\(\int \frac {(5-x) (3+2 x)^6}{(2+3 x^2)^{5/2}} \, dx\) [252]

Optimal result

Integrand size = 24, antiderivative size = 116 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {7 (2-7 x) (3+2 x)^5}{18 \left (2+3 x^2\right )^{3/2}}+\frac {(3+2 x)^3 (158+2427 x)}{54 \sqrt {2+3 x^2}}-\frac {2639}{81} (3+2 x)^2 \sqrt {2+3 x^2}-\frac {70}{243} (2167+801 x) \sqrt {2+3 x^2}+\frac {20720 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{27 \sqrt {3}} \] Output:

-7/18*(2-7*x)*(3+2*x)^5/(3*x^2+2)^(3/2)+1/54*(3+2*x)^3*(158+2427*x)/(3*x^2 
+2)^(1/2)-2639/81*(3+2*x)^2*(3*x^2+2)^(1/2)-70/243*(2167+801*x)*(3*x^2+2)^ 
(1/2)+20720/81*arcsinh(1/2*x*6^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {1798610-139815 x+2363976 x^2-1125999 x^3-130464 x^4+20736 x^5+3456 x^6}{486 \left (2+3 x^2\right )^{3/2}}-\frac {20720 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{27 \sqrt {3}} \] Input:

Integrate[((5 - x)*(3 + 2*x)^6)/(2 + 3*x^2)^(5/2),x]
 

Output:

-1/486*(1798610 - 139815*x + 2363976*x^2 - 1125999*x^3 - 130464*x^4 + 2073 
6*x^5 + 3456*x^6)/(2 + 3*x^2)^(3/2) - (20720*Log[-(Sqrt[3]*x) + Sqrt[2 + 3 
*x^2]])/(27*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {684, 27, 684, 27, 687, 27, 676, 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.

\(\displaystyle \int \frac {(5-x) (2 x+3)^6}{\left (3 x^2+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 684

\(\displaystyle \frac {1}{18} \int \frac {2 (199-159 x) (2 x+3)^4}{\left (3 x^2+2\right )^{3/2}}dx-\frac {7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \frac {(199-159 x) (2 x+3)^4}{\left (3 x^2+2\right )^{3/2}}dx-\frac {7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 684

\(\displaystyle \frac {1}{9} \left (\frac {1}{6} \int -\frac {42 (2 x+3)^2 (377 x+68)}{\sqrt {3 x^2+2}}dx+\frac {(2427 x+158) (2 x+3)^3}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {(2 x+3)^3 (2427 x+158)}{6 \sqrt {3 x^2+2}}-7 \int \frac {(2 x+3)^2 (377 x+68)}{\sqrt {3 x^2+2}}dx\right )-\frac {7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {1}{9} \left (\frac {(2 x+3)^3 (2427 x+158)}{6 \sqrt {3 x^2+2}}-7 \left (\frac {1}{9} \int -\frac {10 (118-801 x) (2 x+3)}{\sqrt {3 x^2+2}}dx+\frac {377}{9} \sqrt {3 x^2+2} (2 x+3)^2\right )\right )-\frac {7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {(2 x+3)^3 (2427 x+158)}{6 \sqrt {3 x^2+2}}-7 \left (\frac {377}{9} (2 x+3)^2 \sqrt {3 x^2+2}-\frac {10}{9} \int \frac {(118-801 x) (2 x+3)}{\sqrt {3 x^2+2}}dx\right )\right )-\frac {7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {1}{9} \left (\frac {(2 x+3)^3 (2427 x+158)}{6 \sqrt {3 x^2+2}}-7 \left (\frac {377}{9} (2 x+3)^2 \sqrt {3 x^2+2}-\frac {10}{9} \left (888 \int \frac {1}{\sqrt {3 x^2+2}}dx-267 \sqrt {3 x^2+2} x-\frac {2167}{3} \sqrt {3 x^2+2}\right )\right )\right )-\frac {7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{9} \left (\frac {(2 x+3)^3 (2427 x+158)}{6 \sqrt {3 x^2+2}}-7 \left (\frac {377}{9} (2 x+3)^2 \sqrt {3 x^2+2}-\frac {10}{9} \left (296 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-267 \sqrt {3 x^2+2} x-\frac {2167}{3} \sqrt {3 x^2+2}\right )\right )\right )-\frac {7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}\)

Input:

Int[((5 - x)*(3 + 2*x)^6)/(2 + 3*x^2)^(5/2),x]
 

Output:

