\(\int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx\) [671]

Optimal result

Integrand size = 24, antiderivative size = 109 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx=\frac {1658 \sqrt {1-2 x}}{3125}+\frac {58}{625} (1-2 x)^{3/2}+\frac {18}{625} (1-2 x)^{5/2}-\frac {121 \sqrt {1-2 x}}{6250 (3+5 x)^2}-\frac {1353 \sqrt {1-2 x}}{6250 (3+5 x)}-\frac {1533 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \] Output:

1658/3125*(1-2*x)^(1/2)+58/625*(1-2*x)^(3/2)+18/625*(1-2*x)^(5/2)-121/6250 
*(1-2*x)^(1/2)/(3+5*x)^2-1353*(1-2*x)^(1/2)/(18750+31250*x)-1533/15625*55^ 
(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx=\frac {\frac {5 \sqrt {1-2 x} \left (32504+98595 x+51980 x^2-25400 x^3+18000 x^4\right )}{(3+5 x)^2}-3066 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{31250} \] Input:

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

Output:

((5*Sqrt[1 - 2*x]*(32504 + 98595*x + 51980*x^2 - 25400*x^3 + 18000*x^4))/( 
3 + 5*x)^2 - 3066*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/31250
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {100, 27, 87, 60, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {1}{550} \int \frac {3 (1-2 x)^{5/2} (330 x+241)}{(5 x+3)^2}dx-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{550} \int \frac {(1-2 x)^{5/2} (330 x+241)}{(5 x+3)^2}dx-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {3}{550} \left (\frac {511}{11} \int \frac {(1-2 x)^{5/2}}{5 x+3}dx-\frac {43 (1-2 x)^{7/2}}{11 (5 x+3)}\right )-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3}{550} \left (\frac {511}{11} \left (\frac {11}{5} \int \frac {(1-2 x)^{3/2}}{5 x+3}dx+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {43 (1-2 x)^{7/2}}{11 (5 x+3)}\right )-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3}{550} \left (\frac {511}{11} \left (\frac {11}{5} \left (\frac {11}{5} \int \frac {\sqrt {1-2 x}}{5 x+3}dx+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {43 (1-2 x)^{7/2}}{11 (5 x+3)}\right )-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3}{550} \left (\frac {511}{11} \left (\frac {11}{5} \left (\frac {11}{5} \left (\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{5} \sqrt {1-2 x}\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {43 (1-2 x)^{7/2}}{11 (5 x+3)}\right )-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {3}{550} \left (\frac {511}{11} \left (\frac {11}{5} \left (\frac {11}{5} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {11}{5} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {43 (1-2 x)^{7/2}}{11 (5 x+3)}\right )-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{550} \left (\frac {511}{11} \left (\frac {11}{5} \left (\frac {11}{5} \left (\frac {2}{5} \sqrt {1-2 x}-\frac {2}{5} \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {2}{15} (1-2 x)^{3/2}\right )+\frac {2}{25} (1-2 x)^{5/2}\right )-\frac {43 (1-2 x)^{7/2}}{11 (5 x+3)}\right )-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}\)

Input:

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

Output:

-1/550*(1 - 2*x)^(7/2)/(3 + 5*x)^2 + (3*((-43*(1 - 2*x)^(7/2))/(11*(3 + 5* 
x)) + (511*((2*(1 - 2*x)^(5/2))/25 + (11*((2*(1 - 2*x)^(3/2))/15 + (11*((2 
*Sqrt[1 - 2*x])/5 - (2*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5))/5 
))/5))/11))/550
 

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 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])
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.56

method	result	size
risch	\(-\frac {36000 x^{5}-68800 x^{4}+129360 x^{3}+145210 x^{2}-33587 x -32504}{6250 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {1533 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{15625}\)	\(61\)
pseudoelliptic	\(\frac {-3066 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}+5 \sqrt {1-2 x}\, \left (18000 x^{4}-25400 x^{3}+51980 x^{2}+98595 x +32504\right )}{31250 \left (3+5 x \right )^{2}}\)	\(65\)
derivativedivides	\(\frac {18 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {58 \left (1-2 x \right )^{\frac {3}{2}}}{625}+\frac {1658 \sqrt {1-2 x}}{3125}+\frac {\frac {1353 \left (1-2 x \right )^{\frac {3}{2}}}{625}-\frac {121 \sqrt {1-2 x}}{25}}{\left (-6-10 x \right )^{2}}-\frac {1533 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{15625}\)	\(75\)
default	\(\frac {18 \left (1-2 x \right )^{\frac {5}{2}}}{625}+\frac {58 \left (1-2 x \right )^{\frac {3}{2}}}{625}+\frac {1658 \sqrt {1-2 x}}{3125}+\frac {\frac {1353 \left (1-2 x \right )^{\frac {3}{2}}}{625}-\frac {121 \sqrt {1-2 x}}{25}}{\left (-6-10 x \right )^{2}}-\frac {1533 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{15625}\)	\(75\)
trager	\(\frac {\left (18000 x^{4}-25400 x^{3}+51980 x^{2}+98595 x +32504\right ) \sqrt {1-2 x}}{6250 \left (3+5 x \right )^{2}}+\frac {1533 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{31250}\)	\(82\)

