\(\int (5-x) (3+2 x)^3 (2+3 x^2)^{3/2} \, dx\) [208]

Optimal result

Integrand size = 24, antiderivative size = 116 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx=\frac {1087}{12} x \sqrt {2+3 x^2}+\frac {1087}{36} x \left (2+3 x^2\right )^{3/2}+\frac {71}{168} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}-\frac {1}{24} (3+2 x)^3 \left (2+3 x^2\right )^{5/2}+\frac {(16973+5405 x) \left (2+3 x^2\right )^{5/2}}{1260}+\frac {1087 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \] Output:

1087/12*x*(3*x^2+2)^(1/2)+1087/36*x*(3*x^2+2)^(3/2)+71/168*(3+2*x)^2*(3*x^ 
2+2)^(5/2)-1/24*(3+2*x)^3*(3*x^2+2)^(5/2)+1/1260*(16973+5405*x)*(3*x^2+2)^ 
(5/2)+1087/18*arcsinh(1/2*x*6^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {\sqrt {2+3 x^2} \left (-81392-226065 x-245136 x^2-219975 x^3-186012 x^4-75600 x^5-2160 x^6+3780 x^7\right )}{1260}-\frac {1087 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{6 \sqrt {3}} \] Input:

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

Output:

-1/1260*(Sqrt[2 + 3*x^2]*(-81392 - 226065*x - 245136*x^2 - 219975*x^3 - 18 
6012*x^4 - 75600*x^5 - 2160*x^6 + 3780*x^7)) - (1087*Log[-(Sqrt[3]*x) + Sq 
rt[2 + 3*x^2]])/(6*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {687, 27, 687, 27, 676, 211, 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.

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{24} \int 3 (2 x+3)^2 (71 x+124) \left (3 x^2+2\right )^{3/2}dx-\frac {1}{24} (2 x+3)^3 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int (2 x+3)^2 (71 x+124) \left (3 x^2+2\right )^{3/2}dx-\frac {1}{24} (2 x+3)^3 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{8} \left (\frac {1}{21} \int 2 (2 x+3) (3243 x+3622) \left (3 x^2+2\right )^{3/2}dx+\frac {71}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {2}{21} \int (2 x+3) (3243 x+3622) \left (3 x^2+2\right )^{3/2}dx+\frac {71}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {1}{8} \left (\frac {2}{21} \left (\frac {30436}{3} \int \left (3 x^2+2\right )^{3/2}dx+\frac {1081}{3} x \left (3 x^2+2\right )^{5/2}+\frac {16973}{15} \left (3 x^2+2\right )^{5/2}\right )+\frac {71}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {1}{8} \left (\frac {2}{21} \left (\frac {30436}{3} \left (\frac {3}{2} \int \sqrt {3 x^2+2}dx+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1081}{3} x \left (3 x^2+2\right )^{5/2}+\frac {16973}{15} \left (3 x^2+2\right )^{5/2}\right )+\frac {71}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {1}{8} \left (\frac {2}{21} \left (\frac {30436}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1081}{3} x \left (3 x^2+2\right )^{5/2}+\frac {16973}{15} \left (3 x^2+2\right )^{5/2}\right )+\frac {71}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+2\right )^{5/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{8} \left (\frac {2}{21} \left (\frac {30436}{3} \left (\frac {3}{2} \left (\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1081}{3} x \left (3 x^2+2\right )^{5/2}+\frac {16973}{15} \left (3 x^2+2\right )^{5/2}\right )+\frac {71}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}\right )-\frac {1}{24} (2 x+3)^3 \left (3 x^2+2\right )^{5/2}\)

Input:

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

Output:

-1/24*((3 + 2*x)^3*(2 + 3*x^2)^(5/2)) + ((71*(3 + 2*x)^2*(2 + 3*x^2)^(5/2) 
)/21 + (2*((16973*(2 + 3*x^2)^(5/2))/15 + (1081*x*(2 + 3*x^2)^(5/2))/3 + ( 
30436*((x*(2 + 3*x^2)^(3/2))/4 + (3*((x*Sqrt[2 + 3*x^2])/2 + ArcSinh[Sqrt[ 
3/2]*x]/Sqrt[3]))/2))/3))/21)/8
 

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 = 0.75 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {\left (3780 x^{7}-2160 x^{6}-75600 x^{5}-186012 x^{4}-219975 x^{3}-245136 x^{2}-226065 x -81392\right ) \sqrt {3 x^{2}+2}}{1260}+\frac {1087 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{18}\)	\(60\)
trager	\(\left (-3 x^{7}+\frac {12}{7} x^{6}+60 x^{5}+\frac {5167}{35} x^{4}+\frac {2095}{12} x^{3}+\frac {20428}{105} x^{2}+\frac {2153}{12} x +\frac {20348}{315}\right ) \sqrt {3 x^{2}+2}-\frac {1087 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{18}\)	\(77\)
default	\(\frac {1087 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{36}+\frac {1087 x \sqrt {3 x^{2}+2}}{12}+\frac {1087 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{18}+\frac {5087 \left (3 x^{2}+2\right )^{\frac {5}{2}}}{315}+\frac {64 x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{9}+\frac {4 x^{2} \left (3 x^{2}+2\right )^{\frac {5}{2}}}{21}-\frac {x^{3} \left (3 x^{2}+2\right )^{\frac {5}{2}}}{3}\)	\(89\)
meijerg	\(\frac {135 \sqrt {3}\, \left (\frac {4 \sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {3 x^{2}}{8}+\frac {5}{8}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{3}+\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{2 \sqrt {\pi }}+\frac {4 \sqrt {2}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {2 \sqrt {\pi }\, \left (-\frac {135}{2} x^{6}-72 x^{4}-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{105}\right )}{3 \sqrt {\pi }}+\frac {42 \sqrt {3}\, \left (\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (18 x^{4}+21 x^{2}+3\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{36}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{6}\right )}{\sqrt {\pi }}+\frac {243 \sqrt {2}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {9}{2} x^{4}+6 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}\right )}{2 \sqrt {\pi }}-\frac {16 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (-270 x^{6}-270 x^{4}-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{480}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{16}\right )}{9 \sqrt {\pi }}\)	\(251\)

Input:

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

Output:

-1/1260*(3780*x^7-2160*x^6-75600*x^5-186012*x^4-219975*x^3-245136*x^2-2260 
65*x-81392)*(3*x^2+2)^(1/2)+1087/18*arcsinh(1/2*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{1260} \, {\left (3780 \, x^{7} - 2160 \, x^{6} - 75600 \, x^{5} - 186012 \, x^{4} - 219975 \, x^{3} - 245136 \, x^{2} - 226065 \, x - 81392\right )} \sqrt {3 \, x^{2} + 2} + \frac {1087}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \] Input:

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

Output:

-1/1260*(3780*x^7 - 2160*x^6 - 75600*x^5 - 186012*x^4 - 219975*x^3 - 24513 
6*x^2 - 226065*x - 81392)*sqrt(3*x^2 + 2) + 1087/36*sqrt(3)*log(-sqrt(3)*s 
qrt(3*x^2 + 2)*x - 3*x^2 - 1)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 1.54 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.24 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx=- 3 x^{7} \sqrt {3 x^{2} + 2} + \frac {12 x^{6} \sqrt {3 x^{2} + 2}}{7} + 60 x^{5} \sqrt {3 x^{2} + 2} + \frac {5167 x^{4} \sqrt {3 x^{2} + 2}}{35} + \frac {2095 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {20428 x^{2} \sqrt {3 x^{2} + 2}}{105} + \frac {2153 x \sqrt {3 x^{2} + 2}}{12} + \frac {20348 \sqrt {3 x^{2} + 2}}{315} + \frac {1087 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \] Input:

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

Output:

-3*x**7*sqrt(3*x**2 + 2) + 12*x**6*sqrt(3*x**2 + 2)/7 + 60*x**5*sqrt(3*x** 
2 + 2) + 5167*x**4*sqrt(3*x**2 + 2)/35 + 2095*x**3*sqrt(3*x**2 + 2)/12 + 2 
0428*x**2*sqrt(3*x**2 + 2)/105 + 2153*x*sqrt(3*x**2 + 2)/12 + 20348*sqrt(3 
*x**2 + 2)/315 + 1087*sqrt(3)*asinh(sqrt(6)*x/2)/18
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{3} + \frac {4}{21} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{2} + \frac {64}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {5087}{315} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {1087}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {1087}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {1087}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \] Input:

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

Output:

-1/3*(3*x^2 + 2)^(5/2)*x^3 + 4/21*(3*x^2 + 2)^(5/2)*x^2 + 64/9*(3*x^2 + 2) 
^(5/2)*x + 5087/315*(3*x^2 + 2)^(5/2) + 1087/36*(3*x^2 + 2)^(3/2)*x + 1087 
/12*sqrt(3*x^2 + 2)*x + 1087/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.57 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{1260} \, {\left (3 \, {\left ({\left ({\left (12 \, {\left (15 \, {\left ({\left (7 \, x - 4\right )} x - 140\right )} x - 5167\right )} x - 73325\right )} x - 81712\right )} x - 75355\right )} x - 81392\right )} \sqrt {3 \, x^{2} + 2} - \frac {1087}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \] Input:

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

Output:

-1/1260*(3*(((12*(15*((7*x - 4)*x - 140)*x - 5167)*x - 73325)*x - 81712)*x 
 - 75355)*x - 81392)*sqrt(3*x^2 + 2) - 1087/18*sqrt(3)*log(-sqrt(3)*x + sq 
rt(3*x^2 + 2))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.52 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx=\frac {1087\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-9\,x^7+\frac {36\,x^6}{7}+180\,x^5+\frac {15501\,x^4}{35}+\frac {2095\,x^3}{4}+\frac {20428\,x^2}{35}+\frac {2153\,x}{4}+\frac {20348}{105}\right )}{3} \] Input:

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

Output:

(1087*3^(1/2)*asinh((6^(1/2)*x)/2))/18 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((2153 
*x)/4 + (20428*x^2)/35 + (2095*x^3)/4 + (15501*x^4)/35 + 180*x^5 + (36*x^6 
)/7 - 9*x^7 + 20348/105))/3
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06 \[ \int (5-x) (3+2 x)^3 \left (2+3 x^2\right )^{3/2} \, dx=-3 \sqrt {3 x^{2}+2}\, x^{7}+\frac {12 \sqrt {3 x^{2}+2}\, x^{6}}{7}+60 \sqrt {3 x^{2}+2}\, x^{5}+\frac {5167 \sqrt {3 x^{2}+2}\, x^{4}}{35}+\frac {2095 \sqrt {3 x^{2}+2}\, x^{3}}{12}+\frac {20428 \sqrt {3 x^{2}+2}\, x^{2}}{105}+\frac {2153 \sqrt {3 x^{2}+2}\, x}{12}+\frac {20348 \sqrt {3 x^{2}+2}}{315}+\frac {1087 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{18} \] Input:

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

Output:

( - 3780*sqrt(3*x**2 + 2)*x**7 + 2160*sqrt(3*x**2 + 2)*x**6 + 75600*sqrt(3 
*x**2 + 2)*x**5 + 186012*sqrt(3*x**2 + 2)*x**4 + 219975*sqrt(3*x**2 + 2)*x 
**3 + 245136*sqrt(3*x**2 + 2)*x**2 + 226065*sqrt(3*x**2 + 2)*x + 81392*sqr 
t(3*x**2 + 2) + 76090*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2))) 
/1260