(-7*(2 - 7*x)*(3 + 2*x)^5)/(18*(2 + 3*x^2)^(3/2)) + (((3 + 2*x)^3*(158 + 2 
427*x))/(6*Sqrt[2 + 3*x^2]) - 7*((377*(3 + 2*x)^2*Sqrt[2 + 3*x^2])/9 - (10 
*((-2167*Sqrt[2 + 3*x^2])/3 - 267*x*Sqrt[2 + 3*x^2] + 296*Sqrt[3]*ArcSinh[ 
Sqrt[3/2]*x]))/9))/9
 

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 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 684

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g 
) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1))   Int[ 
(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 
2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a 
, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] 
 && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])
 

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 = 0.85 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.47

method	result	size
risch	\(-\frac {3456 x^{6}+20736 x^{5}-130464 x^{4}-1125999 x^{3}+2363976 x^{2}-139815 x +1798610}{486 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {20720 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{81}\)	\(55\)
trager	\(-\frac {3456 x^{6}+20736 x^{5}-130464 x^{4}-1125999 x^{3}+2363976 x^{2}-139815 x +1798610}{486 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {20720 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{81}\)	\(73\)
default	\(-\frac {3537 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {55517 x}{54 \sqrt {3 x^{2}+2}}-\frac {899305}{243 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {131332 x^{2}}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {20720 x^{3}}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {20720 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{81}+\frac {2416 x^{4}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {128 x^{5}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {64 x^{6}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\)	\(119\)
meijerg	\(\frac {1215 \sqrt {2}\, x \left (3 x^{2}+3\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}-\frac {1024 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {189}{4} x^{4}+210 x^{2}+105\right )}{56 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}-\frac {15 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{4}\right )}{243 \sqrt {\pi }}+\frac {160 \sqrt {2}\, \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (\frac {27}{2} x^{4}+36 x^{2}+16\right )}{4 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{9 \sqrt {\pi }}+\frac {160 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (30 x^{2}+15\right )}{20 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{\sqrt {\pi }}+\frac {620 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {891 \sqrt {2}\, x^{3}}{\left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {1539 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{2 \sqrt {\pi }}-\frac {256 \sqrt {2}\, \left (8 \sqrt {\pi }-\frac {\sqrt {\pi }\, \left (-27 x^{6}+108 x^{4}+288 x^{2}+128\right )}{16 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{243 \sqrt {\pi }}\)	\(296\)

Input:

int((5-x)*(2*x+3)^6/(3*x^2+2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/486*(3456*x^6+20736*x^5-130464*x^4-1125999*x^3+2363976*x^2-139815*x+179 
8610)/(3*x^2+2)^(3/2)+20720/81*arcsinh(1/2*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {62160 \, \sqrt {3} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (3456 \, x^{6} + 20736 \, x^{5} - 130464 \, x^{4} - 1125999 \, x^{3} + 2363976 \, x^{2} - 139815 \, x + 1798610\right )} \sqrt {3 \, x^{2} + 2}}{486 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \] Input:

integrate((5-x)*(3+2*x)^6/(3*x^2+2)^(5/2),x, algorithm="fricas")
 

Output:

1/486*(62160*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 
 3*x^2 - 1) - (3456*x^6 + 20736*x^5 - 130464*x^4 - 1125999*x^3 + 2363976*x 
^2 - 139815*x + 1798610)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {13851 x}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {21384 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {16740 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {6480 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {720 x^{5}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {256 x^{6}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \frac {64 x^{7}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {3645}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx \] Input:

integrate((5-x)*(3+2*x)**6/(3*x**2+2)**(5/2),x)
 

Output:

-Integral(-13851*x/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4 
*sqrt(3*x**2 + 2)), x) - Integral(-21384*x**2/(9*x**4*sqrt(3*x**2 + 2) + 1 
2*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-16740*x**3/( 
9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), 
x) - Integral(-6480*x**4/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 
2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-720*x**5/(9*x**4*sqrt(3*x**2 + 2) 
 + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(256*x**6/ 
(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), 
 x) - Integral(64*x**7/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) 
 + 4*sqrt(3*x**2 + 2)), x) - Integral(-3645/(9*x**4*sqrt(3*x**2 + 2) + 12* 
x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.15 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {64 \, x^{6}}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {128 \, x^{5}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {2416 \, x^{4}}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {20720}{81} \, x {\left (\frac {9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\right )} + \frac {20720}{81} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {249431 \, x}{162 \, \sqrt {3 \, x^{2} + 2}} - \frac {131332 \, x^{2}}{27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {3537 \, x}{2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {899305}{243 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

integrate((5-x)*(3+2*x)^6/(3*x^2+2)^(5/2),x, algorithm="maxima")
 

