\(\int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx\) [647]

Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {16570}{729} \sqrt {1-2 x}+\frac {2050}{729} (1-2 x)^{3/2}+\frac {50}{81} (1-2 x)^{5/2}+\frac {49 \sqrt {1-2 x}}{2187 (2+3 x)^3}-\frac {1148 \sqrt {1-2 x}}{2187 (2+3 x)^2}+\frac {4153 \sqrt {1-2 x}}{729 (2+3 x)}-\frac {41360 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{243 \sqrt {21}} \] Output:

16570/729*(1-2*x)^(1/2)+2050/729*(1-2*x)^(3/2)+50/81*(1-2*x)^(5/2)+49/2187 
*(1-2*x)^(1/2)/(2+3*x)^3-1148/2187*(1-2*x)^(1/2)/(2+3*x)^2+4153*(1-2*x)^(1 
/2)/(1458+2187*x)-41360/5103*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (56141+229336 x+289719 x^2+87030 x^3-20700 x^4+16200 x^5\right )}{(2+3 x)^3}-41360 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5103} \] Input:

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^4,x]
 

Output:

((21*Sqrt[1 - 2*x]*(56141 + 229336*x + 289719*x^2 + 87030*x^3 - 20700*x^4 
+ 16200*x^5))/(2 + 3*x)^3 - 41360*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x] 
])/5103
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {108, 27, 166, 27, 166, 27, 170, 27, 164, 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.

\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^3}{(3 x+2)^4} \, dx\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{9} \int -\frac {55 (1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^3}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {55}{9} \int \frac {(1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^3}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {55}{9} \left (-\frac {1}{6} \int \frac {6 \sqrt {1-2 x} (5 x+3)^2 (9 x+1)}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {55}{9} \left (-\int \frac {\sqrt {1-2 x} (5 x+3)^2 (9 x+1)}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 166

\(\displaystyle -\frac {55}{9} \left (\frac {1}{3} \int -\frac {3 (38-131 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)}dx+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {55}{9} \left (-\int \frac {(38-131 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)}dx+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 170

\(\displaystyle -\frac {55}{9} \left (\frac {1}{15} \int \frac {2 (62-377 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {55}{9} \left (\frac {2}{15} \int \frac {(62-377 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 164

\(\displaystyle -\frac {55}{9} \left (\frac {2}{15} \left (\frac {1}{9} \sqrt {1-2 x} (1885 x+578)-\frac {940}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {55}{9} \left (\frac {2}{15} \left (\frac {940}{9} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}+\frac {1}{9} \sqrt {1-2 x} (1885 x+578)\right )+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {55}{9} \left (\frac {2}{15} \left (\frac {1880 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}+\frac {1}{9} \sqrt {1-2 x} (1885 x+578)\right )+\frac {5 \sqrt {1-2 x} (5 x+3)^3}{3 x+2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)^2}-\frac {131}{15} \sqrt {1-2 x} (5 x+3)^2\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}\)

Input:

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^4,x]
 

Output:

-1/9*((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^3 - (55*((-131*Sqrt[1 - 2*x]* 
(3 + 5*x)^2)/15 - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(3*(2 + 3*x)^2) + (5*Sqrt[ 
1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x) + (2*((Sqrt[1 - 2*x]*(578 + 1885*x))/9 + ( 
1880*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21])))/15))/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 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 108

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

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

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])
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {32400 x^{6}-57600 x^{5}+194760 x^{4}+492408 x^{3}+168953 x^{2}-117054 x -56141}{243 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {41360 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{5103}\)	\(66\)
pseudoelliptic	\(\frac {-41360 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (16200 x^{5}-20700 x^{4}+87030 x^{3}+289719 x^{2}+229336 x +56141\right )}{5103 \left (2+3 x \right )^{3}}\)	\(70\)
derivativedivides	\(\frac {50 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {2050 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {16570 \sqrt {1-2 x}}{729}+\frac {-\frac {8306 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {344260 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {396410 \sqrt {1-2 x}}{729}}{\left (-4-6 x \right )^{3}}-\frac {41360 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{5103}\)	\(84\)
default	\(\frac {50 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {2050 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {16570 \sqrt {1-2 x}}{729}+\frac {-\frac {8306 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {344260 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {396410 \sqrt {1-2 x}}{729}}{\left (-4-6 x \right )^{3}}-\frac {41360 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{5103}\)	\(84\)
trager	\(\frac {\left (16200 x^{5}-20700 x^{4}+87030 x^{3}+289719 x^{2}+229336 x +56141\right ) \sqrt {1-2 x}}{243 \left (2+3 x \right )^{3}}+\frac {20680 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{5103}\)	\(87\)

