\(\int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx\) [645]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {1526}{729} \sqrt {1-2 x}+\frac {214}{729} (1-2 x)^{3/2}+\frac {2}{27} (1-2 x)^{5/2}-\frac {725}{378} (1-2 x)^{7/2}+\frac {125}{162} (1-2 x)^{9/2}+\frac {49 \sqrt {1-2 x}}{729 (2+3 x)}-\frac {1540}{729} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \] Output:

1526/729*(1-2*x)^(1/2)+214/729*(1-2*x)^(3/2)+2/27*(1-2*x)^(5/2)-725/378*(1 
-2*x)^(7/2)+125/162*(1-2*x)^(9/2)+49*(1-2*x)^(1/2)/(1458+2187*x)-1540/2187 
*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {\frac {3 \sqrt {1-2 x} \left (13759+65558 x+25275 x^2-159714 x^3-17100 x^4+189000 x^5\right )}{2+3 x}-10780 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15309} \] Input:

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

Output:

((3*Sqrt[1 - 2*x]*(13759 + 65558*x + 25275*x^2 - 159714*x^3 - 17100*x^4 + 
189000*x^5))/(2 + 3*x) - 10780*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/ 
15309
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {108, 27, 170, 27, 164, 60, 60, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 170

\(\displaystyle -\frac {55}{3} \left (-\frac {1}{27} \int \frac {2 (1-2 x)^{3/2} (5 x+3) (12 x+5)}{3 x+2}dx-\frac {1}{27} (5 x+3)^2 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {55}{3} \left (-\frac {2}{27} \int \frac {(1-2 x)^{3/2} (5 x+3) (12 x+5)}{3 x+2}dx-\frac {1}{27} (5 x+3)^2 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}\)

\(\Big \downarrow \) 164

\(\displaystyle -\frac {55}{3} \left (-\frac {2}{27} \left (\int \frac {(1-2 x)^{3/2}}{3 x+2}dx-\frac {1}{35} (1-2 x)^{5/2} (100 x+69)\right )-\frac {1}{27} (5 x+3)^2 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {55}{3} \left (-\frac {2}{27} \left (\frac {7}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx-\frac {1}{35} (100 x+69) (1-2 x)^{5/2}+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {1}{27} (5 x+3)^2 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {55}{3} \left (-\frac {2}{27} \left (\frac {7}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )-\frac {1}{35} (100 x+69) (1-2 x)^{5/2}+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {1}{27} (5 x+3)^2 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {55}{3} \left (-\frac {2}{27} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{35} (100 x+69) (1-2 x)^{5/2}+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {1}{27} (5 x+3)^2 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {55}{3} \left (-\frac {2}{27} \left (\frac {7}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {1}{35} (100 x+69) (1-2 x)^{5/2}+\frac {2}{9} (1-2 x)^{3/2}\right )-\frac {1}{27} (5 x+3)^2 (1-2 x)^{5/2}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}\)

Input:

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

Output:

-1/3*((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x) - (55*(-1/27*((1 - 2*x)^(5/2) 
*(3 + 5*x)^2) - (2*((2*(1 - 2*x)^(3/2))/9 - ((1 - 2*x)^(5/2)*(69 + 100*x)) 
/35 + (7*((2*Sqrt[1 - 2*x])/3 - (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2* 
x]])/3))/3))/27))/3
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 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 = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.57

method	result	size
risch	\(-\frac {378000 x^{6}-223200 x^{5}-302328 x^{4}+210264 x^{3}+105841 x^{2}-38040 x -13759}{5103 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {1540 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2187}\)	\(66\)
pseudoelliptic	\(\frac {-10780 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}+3 \sqrt {1-2 x}\, \left (189000 x^{5}-17100 x^{4}-159714 x^{3}+25275 x^{2}+65558 x +13759\right )}{30618+45927 x}\)	\(67\)
derivativedivides	\(\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{162}-\frac {725 \left (1-2 x \right )^{\frac {7}{2}}}{378}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {214 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {1526 \sqrt {1-2 x}}{729}-\frac {98 \sqrt {1-2 x}}{2187 \left (-\frac {4}{3}-2 x \right )}-\frac {1540 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2187}\)	\(81\)
default	\(\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{162}-\frac {725 \left (1-2 x \right )^{\frac {7}{2}}}{378}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {214 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {1526 \sqrt {1-2 x}}{729}-\frac {98 \sqrt {1-2 x}}{2187 \left (-\frac {4}{3}-2 x \right )}-\frac {1540 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2187}\)	\(81\)
trager	\(\frac {\sqrt {1-2 x}\, \left (189000 x^{5}-17100 x^{4}-159714 x^{3}+25275 x^{2}+65558 x +13759\right )}{10206+15309 x}-\frac {770 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2187}\)	\(87\)

Input:

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

Output:

-1/5103*(378000*x^6-223200*x^5-302328*x^4+210264*x^3+105841*x^2-38040*x-13 
759)/(2+3*x)/(1-2*x)^(1/2)-1540/2187*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 = 79, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {5390 \, \sqrt {\frac {7}{3}} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + 3 \, \sqrt {\frac {7}{3}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (189000 \, x^{5} - 17100 \, x^{4} - 159714 \, x^{3} + 25275 \, x^{2} + 65558 \, x + 13759\right )} \sqrt {-2 \, x + 1}}{5103 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

1/5103*(5390*sqrt(7/3)*(3*x + 2)*log((3*x + 3*sqrt(7/3)*sqrt(-2*x + 1) - 5 
)/(3*x + 2)) + (189000*x^5 - 17100*x^4 - 159714*x^3 + 25275*x^2 + 65558*x 
+ 13759)*sqrt(-2*x + 1))/(3*x + 2)
 

Sympy [A] (verification not implemented)

Time = 37.96 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.92 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {125 \left (1 - 2 x\right )^{\frac {9}{2}}}{162} - \frac {725 \left (1 - 2 x\right )^{\frac {7}{2}}}{378} + \frac {2 \left (1 - 2 x\right )^{\frac {5}{2}}}{27} + \frac {214 \left (1 - 2 x\right )^{\frac {3}{2}}}{729} + \frac {1526 \sqrt {1 - 2 x}}{729} + \frac {259 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{729} + \frac {1372 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{729} \] Input:

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

Output:

125*(1 - 2*x)**(9/2)/162 - 725*(1 - 2*x)**(7/2)/378 + 2*(1 - 2*x)**(5/2)/2 
7 + 214*(1 - 2*x)**(3/2)/729 + 1526*sqrt(1 - 2*x)/729 + 259*sqrt(21)*(log( 
sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21)/3))/729 + 1372* 
Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*s 
qrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt( 
21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 
2*x) < sqrt(21)/3)))/729
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {125}{162} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {725}{378} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {214}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {770}{2187} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1526}{729} \, \sqrt {-2 \, x + 1} + \frac {49 \, \sqrt {-2 \, x + 1}}{729 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

125/162*(-2*x + 1)^(9/2) - 725/378*(-2*x + 1)^(7/2) + 2/27*(-2*x + 1)^(5/2 
) + 214/729*(-2*x + 1)^(3/2) + 770/2187*sqrt(21)*log(-(sqrt(21) - 3*sqrt(- 
2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1526/729*sqrt(-2*x + 1) + 49/72 
9*sqrt(-2*x + 1)/(3*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.06 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {125}{162} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {725}{378} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {214}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {770}{2187} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1526}{729} \, \sqrt {-2 \, x + 1} + \frac {49 \, \sqrt {-2 \, x + 1}}{729 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

125/162*(2*x - 1)^4*sqrt(-2*x + 1) + 725/378*(2*x - 1)^3*sqrt(-2*x + 1) + 
2/27*(2*x - 1)^2*sqrt(-2*x + 1) + 214/729*(-2*x + 1)^(3/2) + 770/2187*sqrt 
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 
 1))) + 1526/729*sqrt(-2*x + 1) + 49/729*sqrt(-2*x + 1)/(3*x + 2)
 

Mupad [B] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {98\,\sqrt {1-2\,x}}{2187\,\left (2\,x+\frac {4}{3}\right )}+\frac {1526\,\sqrt {1-2\,x}}{729}+\frac {214\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {725\,{\left (1-2\,x\right )}^{7/2}}{378}+\frac {125\,{\left (1-2\,x\right )}^{9/2}}{162}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,1540{}\mathrm {i}}{2187} \] Input:

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

Output:

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*1540i)/2187 + (98*(1 - 2*x 
)^(1/2))/(2187*(2*x + 4/3)) + (1526*(1 - 2*x)^(1/2))/729 + (214*(1 - 2*x)^ 
(3/2))/729 + (2*(1 - 2*x)^(5/2))/27 - (725*(1 - 2*x)^(7/2))/378 + (125*(1 
- 2*x)^(9/2))/162
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.22 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {567000 \sqrt {-2 x +1}\, x^{5}-51300 \sqrt {-2 x +1}\, x^{4}-479142 \sqrt {-2 x +1}\, x^{3}+75825 \sqrt {-2 x +1}\, x^{2}+196674 \sqrt {-2 x +1}\, x +41277 \sqrt {-2 x +1}+16170 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +10780 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-16170 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -10780 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{45927 x +30618} \] Input:

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

Output:

(567000*sqrt( - 2*x + 1)*x**5 - 51300*sqrt( - 2*x + 1)*x**4 - 479142*sqrt( 
 - 2*x + 1)*x**3 + 75825*sqrt( - 2*x + 1)*x**2 + 196674*sqrt( - 2*x + 1)*x 
 + 41277*sqrt( - 2*x + 1) + 16170*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(2 
1))*x + 10780*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 16170*sqrt(21) 
*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 10780*sqrt(21)*log(3*sqrt( - 2*x + 
 1) + sqrt(21)))/(15309*(3*x + 2))