\(\int (5-x) (3+2 x)^2 (2+3 x^2)^{5/2} \, dx\) [222]

Optimal result

Integrand size = 24, antiderivative size = 110 \[ \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx=\frac {665}{12} x \sqrt {2+3 x^2}+\frac {665}{36} x \left (2+3 x^2\right )^{3/2}+\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \] Output:

665/12*x*(3*x^2+2)^(1/2)+665/36*x*(3*x^2+2)^(3/2)+133/18*x*(3*x^2+2)^(5/2) 
-1/27*(3+2*x)^2*(3*x^2+2)^(7/2)+1/81*(226+63*x)*(3*x^2+2)^(7/2)+665/18*arc 
sinh(1/2*x*6^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.78 \[ \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{324} \sqrt {2+3 x^2} \left (-6368-40365 x-28272 x^2-50571 x^3-41256 x^4-27378 x^5-18900 x^6-2916 x^7+1296 x^8\right )-\frac {665 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{6 \sqrt {3}} \] Input:

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

Output:

-1/324*(Sqrt[2 + 3*x^2]*(-6368 - 40365*x - 28272*x^2 - 50571*x^3 - 41256*x 
^4 - 27378*x^5 - 18900*x^6 - 2916*x^7 + 1296*x^8)) - (665*Log[-(Sqrt[3]*x) 
 + Sqrt[2 + 3*x^2]])/(6*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {687, 27, 676, 211, 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.

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{27} \int 7 (2 x+3) (36 x+59) \left (3 x^2+2\right )^{5/2}dx-\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{27} \int (2 x+3) (36 x+59) \left (3 x^2+2\right )^{5/2}dx-\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {7}{27} \left (171 \int \left (3 x^2+2\right )^{5/2}dx+3 x \left (3 x^2+2\right )^{7/2}+\frac {226}{21} \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {7}{27} \left (171 \left (\frac {5}{3} \int \left (3 x^2+2\right )^{3/2}dx+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+3 x \left (3 x^2+2\right )^{7/2}+\frac {226}{21} \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {7}{27} \left (171 \left (\frac {5}{3} \left (\frac {3}{2} \int \sqrt {3 x^2+2}dx+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+3 x \left (3 x^2+2\right )^{7/2}+\frac {226}{21} \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {7}{27} \left (171 \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+3 x \left (3 x^2+2\right )^{7/2}+\frac {226}{21} \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {7}{27} \left (171 \left (\frac {5}{3} \left (\frac {3}{2} \left (\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+3 x \left (3 x^2+2\right )^{7/2}+\frac {226}{21} \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\)

Input:

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

Output:

-1/27*((3 + 2*x)^2*(2 + 3*x^2)^(7/2)) + (7*((226*(2 + 3*x^2)^(7/2))/21 + 3 
*x*(2 + 3*x^2)^(7/2) + 171*((x*(2 + 3*x^2)^(5/2))/6 + (5*((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 
)))/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 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 = 0.76 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59

method	result	size
risch	\(-\frac {\left (1296 x^{8}-2916 x^{7}-18900 x^{6}-27378 x^{5}-41256 x^{4}-50571 x^{3}-28272 x^{2}-40365 x -6368\right ) \sqrt {3 x^{2}+2}}{324}+\frac {665 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{18}\)	\(65\)
trager	\(\left (-4 x^{8}+9 x^{7}+\frac {175}{3} x^{6}+\frac {169}{2} x^{5}+\frac {382}{3} x^{4}+\frac {1873}{12} x^{3}+\frac {2356}{27} x^{2}+\frac {1495}{12} x +\frac {1592}{81}\right ) \sqrt {3 x^{2}+2}-\frac {665 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{18}\)	\(82\)
default	\(\frac {133 x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{18}+\frac {665 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{36}+\frac {665 x \sqrt {3 x^{2}+2}}{12}+\frac {665 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{18}+\frac {199 \left (3 x^{2}+2\right )^{\frac {7}{2}}}{81}+\frac {x \left (3 x^{2}+2\right )^{\frac {7}{2}}}{3}-\frac {4 x^{2} \left (3 x^{2}+2\right )^{\frac {7}{2}}}{27}\)	\(87\)
meijerg	\(-\frac {225 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {3}{8} x^{4}+\frac {13}{16} x^{2}+\frac {11}{16}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{3}\right )}{2 \sqrt {\pi }}-\frac {40 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (162 x^{6}+306 x^{4}+177 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{720}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{24}\right )}{3 \sqrt {\pi }}-\frac {255 \sqrt {2}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {8 \sqrt {\pi }\, \left (\frac {27}{4} x^{6}+\frac {27}{2} x^{4}+9 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{105}\right )}{2 \sqrt {\pi }}+\frac {20 \sqrt {2}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-\frac {567}{4} x^{8}-\frac {513}{2} x^{6}-135 x^{4}-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{945}\right )}{3 \sqrt {\pi }}\)	\(213\)

Input:

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

Output:

-1/324*(1296*x^8-2916*x^7-18900*x^6-27378*x^5-41256*x^4-50571*x^3-28272*x^ 
2-40365*x-6368)*(3*x^2+2)^(1/2)+665/18*arcsinh(1/2*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.73 \[ \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{324} \, {\left (1296 \, x^{8} - 2916 \, x^{7} - 18900 \, x^{6} - 27378 \, x^{5} - 41256 \, x^{4} - 50571 \, x^{3} - 28272 \, x^{2} - 40365 \, x - 6368\right )} \sqrt {3 \, x^{2} + 2} + \frac {665}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \] Input:

