\(\int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx\) [206]

Optimal result

Integrand size = 24, antiderivative size = 148 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx=-\frac {(647-327 x) \sqrt {2+3 x^2}}{2100 (3+2 x)^6}+\frac {127 \sqrt {2+3 x^2}}{98000 (3+2 x)^4}-\frac {479 \sqrt {2+3 x^2}}{490000 (3+2 x)^3}-\frac {2727 \sqrt {2+3 x^2}}{3430000 (3+2 x)^2}-\frac {53511 \sqrt {2+3 x^2}}{120050000 (3+2 x)}-\frac {6102 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{7503125 \sqrt {35}} \] Output:

-1/2100*(647-327*x)*(3*x^2+2)^(1/2)/(3+2*x)^6+127/98000*(3*x^2+2)^(1/2)/(3 
+2*x)^4-479/490000*(3*x^2+2)^(1/2)/(3+2*x)^3-2727/3430000*(3*x^2+2)^(1/2)/ 
(3+2*x)^2-53511*(3*x^2+2)^(1/2)/(360150000+240100000*x)-6102/262609375*35^ 
(1/2)*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))
 

Mathematica [A] (verified)

Time = 2.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.63 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (22308548+18651300 x+30753930 x^2+18236055 x^3+5388660 x^4+642132 x^5\right )}{(3+2 x)^6}+73224 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{1575656250} \] Input:

Integrate[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^7,x]
 

Output:

((-35*Sqrt[2 + 3*x^2]*(22308548 + 18651300*x + 30753930*x^2 + 18236055*x^3 
 + 5388660*x^4 + 642132*x^5))/(3 + 2*x)^6 + 73224*Sqrt[35]*ArcTanh[(3*Sqrt 
[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/1575656250
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {688, 27, 688, 27, 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.

\(\displaystyle \int \frac {(5-x) \sqrt {3 x^2+2}}{(2 x+3)^7} \, dx\)

\(\Big \downarrow \) 688

\(\displaystyle -\frac {1}{210} \int -\frac {3 (82-39 x) \sqrt {3 x^2+2}}{(2 x+3)^6}dx-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{70} \int \frac {(82-39 x) \sqrt {3 x^2+2}}{(2 x+3)^6}dx-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {1}{70} \left (-\frac {1}{175} \int -\frac {6 (485-281 x) \sqrt {3 x^2+2}}{(2 x+3)^5}dx-\frac {281 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{70} \left (\frac {6}{175} \int \frac {(485-281 x) \sqrt {3 x^2+2}}{(2 x+3)^5}dx-\frac {281 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {1}{70} \left (\frac {6}{175} \left (-\frac {1}{140} \int -\frac {7 (1852-777 x) \sqrt {3 x^2+2}}{(2 x+3)^4}dx-\frac {259 \left (3 x^2+2\right )^{3/2}}{20 (2 x+3)^4}\right )-\frac {281 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{70} \left (\frac {6}{175} \left (\frac {1}{20} \int \frac {(1852-777 x) \sqrt {3 x^2+2}}{(2 x+3)^4}dx-\frac {259 \left (3 x^2+2\right )^{3/2}}{20 (2 x+3)^4}\right )-\frac {281 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {1}{70} \left (\frac {6}{175} \left (\frac {1}{20} \left (\frac {2712}{7} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {1207 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {259 \left (3 x^2+2\right )^{3/2}}{20 (2 x+3)^4}\right )-\frac {281 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {1}{70} \left (\frac {6}{175} \left (\frac {1}{20} \left (\frac {2712}{7} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {1207 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {259 \left (3 x^2+2\right )^{3/2}}{20 (2 x+3)^4}\right )-\frac {281 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{70} \left (\frac {6}{175} \left (\frac {1}{20} \left (\frac {2712}{7} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {1207 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {259 \left (3 x^2+2\right )^{3/2}}{20 (2 x+3)^4}\right )-\frac {281 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{70} \left (\frac {6}{175} \left (\frac {1}{20} \left (\frac {2712}{7} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {1207 \left (3 x^2+2\right )^{3/2}}{21 (2 x+3)^3}\right )-\frac {259 \left (3 x^2+2\right )^{3/2}}{20 (2 x+3)^4}\right )-\frac {281 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}\right )-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}\)