Output:

-64/9*x^6/(3*x^2 + 2)^(3/2) - 128/3*x^5/(3*x^2 + 2)^(3/2) + 2416/9*x^4/(3* 
x^2 + 2)^(3/2) - 20720/81*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2) 
) + 20720/81*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 249431/162*x/sqrt(3*x^2 + 2) 
 - 131332/27*x^2/(3*x^2 + 2)^(3/2) - 3537/2*x/(3*x^2 + 2)^(3/2) - 899305/2 
43/(3*x^2 + 2)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.52 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {20720}{81} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {9 \, {\left ({\left ({\left (96 \, {\left (4 \, {\left (x + 6\right )} x - 151\right )} x - 125111\right )} x + 262664\right )} x - 15535\right )} x + 1798610}{486 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

integrate((5-x)*(3+2*x)^6/(3*x^2+2)^(5/2),x, algorithm="giac")
 

Output:

-20720/81*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/486*(9*(((96*(4*(x 
 + 6)*x - 151)*x - 125111)*x + 262664)*x - 15535)*x + 1798610)/(3*x^2 + 2) 
^(3/2)
 

Mupad [B] (verification not implemented)

Time = 5.98 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.91 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {20720\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{81}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {64\,x^2}{27}+\frac {128\,x}{9}-\frac {7504}{81}\right )}{3}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {206689}{144}+\frac {\sqrt {6}\,81809{}\mathrm {i}}{432}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {206689}{216}+\frac {\sqrt {6}\,81809{}\mathrm {i}}{648}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {206689}{144}+\frac {\sqrt {6}\,81809{}\mathrm {i}}{432}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {206689}{216}+\frac {\sqrt {6}\,81809{}\mathrm {i}}{648}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-3390048+\sqrt {6}\,719421{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{23328\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (3390048+\sqrt {6}\,719421{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{23328\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

int(-((2*x + 3)^6*(x - 5))/(3*x^2 + 2)^(5/2),x)
 

Output:

(20720*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/81 - (3^(1/2)*(x^2 + 2/3)^(1/ 
2)*((128*x)/9 + (64*x^2)/27 - 7504/81))/3 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((( 
6^(1/2)*81809i)/432 - 206689/144)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2 
)*81809i)/648 - 206689/216)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2) 
*(x^2 + 2/3)^(1/2)*(((6^(1/2)*81809i)/432 + 206689/144)/(x + (6^(1/2)*1i)/ 
3) + (6^(1/2)*((6^(1/2)*81809i)/648 + 206689/216)*1i)/(2*(x + (6^(1/2)*1i) 
/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*719421i - 3390048)*(x^2 + 2/3)^(1/ 
2)*1i)/(23328*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*719421i + 
3390048)*(x^2 + 2/3)^(1/2)*1i)/(23328*(x + (6^(1/2)*1i)/3))
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.68 \[ \int \frac {(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {-3456 \sqrt {3 x^{2}+2}\, x^{6}-20736 \sqrt {3 x^{2}+2}\, x^{5}+130464 \sqrt {3 x^{2}+2}\, x^{4}+1125999 \sqrt {3 x^{2}+2}\, x^{3}-2363976 \sqrt {3 x^{2}+2}\, x^{2}+139815 \sqrt {3 x^{2}+2}\, x -1798610 \sqrt {3 x^{2}+2}+1118880 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{4}+1491840 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{2}+497280 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )+857889 \sqrt {3}\, x^{4}+1143852 \sqrt {3}\, x^{2}+381284 \sqrt {3}}{4374 x^{4}+5832 x^{2}+1944} \] Input:

int((5-x)*(3+2*x)^6/(3*x^2+2)^(5/2),x)
 

Output:

( - 3456*sqrt(3*x**2 + 2)*x**6 - 20736*sqrt(3*x**2 + 2)*x**5 + 130464*sqrt 
(3*x**2 + 2)*x**4 + 1125999*sqrt(3*x**2 + 2)*x**3 - 2363976*sqrt(3*x**2 + 
2)*x**2 + 139815*sqrt(3*x**2 + 2)*x - 1798610*sqrt(3*x**2 + 2) + 1118880*s 
qrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2))*x**4 + 1491840*sqrt(3)* 
log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2))*x**2 + 497280*sqrt(3)*log((sqr 
t(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)) + 857889*sqrt(3)*x**4 + 1143852*sqrt(3 
)*x**2 + 381284*sqrt(3))/(486*(9*x**4 + 12*x**2 + 4))