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

Output:

-1/324*(1296*x^8 - 2916*x^7 - 18900*x^6 - 27378*x^5 - 41256*x^4 - 50571*x^ 
3 - 28272*x^2 - 40365*x - 6368)*sqrt(3*x^2 + 2) + 665/36*sqrt(3)*log(-sqrt 
(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 4.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.47 \[ \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx=- 4 x^{8} \sqrt {3 x^{2} + 2} + 9 x^{7} \sqrt {3 x^{2} + 2} + \frac {175 x^{6} \sqrt {3 x^{2} + 2}}{3} + \frac {169 x^{5} \sqrt {3 x^{2} + 2}}{2} + \frac {382 x^{4} \sqrt {3 x^{2} + 2}}{3} + \frac {1873 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {2356 x^{2} \sqrt {3 x^{2} + 2}}{27} + \frac {1495 x \sqrt {3 x^{2} + 2}}{12} + \frac {1592 \sqrt {3 x^{2} + 2}}{81} + \frac {665 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \] Input:

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

Output:

-4*x**8*sqrt(3*x**2 + 2) + 9*x**7*sqrt(3*x**2 + 2) + 175*x**6*sqrt(3*x**2 
+ 2)/3 + 169*x**5*sqrt(3*x**2 + 2)/2 + 382*x**4*sqrt(3*x**2 + 2)/3 + 1873* 
x**3*sqrt(3*x**2 + 2)/12 + 2356*x**2*sqrt(3*x**2 + 2)/27 + 1495*x*sqrt(3*x 
**2 + 2)/12 + 1592*sqrt(3*x**2 + 2)/81 + 665*sqrt(3)*asinh(sqrt(6)*x/2)/18
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.78 \[ \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {4}{27} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{2} + \frac {1}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {199}{81} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {133}{18} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {665}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {665}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {665}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \] Input:

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

Output:

-4/27*(3*x^2 + 2)^(7/2)*x^2 + 1/3*(3*x^2 + 2)^(7/2)*x + 199/81*(3*x^2 + 2) 
^(7/2) + 133/18*(3*x^2 + 2)^(5/2)*x + 665/36*(3*x^2 + 2)^(3/2)*x + 665/12* 
sqrt(3*x^2 + 2)*x + 665/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65 \[ \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{324} \, {\left (3 \, {\left ({\left (9 \, {\left (2 \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 9\right )} x - 175\right )} x - 507\right )} x - 764\right )} x - 1873\right )} x - 9424\right )} x - 13455\right )} x - 6368\right )} \sqrt {3 \, x^{2} + 2} - \frac {665}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \] Input:

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

Output:

-1/324*(3*((9*(2*((2*(3*(4*x - 9)*x - 175)*x - 507)*x - 764)*x - 1873)*x - 
 9424)*x - 13455)*x - 6368)*sqrt(3*x^2 + 2) - 665/18*sqrt(3)*log(-sqrt(3)* 
x + sqrt(3*x^2 + 2))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59 \[ \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx=\frac {665\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-12\,x^8+27\,x^7+175\,x^6+\frac {507\,x^5}{2}+382\,x^4+\frac {1873\,x^3}{4}+\frac {2356\,x^2}{9}+\frac {1495\,x}{4}+\frac {1592}{27}\right )}{3} \] Input:

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

Output:

(665*3^(1/2)*asinh((6^(1/2)*x)/2))/18 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((1495* 
x)/4 + (2356*x^2)/9 + (1873*x^3)/4 + 382*x^4 + (507*x^5)/2 + 175*x^6 + 27* 
x^7 - 12*x^8 + 1592/27))/3
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.24 \[ \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx=-4 \sqrt {3 x^{2}+2}\, x^{8}+9 \sqrt {3 x^{2}+2}\, x^{7}+\frac {175 \sqrt {3 x^{2}+2}\, x^{6}}{3}+\frac {169 \sqrt {3 x^{2}+2}\, x^{5}}{2}+\frac {382 \sqrt {3 x^{2}+2}\, x^{4}}{3}+\frac {1873 \sqrt {3 x^{2}+2}\, x^{3}}{12}+\frac {2356 \sqrt {3 x^{2}+2}\, x^{2}}{27}+\frac {1495 \sqrt {3 x^{2}+2}\, x}{12}+\frac {1592 \sqrt {3 x^{2}+2}}{81}+\frac {665 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{18} \] Input:

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

Output:

( - 1296*sqrt(3*x**2 + 2)*x**8 + 2916*sqrt(3*x**2 + 2)*x**7 + 18900*sqrt(3 
*x**2 + 2)*x**6 + 27378*sqrt(3*x**2 + 2)*x**5 + 41256*sqrt(3*x**2 + 2)*x** 
4 + 50571*sqrt(3*x**2 + 2)*x**3 + 28272*sqrt(3*x**2 + 2)*x**2 + 40365*sqrt 
(3*x**2 + 2)*x + 6368*sqrt(3*x**2 + 2) + 11970*sqrt(3)*log((sqrt(3*x**2 + 
2) + sqrt(3)*x)/sqrt(2)))/324