Input:

Int[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^7,x]
 

Output:

(-13*(2 + 3*x^2)^(3/2))/(210*(3 + 2*x)^6) + ((-281*(2 + 3*x^2)^(3/2))/(175 
*(3 + 2*x)^5) + (6*((-259*(2 + 3*x^2)^(3/2))/(20*(3 + 2*x)^4) + ((-1207*(2 
 + 3*x^2)^(3/2))/(21*(3 + 2*x)^3) + (2712*(-1/70*((4 - 9*x)*Sqrt[2 + 3*x^2 
])/(3 + 2*x)^2 - (3*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqr 
t[35])))/7)/20))/175)/70
 

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 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 = 0.82 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.57

method	result	size
risch	\(-\frac {1926396 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{45018750 \left (2 x +3\right )^{6} \sqrt {3 x^{2}+2}}-\frac {6102 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{262609375}\)	\(85\)
trager	\(-\frac {\left (642132 x^{5}+5388660 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3 x^{2}+2}}{45018750 \left (2 x +3\right )^{6}}-\frac {6102 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{262609375}\)	\(92\)
default	\(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{13440 \left (x +\frac {3}{2}\right )^{6}}-\frac {281 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{392000 \left (x +\frac {3}{2}\right )^{5}}-\frac {111 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{280000 \left (x +\frac {3}{2}\right )^{4}}-\frac {1207 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{6860000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1017 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{15006250 \left (x +\frac {3}{2}\right )^{2}}-\frac {9153 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{262609375 \left (x +\frac {3}{2}\right )}+\frac {6102 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{262609375}-\frac {6102 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{262609375}+\frac {27459 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{262609375}\)	\(191\)

Input:

int((5-x)*(3*x^2+2)^(1/2)/(2*x+3)^7,x,method=_RETURNVERBOSE)
 

Output:

-1/45018750*(1926396*x^7+16165980*x^6+55992429*x^5+103039110*x^4+92426010* 
x^3+128433504*x^2+37302600*x[phone])/(2*x+3)^6/(3*x^2+2)^(1/2)-6102/2626 
09375*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 = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.01 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx=\frac {18306 \, \sqrt {35} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (642132 \, x^{5} + 5388660 \, x^{4} + 18236055 \, x^{3} + 30753930 \, x^{2} + 18651300 \, x + 22308548\right )} \sqrt {3 \, x^{2} + 2}}{1575656250 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^7,x, algorithm="fricas")
 

Output:

1/1575656250*(18306*sqrt(35)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 486 
0*x^2 + 2916*x + 729)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 
36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(642132*x^5 + 5388660*x^4 + 18236055*x 
^3 + 30753930*x^2 + 18651300*x + 22308548)*sqrt(3*x^2 + 2))/(64*x^6 + 576* 
x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx=\text {Timed out} \] Input:

integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**7,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.55 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx=\frac {6102}{262609375} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {3051}{15006250} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{210 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {281 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{12250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {111 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{17500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1207 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{857500 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2034 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{7503125 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {9153 \, \sqrt {3 \, x^{2} + 2}}{15006250 \, {\left (2 \, x + 3\right )}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^7,x, algorithm="maxima")
 

Output:

6102/262609375*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/a 
bs(2*x + 3)) + 3051/15006250*sqrt(3*x^2 + 2) - 13/210*(3*x^2 + 2)^(3/2)/(6 
4*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 281/122 
50*(3*x^2 + 2)^(3/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) 
 - 111/17500*(3*x^2 + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 
1207/857500*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 2034/7503125* 
(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 9153/15006250*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. 367 vs. \(2 (121) = 242\).

