\(\int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx\) [195]

Optimal result

Integrand size = 24, antiderivative size = 122 \[ \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx=\frac {2341}{18} x \sqrt {2+3 x^2}+\frac {923}{315} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}+\frac {29}{63} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^4 \left (2+3 x^2\right )^{3/2}+\frac {2}{405} (13781+4599 x) \left (2+3 x^2\right )^{3/2}+\frac {2341 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \] Output:

2341/18*x*(3*x^2+2)^(1/2)+923/315*(3+2*x)^2*(3*x^2+2)^(3/2)+29/63*(3+2*x)^ 
3*(3*x^2+2)^(3/2)-1/21*(3+2*x)^4*(3*x^2+2)^(3/2)+2/405*(13781+4599*x)*(3*x 
^2+2)^(3/2)+2341/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.62 \[ \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx=-\frac {\sqrt {2+3 x^2} \left (-1167988-1558935 x-1956174 x^2-1222200 x^3-297648 x^4+15120 x^5+12960 x^6\right )}{5670}-\frac {2341 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{9 \sqrt {3}} \] Input:

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

Output:

-1/5670*(Sqrt[2 + 3*x^2]*(-1167988 - 1558935*x - 1956174*x^2 - 1222200*x^3 
 - 297648*x^4 + 15120*x^5 + 12960*x^6)) - (2341*Log[-(Sqrt[3]*x) + Sqrt[2 
+ 3*x^2]])/(9*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {687, 687, 27, 687, 27, 676, 211, 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.

\(\displaystyle \int (5-x) (2 x+3)^4 \sqrt {3 x^2+2} \, dx\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{21} \int (2 x+3)^3 (174 x+331) \sqrt {3 x^2+2}dx-\frac {1}{21} (2 x+3)^4 \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{21} \left (\frac {1}{18} \int 18 (2 x+3)^2 (923 x+877) \sqrt {3 x^2+2}dx+\frac {29}{3} \left (3 x^2+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \left (\int (2 x+3)^2 (923 x+877) \sqrt {3 x^2+2}dx+\frac {29}{3} \left (3 x^2+2\right )^{3/2} (2 x+3)^3\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{21} \left (\frac {1}{15} \int 7 (2 x+3) (6132 x+4583) \sqrt {3 x^2+2}dx+\frac {29}{3} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {923}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \left (\frac {7}{15} \int (2 x+3) (6132 x+4583) \sqrt {3 x^2+2}dx+\frac {29}{3} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {923}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {1}{21} \left (\frac {7}{15} \left (11705 \int \sqrt {3 x^2+2}dx+1022 x \left (3 x^2+2\right )^{3/2}+\frac {27562}{9} \left (3 x^2+2\right )^{3/2}\right )+\frac {29}{3} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {923}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {1}{21} \left (\frac {7}{15} \left (11705 \left (\int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \sqrt {3 x^2+2} x\right )+1022 x \left (3 x^2+2\right )^{3/2}+\frac {27562}{9} \left (3 x^2+2\right )^{3/2}\right )+\frac {29}{3} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {923}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+2\right )^{3/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{21} \left (\frac {7}{15} \left (11705 \left (\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3 x^2+2} x\right )+1022 x \left (3 x^2+2\right )^{3/2}+\frac {27562}{9} \left (3 x^2+2\right )^{3/2}\right )+\frac {29}{3} \left (3 x^2+2\right )^{3/2} (2 x+3)^3+\frac {923}{15} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{21} (2 x+3)^4 \left (3 x^2+2\right )^{3/2}\)

Input:

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

Output:

-1/21*((3 + 2*x)^4*(2 + 3*x^2)^(3/2)) + ((923*(3 + 2*x)^2*(2 + 3*x^2)^(3/2 
))/15 + (29*(3 + 2*x)^3*(2 + 3*x^2)^(3/2))/3 + (7*((27562*(2 + 3*x^2)^(3/2 
))/9 + 1022*x*(2 + 3*x^2)^(3/2) + 11705*((x*Sqrt[2 + 3*x^2])/2 + ArcSinh[S 
qrt[3/2]*x]/Sqrt[3])))/15)/21
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 676

