\(\int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx\) [205]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx=-\frac {(118-73 x) \sqrt {2+3 x^2}}{350 (3+2 x)^5}+\frac {\sqrt {2+3 x^2}}{875 (3+2 x)^3}-\frac {12 \sqrt {2+3 x^2}}{6125 (3+2 x)^2}-\frac {366 \sqrt {2+3 x^2}}{214375 (3+2 x)}-\frac {1017 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{214375 \sqrt {35}} \] Output:

-1/350*(118-73*x)*(3*x^2+2)^(1/2)/(3+2*x)^5+1/875*(3*x^2+2)^(1/2)/(3+2*x)^ 
3-12/6125*(3*x^2+2)^(1/2)/(3+2*x)^2-366*(3*x^2+2)^(1/2)/(643125+428750*x)- 
1017/7503125*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 2.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (222112+108167 x+186392 x^2+76992 x^3+11712 x^4\right )}{(3+2 x)^5}+4068 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{15006250} \] Input:

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

Output:

((-35*Sqrt[2 + 3*x^2]*(222112 + 108167*x + 186392*x^2 + 76992*x^3 + 11712* 
x^4))/(3 + 2*x)^5 + 4068*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqr 
t[2 + 3*x^2])/Sqrt[35]])/15006250
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {688, 25, 688, 27, 679, 486, 488, 219}

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.

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

\(\Big \downarrow \) 688

\(\displaystyle -\frac {1}{175} \int -\frac {(205-78 x) \sqrt {3 x^2+2}}{(2 x+3)^5}dx-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{175} \int \frac {(205-78 x) \sqrt {3 x^2+2}}{(2 x+3)^5}dx-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {1}{175} \left (-\frac {1}{140} \int -\frac {84 (73-23 x) \sqrt {3 x^2+2}}{(2 x+3)^4}dx-\frac {23 \left (3 x^2+2\right )^{3/2}}{5 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{175} \left (\frac {3}{5} \int \frac {(73-23 x) \sqrt {3 x^2+2}}{(2 x+3)^4}dx-\frac {23 \left (3 x^2+2\right )^{3/2}}{5 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {1}{175} \left (\frac {3}{5} \left (\frac {113}{7} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {43 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {23 \left (3 x^2+2\right )^{3/2}}{5 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {1}{175} \left (\frac {3}{5} \left (\frac {113}{7} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {43 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {23 \left (3 x^2+2\right )^{3/2}}{5 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{175} \left (\frac {3}{5} \left (\frac {113}{7} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {43 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {23 \left (3 x^2+2\right )^{3/2}}{5 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{175} \left (\frac {3}{5} \left (\frac {113}{7} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {43 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {23 \left (3 x^2+2\right )^{3/2}}{5 (2 x+3)^4}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\)

Input:

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

Output:

(-13*(2 + 3*x^2)^(3/2))/(175*(3 + 2*x)^5) + ((-23*(2 + 3*x^2)^(3/2))/(5*(3 
 + 2*x)^4) + (3*((-43*(2 + 3*x^2)^(3/2))/(21*(3 + 2*x)^3) + (113*(-1/70*(( 
4 - 9*x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 - (3*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqr 
t[2 + 3*x^2])])/(35*Sqrt[35])))/7))/5)/175
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 486

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), 
x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2)))   Int[(c + d*x)^(n + 2)*(a + 
b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && 
GtQ[p, 0]
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 679

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

rule 688

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

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {35136 x^{6}+230976 x^{5}+582600 x^{4}+478485 x^{3}+1039120 x^{2}+216334 x +444224}{428750 \left (2 x +3\right )^{5} \sqrt {3 x^{2}+2}}-\frac {1017 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{7503125}\)	\(80\)
trager	\(-\frac {\left (11712 x^{4}+76992 x^{3}+186392 x^{2}+108167 x +222112\right ) \sqrt {3 x^{2}+2}}{428750 \left (2 x +3\right )^{5}}+\frac {1017 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{7503125}\)	\(86\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{5600 \left (x +\frac {3}{2}\right )^{5}}-\frac {23 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{14000 \left (x +\frac {3}{2}\right )^{4}}-\frac {43 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{49000 \left (x +\frac {3}{2}\right )^{3}}-\frac {339 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{857500 \left (x +\frac {3}{2}\right )^{2}}-\frac {3051 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{15006250 \left (x +\frac {3}{2}\right )}+\frac {1017 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{7503125}-\frac {1017 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{7503125}+\frac {9153 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{15006250}\)	\(170\)

Input:

int((5-x)*(3*x^2+2)^(1/2)/(2*x+3)^6,x,method=_RETURNVERBOSE)
 

Output:

-1/428750*(35136*x^6+230976*x^5+582600*x^4+478485*x^3+1039120*x^2+216334*x 
+444224)/(2*x+3)^5/(3*x^2+2)^(1/2)-1017/7503125*35^(1/2)*arctanh(2/35*(4-9 
*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx=\frac {1017 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (11712 \, x^{4} + 76992 \, x^{3} + 186392 \, x^{2} + 108167 \, x + 222112\right )} \sqrt {3 \, x^{2} + 2}}{15006250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^6,x, algorithm="fricas")
 