Input:

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

Output:

-1/6250*(36000*x^5-68800*x^4+129360*x^3+145210*x^2-33587*x-32504)/(3+5*x)^ 
2/(1-2*x)^(1/2)-1533/15625*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx=\frac {1533 \, \sqrt {\frac {11}{5}} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + 5 \, \sqrt {\frac {11}{5}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + {\left (18000 \, x^{4} - 25400 \, x^{3} + 51980 \, x^{2} + 98595 \, x + 32504\right )} \sqrt {-2 \, x + 1}}{6250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:

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

Output:

1/6250*(1533*sqrt(11/5)*(25*x^2 + 30*x + 9)*log((5*x + 5*sqrt(11/5)*sqrt(- 
2*x + 1) - 8)/(5*x + 3)) + (18000*x^4 - 25400*x^3 + 51980*x^2 + 98595*x + 
32504)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
 

Sympy [A] (verification not implemented)

Time = 114.66 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.36 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx=\frac {18 \left (1 - 2 x\right )^{\frac {5}{2}}}{625} + \frac {58 \left (1 - 2 x\right )^{\frac {3}{2}}}{625} + \frac {1658 \sqrt {1 - 2 x}}{3125} + \frac {141 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{3125} - \frac {5808 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{625} + \frac {10648 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{3125} \] Input:

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

Output:

18*(1 - 2*x)**(5/2)/625 + 58*(1 - 2*x)**(3/2)/625 + 1658*sqrt(1 - 2*x)/312 
5 + 141*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sq 
rt(55)/5))/3125 - 5808*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 
 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2 
*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) 
> -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/625 + 10648*Piecewise((sqr 
t(55)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2 
*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt(55)* 
sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/(16 
*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(55)/5) 
& (sqrt(1 - 2*x) < sqrt(55)/5)))/3125
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx=\frac {18}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {58}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1533}{31250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1658}{3125} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (123 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 275 \, \sqrt {-2 \, x + 1}\right )}}{625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \] Input:

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

Output:

18/625*(-2*x + 1)^(5/2) + 58/625*(-2*x + 1)^(3/2) + 1533/31250*sqrt(55)*lo 
g(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1658/312 
5*sqrt(-2*x + 1) + 11/625*(123*(-2*x + 1)^(3/2) - 275*sqrt(-2*x + 1))/(25* 
(2*x - 1)^2 + 220*x + 11)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx=\frac {18}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {58}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1533}{31250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1658}{3125} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (123 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 275 \, \sqrt {-2 \, x + 1}\right )}}{2500 \, {\left (5 \, x + 3\right )}^{2}} \] Input:

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

Output:

18/625*(2*x - 1)^2*sqrt(-2*x + 1) + 58/625*(-2*x + 1)^(3/2) + 1533/31250*s 
qrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 
*x + 1))) + 1658/3125*sqrt(-2*x + 1) + 11/2500*(123*(-2*x + 1)^(3/2) - 275 
*sqrt(-2*x + 1))/(5*x + 3)^2
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx=\frac {1658\,\sqrt {1-2\,x}}{3125}+\frac {58\,{\left (1-2\,x\right )}^{3/2}}{625}+\frac {18\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {\frac {121\,\sqrt {1-2\,x}}{625}-\frac {1353\,{\left (1-2\,x\right )}^{3/2}}{15625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,1533{}\mathrm {i}}{15625} \] Input:

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

Output:

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*1533i)/15625 + (1658*(1 - 
 2*x)^(1/2))/3125 + (58*(1 - 2*x)^(3/2))/625 + (18*(1 - 2*x)^(5/2))/625 - 
((121*(1 - 2*x)^(1/2))/625 - (1353*(1 - 2*x)^(3/2))/15625)/((44*x)/5 + (2* 
x - 1)^2 + 11/25)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.60 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx=\frac {90000 \sqrt {-2 x +1}\, x^{4}-127000 \sqrt {-2 x +1}\, x^{3}+259900 \sqrt {-2 x +1}\, x^{2}+492975 \sqrt {-2 x +1}\, x +162520 \sqrt {-2 x +1}+38325 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+45990 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +13797 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-38325 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-45990 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -13797 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )}{781250 x^{2}+937500 x +281250} \] Input:

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

Output:

(90000*sqrt( - 2*x + 1)*x**4 - 127000*sqrt( - 2*x + 1)*x**3 + 259900*sqrt( 
 - 2*x + 1)*x**2 + 492975*sqrt( - 2*x + 1)*x + 162520*sqrt( - 2*x + 1) + 3 
8325*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**2 + 45990*sqrt(55)*log 
(5*sqrt( - 2*x + 1) - sqrt(55))*x + 13797*sqrt(55)*log(5*sqrt( - 2*x + 1) 
- sqrt(55)) - 38325*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 459 
90*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x - 13797*sqrt(55)*log(5*sq 
rt( - 2*x + 1) + sqrt(55)))/(31250*(25*x**2 + 30*x + 9))