3.19.81 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx\) [1881]

3.19.81.1 Optimal result

Integrand size = 24, antiderivative size = 148 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx=-\frac {(1-2 x)^{5/2}}{378 (2+3 x)^6}+\frac {59 (1-2 x)^{5/2}}{1890 (2+3 x)^5}-\frac {991 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {991 \sqrt {1-2 x}}{13608 (2+3 x)^3}-\frac {991 \sqrt {1-2 x}}{190512 (2+3 x)^2}-\frac {991 \sqrt {1-2 x}}{444528 (2+3 x)}-\frac {991 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{222264 \sqrt {21}} \]

output

-1/378*(1-2*x)^(5/2)/(2+3*x)^6+59/1890*(1-2*x)^(5/2)/(2+3*x)^5-991/4536*(1 
-2*x)^(3/2)/(2+3*x)^4-991/4667544*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^( 
1/2)+991/13608*(1-2*x)^(1/2)/(2+3*x)^3-991/190512*(1-2*x)^(1/2)/(2+3*x)^2- 
991/444528*(1-2*x)^(1/2)/(2+3*x)
 

3.19.81.2 Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (858112-1262200 x-9658494 x^2-6094818 x^3+4950045 x^4+1204065 x^5\right )}{2 (2+3 x)^6}-4955 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{23337720} \]

input

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

output

((-21*Sqrt[1 - 2*x]*(858112 - 1262200*x - 9658494*x^2 - 6094818*x^3 + 4950 
045*x^4 + 1204065*x^5))/(2*(2 + 3*x)^6) - 4955*Sqrt[21]*ArcTanh[Sqrt[3/7]* 
Sqrt[1 - 2*x]])/23337720
 

3.19.81.3 Rubi [A] (verified)

Time = 0.22 (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, 52, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{378} \int \frac {7 (1-2 x)^{3/2} (450 x+241)}{(3 x+2)^6}dx-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{54} \int \frac {(1-2 x)^{3/2} (450 x+241)}{(3 x+2)^6}dx-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{54} \left (\frac {991}{7} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^5}dx+\frac {59 (1-2 x)^{5/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{54} \left (\frac {991}{7} \left (-\frac {1}{4} \int \frac {\sqrt {1-2 x}}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {59 (1-2 x)^{5/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{54} \left (\frac {991}{7} \left (\frac {1}{4} \left (\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {59 (1-2 x)^{5/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{54} \left (\frac {991}{7} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {59 (1-2 x)^{5/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{54} \left (\frac {991}{7} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {59 (1-2 x)^{5/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{54} \left (\frac {991}{7} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {59 (1-2 x)^{5/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{54} \left (\frac {991}{7} \left (\frac {1}{4} \left (\frac {1}{9} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{3/2}}{12 (3 x+2)^4}\right )+\frac {59 (1-2 x)^{5/2}}{35 (3 x+2)^5}\right )-\frac {(1-2 x)^{5/2}}{378 (3 x+2)^6}\)

input

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

output

-1/378*(1 - 2*x)^(5/2)/(2 + 3*x)^6 + ((59*(1 - 2*x)^(5/2))/(35*(2 + 3*x)^5 
) + (991*(-1/12*(1 - 2*x)^(3/2)/(2 + 3*x)^4 + (Sqrt[1 - 2*x]/(9*(2 + 3*x)^ 
3) + (-1/14*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - 
 (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14)/9)/4))/7)/54
 

3.19.81.3.1 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])
 