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

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.45

method	result	size
risch	\(-\frac {\left (12960 x^{6}+15120 x^{5}-297648 x^{4}-1222200 x^{3}-1956174 x^{2}-1558935 x -1167988\right ) \sqrt {3 x^{2}+2}}{5670}+\frac {2341 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{27}\)	\(55\)
trager	\(\left (-\frac {16}{7} x^{6}-\frac {8}{3} x^{5}+\frac {5512}{105} x^{4}+\frac {1940}{9} x^{3}+\frac {326029}{945} x^{2}+\frac {4949}{18} x +\frac {583994}{2835}\right ) \sqrt {3 x^{2}+2}+\frac {2341 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{27}\)	\(71\)
default	\(\frac {2341 x \sqrt {3 x^{2}+2}}{18}+\frac {2341 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{27}+\frac {291997 \left (3 x^{2}+2\right )^{\frac {3}{2}}}{2835}+\frac {652 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{9}+\frac {5672 x^{2} \left (3 x^{2}+2\right )^{\frac {3}{2}}}{315}-\frac {8 x^{3} \left (3 x^{2}+2\right )^{\frac {3}{2}}}{9}-\frac {16 x^{4} \left (3 x^{2}+2\right )^{\frac {3}{2}}}{21}\)	\(91\)
meijerg	\(-\frac {135 \sqrt {3}\, \left (-\sqrt {6}\, \sqrt {\pi }\, x \sqrt {\frac {3 x^{2}}{2}+1}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{2 \sqrt {\pi }}+\frac {32 \sqrt {3}\, \left (\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (-90 x^{4}-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{120}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{4}\right )}{27 \sqrt {\pi }}-\frac {88 \sqrt {2}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}} \left (-\frac {9 x^{2}}{2}+2\right )}{15}\right )}{3 \sqrt {\pi }}-\frac {96 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (9 x^{2}+3\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{12}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{\sqrt {\pi }}-\frac {333 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (3 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{3}\right )}{2 \sqrt {\pi }}+\frac {32 \sqrt {2}\, \left (\frac {32 \sqrt {\pi }}{105}-\frac {4 \sqrt {\pi }\, \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}} \left (\frac {135}{4} x^{4}-18 x^{2}+8\right )}{105}\right )}{27 \sqrt {\pi }}\)	\(257\)

Input:

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

Output:

-1/5670*(12960*x^6+15120*x^5-297648*x^4-1222200*x^3-1956174*x^2-1558935*x- 
1167988)*(3*x^2+2)^(1/2)+2341/27*arcsinh(1/2*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.57 \[ \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx=-\frac {1}{5670} \, {\left (12960 \, x^{6} + 15120 \, x^{5} - 297648 \, x^{4} - 1222200 \, x^{3} - 1956174 \, x^{2} - 1558935 \, x - 1167988\right )} \sqrt {3 \, x^{2} + 2} + \frac {2341}{54} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \] Input:

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

Output:

-1/5670*(12960*x^6 + 15120*x^5 - 297648*x^4 - 1222200*x^3 - 1956174*x^2 - 
1558935*x - 1167988)*sqrt(3*x^2 + 2) + 2341/54*sqrt(3)*log(-sqrt(3)*sqrt(3 
*x^2 + 2)*x - 3*x^2 - 1)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx=- \frac {16 x^{6} \sqrt {3 x^{2} + 2}}{7} - \frac {8 x^{5} \sqrt {3 x^{2} + 2}}{3} + \frac {5512 x^{4} \sqrt {3 x^{2} + 2}}{105} + \frac {1940 x^{3} \sqrt {3 x^{2} + 2}}{9} + \frac {326029 x^{2} \sqrt {3 x^{2} + 2}}{945} + \frac {4949 x \sqrt {3 x^{2} + 2}}{18} + \frac {583994 \sqrt {3 x^{2} + 2}}{2835} + \frac {2341 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27} \] Input:

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

Output:

-16*x**6*sqrt(3*x**2 + 2)/7 - 8*x**5*sqrt(3*x**2 + 2)/3 + 5512*x**4*sqrt(3 
*x**2 + 2)/105 + 1940*x**3*sqrt(3*x**2 + 2)/9 + 326029*x**2*sqrt(3*x**2 + 
2)/945 + 4949*x*sqrt(3*x**2 + 2)/18 + 583994*sqrt(3*x**2 + 2)/2835 + 2341* 
sqrt(3)*asinh(sqrt(6)*x/2)/27
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.74 \[ \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx=-\frac {16}{21} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{4} - \frac {8}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{3} + \frac {5672}{315} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + \frac {652}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {291997}{2835} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {2341}{18} \, \sqrt {3 \, x^{2} + 2} x + \frac {2341}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \] Input:

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

Output:

-16/21*(3*x^2 + 2)^(3/2)*x^4 - 8/9*(3*x^2 + 2)^(3/2)*x^3 + 5672/315*(3*x^2 
 + 2)^(3/2)*x^2 + 652/9*(3*x^2 + 2)^(3/2)*x + 291997/2835*(3*x^2 + 2)^(3/2 
) + 2341/18*sqrt(3*x^2 + 2)*x + 2341/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.52 \[ \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx=-\frac {1}{5670} \, {\left (3 \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (5 \, {\left (6 \, x + 7\right )} x - 689\right )} x - 16975\right )} x - 326029\right )} x - 519645\right )} x - 1167988\right )} \sqrt {3 \, x^{2} + 2} - \frac {2341}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \] Input:

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

Output:

-1/5670*(3*(2*(12*(6*(5*(6*x + 7)*x - 689)*x - 16975)*x - 326029)*x - 5196 
45)*x - 1167988)*sqrt(3*x^2 + 2) - 2341/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3 
*x^2 + 2))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.45 \[ \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx=\frac {2341\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {48\,x^6}{7}-8\,x^5+\frac {5512\,x^4}{35}+\frac {1940\,x^3}{3}+\frac {326029\,x^2}{315}+\frac {4949\,x}{6}+\frac {583994}{945}\right )}{3} \] Input:

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

Output:

(2341*3^(1/2)*asinh((6^(1/2)*x)/2))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((4949 
*x)/6 + (326029*x^2)/315 + (1940*x^3)/3 + (5512*x^4)/35 - 8*x^5 - (48*x^6) 
/7 + 583994/945))/3
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.90 \[ \int (5-x) (3+2 x)^4 \sqrt {2+3 x^2} \, dx=-\frac {16 \sqrt {3 x^{2}+2}\, x^{6}}{7}-\frac {8 \sqrt {3 x^{2}+2}\, x^{5}}{3}+\frac {5512 \sqrt {3 x^{2}+2}\, x^{4}}{105}+\frac {1940 \sqrt {3 x^{2}+2}\, x^{3}}{9}+\frac {326029 \sqrt {3 x^{2}+2}\, x^{2}}{945}+\frac {4949 \sqrt {3 x^{2}+2}\, x}{18}+\frac {583994 \sqrt {3 x^{2}+2}}{2835}+\frac {2341 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{27} \] Input:

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

Output:

( - 12960*sqrt(3*x**2 + 2)*x**6 - 15120*sqrt(3*x**2 + 2)*x**5 + 297648*sqr 
t(3*x**2 + 2)*x**4 + 1222200*sqrt(3*x**2 + 2)*x**3 + 1956174*sqrt(3*x**2 + 
 2)*x**2 + 1558935*sqrt(3*x**2 + 2)*x + 1167988*sqrt(3*x**2 + 2) + 491610* 
sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)))/5670