Time = 0.14 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.48 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx=\frac {6102}{262609375} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3 \, \sqrt {3} {\left (21696 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 1073952 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 6978880 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 87678735 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 66333990 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 258582989 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 426764436 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 755892540 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 355133440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 207134880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 19853952 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 2283136\right )}}{240100000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \] Input:

integrate((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^7,x, algorithm="giac")
 

Output:

6102/262609375*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*s 
qrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) 
- 3/240100000*sqrt(3)*(21696*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 + 10 
73952*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 + 6978880*sqrt(3)*(sqrt(3)*x - sqrt 
(3*x^2 + 2))^9 + 87678735*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 66333990*sqrt( 
3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 258582989*(sqrt(3)*x - sqrt(3*x^2 + 2 
))^6 - 426764436*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 755892540*(sqrt 
(3)*x - sqrt(3*x^2 + 2))^4 - 355133440*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2 
))^3 + 207134880*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 19853952*sqrt(3)*(sqrt( 
3)*x - sqrt(3*x^2 + 2)) + 2283136)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sq 
rt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^6
 

Mupad [B] (verification not implemented)

Time = 5.93 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.51 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx=\frac {6102\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{262609375}-\frac {6102\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{262609375}+\frac {127\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1568000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {109\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{44800\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {53511\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{240100000\,\left (x+\frac {3}{2}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1536\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}-\frac {2727\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{13720000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {479\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3920000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \] Input:

int(-((3*x^2 + 2)^(1/2)*(x - 5))/(2*x + 3)^7,x)
 

Output:

(6102*35^(1/2)*log(x + 3/2))/262609375 - (6102*35^(1/2)*log(x - (3^(1/2)*3 
5^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/262609375 + (127*3^(1/2)*(x^2 + 2/3)^ 
(1/2))/(1568000*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (109*3^(1 
/2)*(x^2 + 2/3)^(1/2))/(44800*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15 
*x^4)/2 + x^5 + 243/32)) - (53511*3^(1/2)*(x^2 + 2/3)^(1/2))/(240100000*(x 
 + 3/2)) - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(1536*((729*x)/16 + (1215*x^2)/1 
6 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) - (2727*3^(1/2)*(x^ 
2 + 2/3)^(1/2))/(13720000*(3*x + x^2 + 9/4)) - (479*3^(1/2)*(x^2 + 2/3)^(1 
/2))/(3920000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.41 \[ \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx=\frac {-22474620 \sqrt {3 x^{2}+2}\, x^{5}-188603100 \sqrt {3 x^{2}+2}\, x^{4}-638261925 \sqrt {3 x^{2}+2}\, x^{3}-1076387550 \sqrt {3 x^{2}+2}\, x^{2}-652795500 \sqrt {3 x^{2}+2}\, x -780799180 \sqrt {3 x^{2}+2}+2343168 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{6}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{5}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{4}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{3}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x^{2}[phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x [phone] \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-2343168 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{6}-21088512 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{5}-79081920 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{4}-158163840 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{3}-177934320 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x^{2}-106760592 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -26690148 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{100842000000 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

int((5-x)*(3*x^2+2)^(1/2)/(3+2*x)^7,x)
 

Output:

( - 22474620*sqrt(3*x**2 + 2)*x**5 - 188603100*sqrt(3*x**2 + 2)*x**4 - 638 
261925*sqrt(3*x**2 + 2)*x**3 - 1076387550*sqrt(3*x**2 + 2)*x**2 - 65279550 
0*sqrt(3*x**2 + 2)*x - 780799180*sqrt(3*x**2 + 2) + 2343168*sqrt(35)*log(s 
qrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x**6 + 21088512*sqrt(35)*log(sqrt(3*x* 
*2 + 2)*sqrt(35) + 9*x - 4)*x**5 + 79081920*sqrt(35)*log(sqrt(3*x**2 + 2)* 
sqrt(35) + 9*x - 4)*x**4 + 158163840*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35 
) + 9*x - 4)*x**3 + 177934320*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x 
 - 4)*x**2 + 106760592*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x 
 + 26690148*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) - 2343168*sq 
rt(35)*log(2*x + 3)*x**6 - 21088512*sqrt(35)*log(2*x + 3)*x**5 - 79081920* 
sqrt(35)*log(2*x + 3)*x**4 - 158163840*sqrt(35)*log(2*x + 3)*x**3 - 177934 
320*sqrt(35)*log(2*x + 3)*x**2 - 106760592*sqrt(35)*log(2*x + 3)*x - 26690 
148*sqrt(35)*log(2*x + 3))/(1575656250*(64*x**6 + 576*x**5 + 2160*x**4 + 4 
320*x**3 + 4860*x**2 + 2916*x + 729))