684416*x^3 + 67272352*x^2 + 148957444*x + 58536675)*sqrt(2*x^2 - x + 3) + 
 16851295/4194304*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 3 
2*x^2 + 16*x - 25)
 

Sympy [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {5 x^{7}}{2} - \frac {69 x^{6}}{56} + \frac {14503 x^{5}}{1344} - \frac {113 x^{4}}{768} + \frac {449699 x^{3}}{28672} + \frac {300323 x^{2}}{49152} + \frac {37239361 x}{2752512} + \frac {19512225}{3670016}\right ) + \frac {16851295 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{2097152} \] Input:

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

Output:

sqrt(2*x**2 - x + 3)*(5*x**7/2 - 69*x**6/56 + 14503*x**5/1344 - 113*x**4/7 
68 + 449699*x**3/28672 + 300323*x**2/49152 + 37239361*x/2752512 + 19512225 
/3670016) + 16851295*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/2097152
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {5}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x + \frac {141}{448} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {277}{768} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {277}{3072} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {31855}{24576} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {31855}{98304} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {732665}{131072} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {16851295}{2097152} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {732665}{524288} \, \sqrt {2 \, x^{2} - x + 3} \] Input:

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

Output:

5/16*(2*x^2 - x + 3)^(7/2)*x + 141/448*(2*x^2 - x + 3)^(7/2) + 277/768*(2* 
x^2 - x + 3)^(5/2)*x - 277/3072*(2*x^2 - x + 3)^(5/2) + 31855/24576*(2*x^2 
 - x + 3)^(3/2)*x - 31855/98304*(2*x^2 - x + 3)^(3/2) + 732665/131072*sqrt 
(2*x^2 - x + 3)*x + 16851295/2097152*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 
1)) - 732665/524288*sqrt(2*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.65 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {1}{11010048} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 69\right )} x + 14503\right )} x - 791\right )} x + 1349097\right )} x + 2102261\right )} x + 37239361\right )} x + 58536675\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {16851295}{2097152} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:

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

Output:

1/11010048*(4*(8*(4*(16*(4*(24*(140*x - 69)*x + 14503)*x - 791)*x + 134909 
7)*x + 2102261)*x + 37239361)*x + 58536675)*sqrt(2*x^2 - x + 3) - 16851295 
/2097152*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.20 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx=\frac {5 \sqrt {2 x^{2}-x +3}\, x^{7}}{2}-\frac {69 \sqrt {2 x^{2}-x +3}\, x^{6}}{56}+\frac {14503 \sqrt {2 x^{2}-x +3}\, x^{5}}{1344}-\frac {113 \sqrt {2 x^{2}-x +3}\, x^{4}}{768}+\frac {449699 \sqrt {2 x^{2}-x +3}\, x^{3}}{28672}+\frac {300323 \sqrt {2 x^{2}-x +3}\, x^{2}}{49152}+\frac {37239361 \sqrt {2 x^{2}-x +3}\, x}{2752512}+\frac {19512225 \sqrt {2 x^{2}-x +3}}{3670016}+\frac {16851295 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{2097152} \] Input:

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

Output:

(110100480*sqrt(2*x**2 - x + 3)*x**7 - 54263808*sqrt(2*x**2 - x + 3)*x**6 
+ 475234304*sqrt(2*x**2 - x + 3)*x**5 - 6479872*sqrt(2*x**2 - x + 3)*x**4 
+ 690737664*sqrt(2*x**2 - x + 3)*x**3 + 269089408*sqrt(2*x**2 - x + 3)*x** 
2 + 595829776*sqrt(2*x**2 - x + 3)*x + 234146700*sqrt(2*x**2 - x + 3) + 35 
3877195*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23)))/ 
44040192