\(\int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\) [815]

Optimal result

Integrand size = 24, antiderivative size = 94 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {1331}{1029 (1-2 x)^{3/2}}-\frac {726}{2401 \sqrt {1-2 x}}+\frac {\sqrt {1-2 x}}{2058 (2+3 x)^2}-\frac {199 \sqrt {1-2 x}}{14406 (2+3 x)}+\frac {905 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \] Output:

1331/1029/(1-2*x)^(3/2)-726/2401/(1-2*x)^(1/2)+1/2058*(1-2*x)^(1/2)/(2+3*x 
)^2-199*(1-2*x)^(1/2)/(28812+43218*x)+905/21609*21^(1/2)*arctanh(1/7*21^(1 
/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.69 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {\frac {21 \left (8103+29593 x+33410 x^2+10860 x^3\right )}{2 (1-2 x)^{3/2} (2+3 x)^2}+905 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609} \] Input:

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

Output:

((21*(8103 + 29593*x + 33410*x^2 + 10860*x^3))/(2*(1 - 2*x)^(3/2)*(2 + 3*x 
)^2) + 905*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/21609
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {109, 161, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {1}{21} \int \frac {(5 x+3) (15 x+31)}{(1-2 x)^{3/2} (3 x+2)^3}dx\)

\(\Big \downarrow \) 161

\(\displaystyle \frac {1}{21} \left (-\frac {905}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {720 x+487}{14 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {1}{21} \left (-\frac {905}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {720 x+487}{14 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{21} \left (-\frac {905}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {720 x+487}{14 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{21} \left (-\frac {905}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {720 x+487}{14 \sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}\)

Input:

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

Output:

(11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (-1/14*(487 + 720*x)/( 
Sqrt[1 - 2*x]*(2 + 3*x)^2) - (905*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTa 
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14)/21
 

Defintions of rubi rules used

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 109

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

rule 161

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_)) 
*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 
 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) 
 + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 
) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( 
m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* 
h*(2 + 3*n + 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*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
 d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( 
n + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -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 = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.62

method	result	size
risch	\(-\frac {10860 x^{3}+33410 x^{2}+29593 x +8103}{2058 \left (2+3 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right )}+\frac {905 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{21609}\)	\(58\)
pseudoelliptic	\(\frac {\frac {905 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {21}\, \left (2+3 x \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{21609}+\frac {1810 x^{3}}{343}+\frac {16705 x^{2}}{1029}+\frac {29593 x}{2058}+\frac {2701}{686}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{2}}\)	\(64\)
derivativedivides	\(\frac {1331}{1029 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {726}{2401 \sqrt {1-2 x}}-\frac {18 \left (-\frac {199 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {1379 \sqrt {1-2 x}}{54}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {905 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{21609}\)	\(66\)
default	\(\frac {1331}{1029 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {726}{2401 \sqrt {1-2 x}}-\frac {18 \left (-\frac {199 \left (1-2 x \right )^{\frac {3}{2}}}{18}+\frac {1379 \sqrt {1-2 x}}{54}\right )}{2401 \left (-4-6 x \right )^{2}}+\frac {905 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{21609}\)	\(66\)
trager	\(\frac {\left (10860 x^{3}+33410 x^{2}+29593 x +8103\right ) \sqrt {1-2 x}}{2058 \left (6 x^{2}+x -2\right )^{2}}-\frac {905 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{43218}\)	\(80\)

Input:

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

Output:

-1/2058*(10860*x^3+33410*x^2+29593*x+8103)/(2+3*x)^2/(1-2*x)^(1/2)/(-1+2*x 
)+905/21609*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.06 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905 \, \sqrt {21} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (10860 \, x^{3} + 33410 \, x^{2} + 29593 \, x + 8103\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \] Input:

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

Output:

1/43218*(905*sqrt(21)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log((3*x - sqrt 
(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(10860*x^3 + 33410*x^2 + 29593*x 
+ 8103)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
 

Sympy [A] (verification not implemented)

Time = 162.24 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.77 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=- \frac {363 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{16807} - \frac {404 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{1029} - \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{147} - \frac {726}{2401 \sqrt {1 - 2 x}} + \frac {1331}{1029 \left (1 - 2 x\right )^{\frac {3}{2}}} \] Input:

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

Output:

-363*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt( 
21)/3))/16807 - 404*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1 
)/4 + log(sqrt(21)*sqrt(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)))/1029 - 8*Piecewise((sqrt(21)*(3*l 
og(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/ 
16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x) 
/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt 
(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) 
 < sqrt(21)/3)))/147 - 726/(2401*sqrt(1 - 2*x)) + 1331/(1029*(1 - 2*x)**(3 
/2))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {905}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2715 \, {\left (2 \, x - 1\right )}^{3} + 24850 \, {\left (2 \, x - 1\right )}^{2} + 142296 \, x - 5929}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \] Input:

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

Output:

-905/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( 
-2*x + 1))) + 1/1029*(2715*(2*x - 1)^3 + 24850*(2*x - 1)^2 + 142296*x - 59 
29)/(9*(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {905}{43218} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {121 \, {\left (36 \, x + 59\right )}}{7203 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {597 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1379 \, \sqrt {-2 \, x + 1}}{28812 \, {\left (3 \, x + 2\right )}^{2}} \] Input:

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

Output:

-905/43218*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) 
+ 3*sqrt(-2*x + 1))) - 121/7203*(36*x + 59)/((2*x - 1)*sqrt(-2*x + 1)) + 1 
/28812*(597*(-2*x + 1)^(3/2) - 1379*sqrt(-2*x + 1))/(3*x + 2)^2
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {905\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}+\frac {\frac {968\,x}{63}+\frac {3550\,{\left (2\,x-1\right )}^2}{1323}+\frac {905\,{\left (2\,x-1\right )}^3}{3087}-\frac {121}{189}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \] Input:

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

Output:

(905*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/21609 + ((968*x)/63 + ( 
3550*(2*x - 1)^2)/1323 + (905*(2*x - 1)^3)/3087 - 121/189)/((49*(1 - 2*x)^ 
(3/2))/9 - (14*(1 - 2*x)^(5/2))/3 + (1 - 2*x)^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.55 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {-16290 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-13575 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+3620 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +3620 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+16290 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}+13575 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-3620 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -3620 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )-228060 x^{3}-701610 x^{2}-621453 x -170163}{43218 \sqrt {-2 x +1}\, \left (18 x^{3}+15 x^{2}-4 x -4\right )} \] Input:

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

Output:

( - 16290*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x** 
3 - 13575*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x** 
2 + 3620*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 
3620*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 16290* 
sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**3 + 13575* 
sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 - 3620*s 
qrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 3620*sqrt( 
 - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) - 228060*x**3 - 70 
1610*x**2 - 621453*x - 170163)/(43218*sqrt( - 2*x + 1)*(18*x**3 + 15*x**2 
- 4*x - 4))