Input:

int((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/243*(32400*x^6-57600*x^5+194760*x^4+492408*x^3+168953*x^2-117054*x-5614 
1)/(2+3*x)^3/(1-2*x)^(1/2)-41360/5103*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x 
)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {20680 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (16200 \, x^{5} - 20700 \, x^{4} + 87030 \, x^{3} + 289719 \, x^{2} + 229336 \, x + 56141\right )} \sqrt {-2 \, x + 1}}{5103 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="fricas")
 

Output:

1/5103*(20680*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sq 
rt(-2*x + 1) - 5)/(3*x + 2)) + 21*(16200*x^5 - 20700*x^4 + 87030*x^3 + 289 
719*x^2 + 229336*x + 56141)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\text {Timed out} \] Input:

integrate((1-2*x)**(5/2)*(3+5*x)**3/(2+3*x)**4,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2050}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {20680}{5103} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {16570}{729} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (37377 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 172130 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 198205 \, \sqrt {-2 \, x + 1}\right )}}{729 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="maxima")
 

Output:

50/81*(-2*x + 1)^(5/2) + 2050/729*(-2*x + 1)^(3/2) + 20680/5103*sqrt(21)*l 
og(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16570/7 
29*sqrt(-2*x + 1) + 2/729*(37377*(-2*x + 1)^(5/2) - 172130*(-2*x + 1)^(3/2 
) + 198205*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {50}{81} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2050}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {20680}{5103} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16570}{729} \, \sqrt {-2 \, x + 1} + \frac {37377 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 172130 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 198205 \, \sqrt {-2 \, x + 1}}{2916 \, {\left (3 \, x + 2\right )}^{3}} \] Input:

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="giac")
 

Output:

50/81*(2*x - 1)^2*sqrt(-2*x + 1) + 2050/729*(-2*x + 1)^(3/2) + 20680/5103* 
sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 
*x + 1))) + 16570/729*sqrt(-2*x + 1) + 1/2916*(37377*(2*x - 1)^2*sqrt(-2*x 
 + 1) - 172130*(-2*x + 1)^(3/2) + 198205*sqrt(-2*x + 1))/(3*x + 2)^3
 

Mupad [B] (verification not implemented)

Time = 1.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {16570\,\sqrt {1-2\,x}}{729}+\frac {2050\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {50\,{\left (1-2\,x\right )}^{5/2}}{81}+\frac {\frac {396410\,\sqrt {1-2\,x}}{19683}-\frac {344260\,{\left (1-2\,x\right )}^{3/2}}{19683}+\frac {8306\,{\left (1-2\,x\right )}^{5/2}}{2187}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,41360{}\mathrm {i}}{5103} \] Input:

int(((1 - 2*x)^(5/2)*(5*x + 3)^3)/(3*x + 2)^4,x)
                                                                                    
                                                                                    
 

Output:

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*41360i)/5103 + (16570*(1 - 
 2*x)^(1/2))/729 + (2050*(1 - 2*x)^(3/2))/729 + (50*(1 - 2*x)^(5/2))/81 + 
((396410*(1 - 2*x)^(1/2))/19683 - (344260*(1 - 2*x)^(3/2))/19683 + (8306*( 
1 - 2*x)^(5/2))/2187)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 98/27)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.81 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {340200 \sqrt {-2 x +1}\, x^{5}-434700 \sqrt {-2 x +1}\, x^{4}+1827630 \sqrt {-2 x +1}\, x^{3}+6084099 \sqrt {-2 x +1}\, x^{2}+4816056 \sqrt {-2 x +1}\, x +1178961 \sqrt {-2 x +1}+558360 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}+1116720 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+744480 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +165440 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-558360 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}-1116720 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-744480 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -165440 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{137781 x^{3}+275562 x^{2}+183708 x +40824} \] Input:

int((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x)
 

Output:

(340200*sqrt( - 2*x + 1)*x**5 - 434700*sqrt( - 2*x + 1)*x**4 + 1827630*sqr 
t( - 2*x + 1)*x**3 + 6084099*sqrt( - 2*x + 1)*x**2 + 4816056*sqrt( - 2*x + 
 1)*x + 1178961*sqrt( - 2*x + 1) + 558360*sqrt(21)*log(3*sqrt( - 2*x + 1) 
- sqrt(21))*x**3 + 1116720*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x** 
2 + 744480*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 165440*sqrt(21) 
*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 558360*sqrt(21)*log(3*sqrt( - 2*x + 
1) + sqrt(21))*x**3 - 1116720*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))* 
x**2 - 744480*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 165440*sqrt( 
21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(5103*(27*x**3 + 54*x**2 + 36*x + 
8))