3.20.53 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx\) [1953]

3.20.53.1 Optimal result

Integrand size = 24, antiderivative size = 148 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {(1-2 x)^{7/2}}{378 (2+3 x)^6}+\frac {83 (1-2 x)^{7/2}}{2646 (2+3 x)^5}-\frac {263 (1-2 x)^{5/2}}{1176 (2+3 x)^4}+\frac {1315 (1-2 x)^{3/2}}{10584 (2+3 x)^3}-\frac {1315 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {1315 \sqrt {1-2 x}}{148176 (2+3 x)}+\frac {1315 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{74088 \sqrt {21}} \]

output

-1/378*(1-2*x)^(7/2)/(2+3*x)^6+83/2646*(1-2*x)^(7/2)/(2+3*x)^5-263/1176*(1 
-2*x)^(5/2)/(2+3*x)^4+1315/10584*(1-2*x)^(3/2)/(2+3*x)^3+1315/1555848*arct 
anh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1315/21168*(1-2*x)^(1/2)/(2+3*x)^ 
2+1315/148176*(1-2*x)^(1/2)/(2+3*x)
 

3.20.53.2 Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-81568-106808 x-587502 x^2-2360850 x^3-1979115 x^4+319545 x^5\right )}{2 (2+3 x)^6}+1315 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1555848} \]

input

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^7,x]
 

output

((21*Sqrt[1 - 2*x]*(-81568 - 106808*x - 587502*x^2 - 2360850*x^3 - 1979115 
*x^4 + 319545*x^5))/(2*(2 + 3*x)^6) + 1315*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt 
[1 - 2*x]])/1555848
 

3.20.53.3 Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {100, 27, 87, 51, 51, 51, 52, 73, 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.

\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^2}{(3 x+2)^7} \, dx\)

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{378} \int \frac {5 (1-2 x)^{5/2} (630 x+337)}{(3 x+2)^6}dx-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{378} \int \frac {(1-2 x)^{5/2} (630 x+337)}{(3 x+2)^6}dx-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {5}{378} \left (\frac {7101}{35} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^5}dx+\frac {83 (1-2 x)^{7/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {5}{378} \left (\frac {7101}{35} \left (-\frac {5}{12} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^4}dx-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {83 (1-2 x)^{7/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {5}{378} \left (\frac {7101}{35} \left (-\frac {5}{12} \left (-\frac {1}{3} \int \frac {\sqrt {1-2 x}}{(3 x+2)^3}dx-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {83 (1-2 x)^{7/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {5}{378} \left (\frac {7101}{35} \left (-\frac {5}{12} \left (\frac {1}{3} \left (\frac {1}{6} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {83 (1-2 x)^{7/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {5}{378} \left (\frac {7101}{35} \left (-\frac {5}{12} \left (\frac {1}{3} \left (\frac {1}{6} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {83 (1-2 x)^{7/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {5}{378} \left (\frac {7101}{35} \left (-\frac {5}{12} \left (\frac {1}{3} \left (\frac {1}{6} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {83 (1-2 x)^{7/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5}{378} \left (\frac {7101}{35} \left (-\frac {5}{12} \left (\frac {1}{3} \left (\frac {1}{6} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2}}{12 (3 x+2)^4}\right )+\frac {83 (1-2 x)^{7/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{7/2}}{378 (3 x+2)^6}\)

input

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^7,x]
 

output

-1/378*(1 - 2*x)^(7/2)/(2 + 3*x)^6 + (5*((83*(1 - 2*x)^(7/2))/(35*(2 + 3*x 
)^5) + (7101*(-1/12*(1 - 2*x)^(5/2)/(2 + 3*x)^4 - (5*(-1/9*(1 - 2*x)^(3/2) 
/(2 + 3*x)^3 + (Sqrt[1 - 2*x]/(6*(2 + 3*x)^2) + (-1/7*Sqrt[1 - 2*x]/(2 + 3 
*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21]))/6)/3))/12))/35))/ 
378
 

3.20.53.3.1 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 51

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

rule 52

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

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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])
 