Output:

1/15006250*(1017*sqrt(35)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 
 243)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^ 
2 + 12*x + 9)) - 35*(11712*x^4 + 76992*x^3 + 186392*x^2 + 108167*x + 22211 
2)*sqrt(3*x^2 + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**6,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.48 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx=\frac {1017}{7503125} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {1017}{857500} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {23 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{875 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {43 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{6125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {339 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{214375 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {3051 \, \sqrt {3 \, x^{2} + 2}}{857500 \, {\left (2 \, x + 3\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^6,x, algorithm="maxima")
 

Output:

1017/7503125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs 
(2*x + 3)) + 1017/857500*sqrt(3*x^2 + 2) - 13/175*(3*x^2 + 2)^(3/2)/(32*x^ 
5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 23/875*(3*x^2 + 2)^(3/2) 
/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 43/6125*(3*x^2 + 2)^(3/2)/(8*x 
^3 + 36*x^2 + 54*x + 27) - 339/214375*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) 
 - 3051/857500*sqrt(3*x^2 + 2)/(2*x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (103) = 206\).

Time = 0.14 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.56 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx=\frac {1017}{7503125} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3 \, \sqrt {3} {\left (904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 36612 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 254217 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 142464 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 338184 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 4315808 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 1676892 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 1737184 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 219776 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 31232\right )}}{1715000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^6,x, algorithm="giac")
 

Output:

1017/7503125*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqr 
t(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 
3/1715000*sqrt(3)*(904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 36612*(sq 
rt(3)*x - sqrt(3*x^2 + 2))^8 + 254217*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2) 
)^7 - 142464*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 338184*sqrt(3)*(sqrt(3)*x - 
 sqrt(3*x^2 + 2))^5 - 4315808*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 1676892*sq 
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 1737184*(sqrt(3)*x - sqrt(3*x^2 + 
2))^2 + 219776*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 31232)/((sqrt(3)*x 
- sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^5
 

Mupad [B] (verification not implemented)

Time = 6.03 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.41 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx=\frac {1017\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{7503125}-\frac {1017\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{7503125}+\frac {73\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{11200\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{640\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {183\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{214375\,\left (x+\frac {3}{2}\right )}-\frac {3\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6125\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{7000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-((3*x^2 + 2)^(1/2)*(x - 5))/(2*x + 3)^6,x)
 

Output:

(1017*35^(1/2)*log(x + 3/2))/7503125 - (1017*35^(1/2)*log(x - (3^(1/2)*35^ 
(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/7503125 + (73*3^(1/2)*(x^2 + 2/3)^(1/2) 
)/(11200*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (13*3^(1/2)*(x^2 
 + 2/3)^(1/2))/(640*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + 
x^5 + 243/32)) - (183*3^(1/2)*(x^2 + 2/3)^(1/2))/(214375*(x + 3/2)) - (3*3 
^(1/2)*(x^2 + 2/3)^(1/2))/(6125*(3*x + x^2 + 9/4)) + (3^(1/2)*(x^2 + 2/3)^ 
(1/2))/(7000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.39 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx=\frac {-409920 \sqrt {3 x^{2}+2}\, x^{4}-2694720 \sqrt {3 x^{2}+2}\, x^{3}-6523720 \sqrt {3 x^{2}+2}\, x^{2}-3785845 \sqrt {3 x^{2}+2}\, x -7773920 \sqrt {3 x^{2}+2}+65088 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{5}+488160 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}+1464480 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}+2196720 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}+1647540 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +494262 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-65088 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{5}-488160 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}-1464480 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-2196720 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-1647540 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -494262 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{480200000 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

int((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^6,x)
 

Output:

( - 409920*sqrt(3*x**2 + 2)*x**4 - 2694720*sqrt(3*x**2 + 2)*x**3 - 6523720 
*sqrt(3*x**2 + 2)*x**2 - 3785845*sqrt(3*x**2 + 2)*x - 7773920*sqrt(3*x**2 
+ 2) + 65088*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**5 + 4881 
60*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**4 + 1464480*sqrt(3 
5)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**3 + 2196720*sqrt(35)*log(sq 
rt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**2 + 1647540*sqrt(35)*log(sqrt(3*x**2 
 + 2)*sqrt(35) + 9*x - 4)*x + 494262*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35 
) + 9*x - 4) - 65088*sqrt(35)*log(2*x + 3)*x**5 - 488160*sqrt(35)*log(2*x 
+ 3)*x**4 - 1464480*sqrt(35)*log(2*x + 3)*x**3 - 2196720*sqrt(35)*log(2*x 
+ 3)*x**2 - 1647540*sqrt(35)*log(2*x + 3)*x - 494262*sqrt(35)*log(2*x + 3) 
)/(15006250*(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243))