3.19.81.4 Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.45

method	result	size
risch	\(\frac {2408130 x^{6}+8696025 x^{5}-17139681 x^{4}-13222170 x^{3}+7134094 x^{2}+2978424 x -858112}{2222640 \left (2+3 x \right )^{6} \sqrt {1-2 x}}-\frac {991 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\)	\(66\)
pseudoelliptic	\(\frac {-9910 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{6} \sqrt {21}-21 \sqrt {1-2 x}\, \left (1204065 x^{5}+4950045 x^{4}-6094818 x^{3}-9658494 x^{2}-1262200 x +858112\right )}{46675440 \left (2+3 x \right )^{6}}\)	\(70\)
derivativedivides	\(\frac {\frac {2973 \left (1-2 x \right )^{\frac {11}{2}}}{2744}-\frac {16847 \left (1-2 x \right )^{\frac {9}{2}}}{1176}+\frac {10303 \left (1-2 x \right )^{\frac {7}{2}}}{420}+\frac {29843 \left (1-2 x \right )^{\frac {5}{2}}}{420}-\frac {117929 \left (1-2 x \right )^{\frac {3}{2}}}{648}+\frac {48559 \sqrt {1-2 x}}{648}}{\left (-4-6 x \right )^{6}}-\frac {991 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\)	\(84\)
default	\(\frac {\frac {2973 \left (1-2 x \right )^{\frac {11}{2}}}{2744}-\frac {16847 \left (1-2 x \right )^{\frac {9}{2}}}{1176}+\frac {10303 \left (1-2 x \right )^{\frac {7}{2}}}{420}+\frac {29843 \left (1-2 x \right )^{\frac {5}{2}}}{420}-\frac {117929 \left (1-2 x \right )^{\frac {3}{2}}}{648}+\frac {48559 \sqrt {1-2 x}}{648}}{\left (-4-6 x \right )^{6}}-\frac {991 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{4667544}\)	\(84\)
trager	\(-\frac {\left (1204065 x^{5}+4950045 x^{4}-6094818 x^{3}-9658494 x^{2}-1262200 x +858112\right ) \sqrt {1-2 x}}{2222640 \left (2+3 x \right )^{6}}-\frac {991 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{9335088}\)	\(87\)

input

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

output

1/2222640*(2408130*x^6+8696025*x^5-17139681*x^4-13222170*x^3+7134094*x^2+2 
978424*x-858112)/(2+3*x)^6/(1-2*x)^(1/2)-991/4667544*arctanh(1/7*21^(1/2)* 
(1-2*x)^(1/2))*21^(1/2)
 

3.19.81.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {4955 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (1204065 \, x^{5} + 4950045 \, x^{4} - 6094818 \, x^{3} - 9658494 \, x^{2} - 1262200 \, x + 858112\right )} \sqrt {-2 \, x + 1}}{46675440 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

input

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

output

1/46675440*(4955*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 
*(1204065*x^5 + 4950045*x^4 - 6094818*x^3 - 9658494*x^2 - 1262200*x + 8581 
12)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 
 576*x + 64)
 

3.19.81.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.19.81.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {991}{9335088} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1204065 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 15920415 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 27261738 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 78964578 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 202248235 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 83278685 \, \sqrt {-2 \, x + 1}}{1111320 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

input

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

output

991/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt 
(-2*x + 1))) + 1/1111320*(1204065*(-2*x + 1)^(11/2) - 15920415*(-2*x + 1)^ 
(9/2) + 27261738*(-2*x + 1)^(7/2) + 78964578*(-2*x + 1)^(5/2) - 202248235* 
(-2*x + 1)^(3/2) + 83278685*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 + 605 
052*x - 184877)
 

3.19.81.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {991}{9335088} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {1204065 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 15920415 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 27261738 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 78964578 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 202248235 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 83278685 \, \sqrt {-2 \, x + 1}}{71124480 \, {\left (3 \, x + 2\right )}^{6}} \]

input

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

output

991/9335088*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) 
 + 3*sqrt(-2*x + 1))) - 1/71124480*(1204065*(2*x - 1)^5*sqrt(-2*x + 1) + 1 
5920415*(2*x - 1)^4*sqrt(-2*x + 1) + 27261738*(2*x - 1)^3*sqrt(-2*x + 1) - 
 78964578*(2*x - 1)^2*sqrt(-2*x + 1) + 202248235*(-2*x + 1)^(3/2) - 832786 
85*sqrt(-2*x + 1))/(3*x + 2)^6
 

3.19.81.9 Mupad [B] (verification not implemented)

Time = 1.36 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx=\frac {\frac {48559\,\sqrt {1-2\,x}}{472392}-\frac {117929\,{\left (1-2\,x\right )}^{3/2}}{472392}+\frac {29843\,{\left (1-2\,x\right )}^{5/2}}{306180}+\frac {10303\,{\left (1-2\,x\right )}^{7/2}}{306180}-\frac {16847\,{\left (1-2\,x\right )}^{9/2}}{857304}+\frac {991\,{\left (1-2\,x\right )}^{11/2}}{666792}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}}-\frac {991\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{4667544} \]

input

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

output

((48559*(1 - 2*x)^(1/2))/472392 - (117929*(1 - 2*x)^(3/2))/472392 + (29843 
*(1 - 2*x)^(5/2))/306180 + (10303*(1 - 2*x)^(7/2))/306180 - (16847*(1 - 2* 
x)^(9/2))/857304 + (991*(1 - 2*x)^(11/2))/666792)/((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) - (991*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^ 
(1/2))/7))/4667544