3.20.53.4 Maple [A] (verified)

Time = 3.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45

method	result	size
risch	\(-\frac {639090 x^{6}-4277775 x^{5}-2742585 x^{4}+1185846 x^{3}+373886 x^{2}-56328 x +81568}{148176 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {1315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1555848}\)	\(66\)
pseudoelliptic	\(\frac {2630 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}+21 \sqrt {1-2 x}\, \left (319545 x^{5}-1979115 x^{4}-2360850 x^{3}-587502 x^{2}-106808 x -81568\right )}{3111696 \left (2+3 x \right )^{6}}\)	\(70\)
derivativedivides	\(\frac {-\frac {11835 \left (1-2 x \right )^{\frac {11}{2}}}{2744}-\frac {112405 \left (1-2 x \right )^{\frac {9}{2}}}{3528}+\frac {8345 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {2893 \left (1-2 x \right )^{\frac {5}{2}}}{4}+\frac {156485 \left (1-2 x \right )^{\frac {3}{2}}}{216}-\frac {64435 \sqrt {1-2 x}}{216}}{\left (-4-6 x \right )^{6}}+\frac {1315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1555848}\)	\(84\)
default	\(\frac {-\frac {11835 \left (1-2 x \right )^{\frac {11}{2}}}{2744}-\frac {112405 \left (1-2 x \right )^{\frac {9}{2}}}{3528}+\frac {8345 \left (1-2 x \right )^{\frac {7}{2}}}{28}-\frac {2893 \left (1-2 x \right )^{\frac {5}{2}}}{4}+\frac {156485 \left (1-2 x \right )^{\frac {3}{2}}}{216}-\frac {64435 \sqrt {1-2 x}}{216}}{\left (-4-6 x \right )^{6}}+\frac {1315 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1555848}\)	\(84\)
trager	\(\frac {\left (319545 x^{5}-1979115 x^{4}-2360850 x^{3}-587502 x^{2}-106808 x -81568\right ) \sqrt {1-2 x}}{148176 \left (2+3 x \right )^{6}}+\frac {1315 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{3111696}\)	\(87\)

input

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

output

-1/148176*(639090*x^6-4277775*x^5-2742585*x^4+1185846*x^3+373886*x^2-56328 
*x+81568)/(2+3*x)^6/(1-2*x)^(1/2)+1315/1555848*arctanh(1/7*21^(1/2)*(1-2*x 
)^(1/2))*21^(1/2)
 

3.20.53.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {1315 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (319545 \, x^{5} - 1979115 \, x^{4} - 2360850 \, x^{3} - 587502 \, x^{2} - 106808 \, x - 81568\right )} \sqrt {-2 \, x + 1}}{3111696 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

input

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

output

1/3111696*(1315*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160* 
x^2 + 576*x + 64)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21* 
(319545*x^5 - 1979115*x^4 - 2360850*x^3 - 587502*x^2 - 106808*x - 81568)*s 
qrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576* 
x + 64)
 

3.20.53.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.20.53.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {1315}{3111696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {319545 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 2360505 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 22080870 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 53584146 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 53674355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 22101205 \, \sqrt {-2 \, x + 1}}{74088 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

input

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

output

-1315/3111696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sq 
rt(-2*x + 1))) - 1/74088*(319545*(-2*x + 1)^(11/2) + 2360505*(-2*x + 1)^(9 
/2) - 22080870*(-2*x + 1)^(7/2) + 53584146*(-2*x + 1)^(5/2) - 53674355*(-2 
*x + 1)^(3/2) + 22101205*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1 
)^5 + 59535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052 
*x - 184877)
 

3.20.53.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {1315}{3111696} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {319545 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - 2360505 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - 22080870 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 53584146 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 53674355 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 22101205 \, \sqrt {-2 \, x + 1}}{4741632 \, {\left (3 \, x + 2\right )}^{6}} \]

input

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

output

-1315/3111696*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(2 
1) + 3*sqrt(-2*x + 1))) + 1/4741632*(319545*(2*x - 1)^5*sqrt(-2*x + 1) - 2 
360505*(2*x - 1)^4*sqrt(-2*x + 1) - 22080870*(2*x - 1)^3*sqrt(-2*x + 1) - 
53584146*(2*x - 1)^2*sqrt(-2*x + 1) + 53674355*(-2*x + 1)^(3/2) - 22101205 
*sqrt(-2*x + 1))/(3*x + 2)^6
 

3.20.53.9 Mupad [B] (verification not implemented)

Time = 1.72 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {1315\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1555848}-\frac {\frac {64435\,\sqrt {1-2\,x}}{157464}-\frac {156485\,{\left (1-2\,x\right )}^{3/2}}{157464}+\frac {2893\,{\left (1-2\,x\right )}^{5/2}}{2916}-\frac {8345\,{\left (1-2\,x\right )}^{7/2}}{20412}+\frac {112405\,{\left (1-2\,x\right )}^{9/2}}{2571912}+\frac {1315\,{\left (1-2\,x\right )}^{11/2}}{222264}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]

input

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

output

(1315*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1555848 - ((64435*(1 - 
 2*x)^(1/2))/157464 - (156485*(1 - 2*x)^(3/2))/157464 + (2893*(1 - 2*x)^(5 
/2))/2916 - (8345*(1 - 2*x)^(7/2))/20412 + (112405*(1 - 2*x)^(9/2))/257191 
2 + (1315*(1 - 2*x)^(11/2))/222264)/((67228*x)/81 + (12005*(2*x - 1)^2)/27 
 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 14*(2*x - 1)^5 + (2*x - 1 
)^6 - 184877/729)