\(\int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx\) [587]

Optimal result

Integrand size = 24, antiderivative size = 102 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {214}{243} \sqrt {1-2 x}+\frac {10}{81} (1-2 x)^{3/2}-\frac {145}{54} (1-2 x)^{5/2}+\frac {125}{126} (1-2 x)^{7/2}+\frac {7 \sqrt {1-2 x}}{243 (2+3 x)}-\frac {8}{9} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \] Output:

214/243*(1-2*x)^(1/2)+10/81*(1-2*x)^(3/2)-145/54*(1-2*x)^(5/2)+125/126*(1- 
2*x)^(7/2)+7*(1-2*x)^(1/2)/(486+729*x)-8/27*21^(1/2)*arctanh(1/7*21^(1/2)* 
(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (85-442 x-1005 x^2+780 x^3+1500 x^4\right )}{63 (2+3 x)}-\frac {8}{9} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \] Input:

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

Output:

-1/63*(Sqrt[1 - 2*x]*(85 - 442*x - 1005*x^2 + 780*x^3 + 1500*x^4))/(2 + 3* 
x) - (8*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {108, 27, 170, 27, 164, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 170

\(\displaystyle -\frac {1}{21} \int -\frac {6 \sqrt {1-2 x} (5 x+3) (45 x+16)}{3 x+2}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {5}{7} (1-2 x)^{3/2} (5 x+3)^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{7} \int \frac {\sqrt {1-2 x} (5 x+3) (45 x+16)}{3 x+2}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {5}{7} (1-2 x)^{3/2} (5 x+3)^2\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {2}{7} \left (\frac {14}{3} \int \frac {\sqrt {1-2 x}}{3 x+2}dx-\frac {5}{9} (1-2 x)^{3/2} (27 x+22)\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {5}{7} (1-2 x)^{3/2} (5 x+3)^2\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2}{7} \left (\frac {14}{3} \left (\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{3} \sqrt {1-2 x}\right )-\frac {5}{9} (1-2 x)^{3/2} (27 x+22)\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {5}{7} (1-2 x)^{3/2} (5 x+3)^2\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2}{7} \left (\frac {14}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {7}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {5}{9} (1-2 x)^{3/2} (27 x+22)\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {5}{7} (1-2 x)^{3/2} (5 x+3)^2\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2}{7} \left (\frac {14}{3} \left (\frac {2}{3} \sqrt {1-2 x}-\frac {2}{3} \sqrt {\frac {7}{3}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {5}{9} (1-2 x)^{3/2} (27 x+22)\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac {5}{7} (1-2 x)^{3/2} (5 x+3)^2\)

Input:

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

Output:

(5*(1 - 2*x)^(3/2)*(3 + 5*x)^2)/7 - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(3*(2 + 
3*x)) + (2*((-5*(1 - 2*x)^(3/2)*(22 + 27*x))/9 + (14*((2*Sqrt[1 - 2*x])/3 
- (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3))/3))/7
 

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 = 61, normalized size of antiderivative = 0.60

method	result	size
risch	\(\frac {3000 x^{5}+60 x^{4}-2790 x^{3}+121 x^{2}+612 x -85}{63 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {8 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{27}\)	\(61\)
pseudoelliptic	\(\frac {-56 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right ) \sqrt {21}-3 \sqrt {1-2 x}\, \left (1500 x^{4}+780 x^{3}-1005 x^{2}-442 x +85\right )}{378+567 x}\)	\(62\)
derivativedivides	\(\frac {125 \left (1-2 x \right )^{\frac {7}{2}}}{126}-\frac {145 \left (1-2 x \right )^{\frac {5}{2}}}{54}+\frac {10 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {214 \sqrt {1-2 x}}{243}-\frac {14 \sqrt {1-2 x}}{729 \left (-\frac {4}{3}-2 x \right )}-\frac {8 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{27}\)	\(72\)
default	\(\frac {125 \left (1-2 x \right )^{\frac {7}{2}}}{126}-\frac {145 \left (1-2 x \right )^{\frac {5}{2}}}{54}+\frac {10 \left (1-2 x \right )^{\frac {3}{2}}}{81}+\frac {214 \sqrt {1-2 x}}{243}-\frac {14 \sqrt {1-2 x}}{729 \left (-\frac {4}{3}-2 x \right )}-\frac {8 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{27}\)	\(72\)
trager	\(-\frac {\left (1500 x^{4}+780 x^{3}-1005 x^{2}-442 x +85\right ) \sqrt {1-2 x}}{63 \left (2+3 x \right )}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{27}\)	\(82\)

Input:

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

Output:

1/63*(3000*x^5+60*x^4-2790*x^3+121*x^2+612*x-85)/(2+3*x)/(1-2*x)^(1/2)-8/2 
7*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {28 \, \sqrt {\frac {7}{3}} {\left (3 \, x + 2\right )} \log \left (\frac {3 \, x + 3 \, \sqrt {\frac {7}{3}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - {\left (1500 \, x^{4} + 780 \, x^{3} - 1005 \, x^{2} - 442 \, x + 85\right )} \sqrt {-2 \, x + 1}}{63 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

1/63*(28*sqrt(7/3)*(3*x + 2)*log((3*x + 3*sqrt(7/3)*sqrt(-2*x + 1) - 5)/(3 
*x + 2)) - (1500*x^4 + 780*x^3 - 1005*x^2 - 442*x + 85)*sqrt(-2*x + 1))/(3 
*x + 2)
 

Sympy [A] (verification not implemented)

Time = 43.36 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.05 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {125 \left (1 - 2 x\right )^{\frac {7}{2}}}{126} - \frac {145 \left (1 - 2 x\right )^{\frac {5}{2}}}{54} + \frac {10 \left (1 - 2 x\right )^{\frac {3}{2}}}{81} + \frac {214 \sqrt {1 - 2 x}}{243} + \frac {109 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{729} + \frac {196 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{243} \] Input:

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

Output:

125*(1 - 2*x)**(7/2)/126 - 145*(1 - 2*x)**(5/2)/54 + 10*(1 - 2*x)**(3/2)/8 
1 + 214*sqrt(1 - 2*x)/243 + 109*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) 
- log(sqrt(1 - 2*x) + sqrt(21)/3))/729 + 196*Piecewise((sqrt(21)*(-log(sqr 
t(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)))/14 
7, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/243
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {125}{126} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {145}{54} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {10}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {4}{27} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {214}{243} \, \sqrt {-2 \, x + 1} + \frac {7 \, \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

125/126*(-2*x + 1)^(7/2) - 145/54*(-2*x + 1)^(5/2) + 10/81*(-2*x + 1)^(3/2 
) + 4/27*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 
*x + 1))) + 214/243*sqrt(-2*x + 1) + 7/243*sqrt(-2*x + 1)/(3*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx=-\frac {125}{126} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {145}{54} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {10}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {4}{27} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {214}{243} \, \sqrt {-2 \, x + 1} + \frac {7 \, \sqrt {-2 \, x + 1}}{243 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

-125/126*(2*x - 1)^3*sqrt(-2*x + 1) - 145/54*(2*x - 1)^2*sqrt(-2*x + 1) + 
10/81*(-2*x + 1)^(3/2) + 4/27*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2 
*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 214/243*sqrt(-2*x + 1) + 7/243*s 
qrt(-2*x + 1)/(3*x + 2)
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {14\,\sqrt {1-2\,x}}{729\,\left (2\,x+\frac {4}{3}\right )}+\frac {214\,\sqrt {1-2\,x}}{243}+\frac {10\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {145\,{\left (1-2\,x\right )}^{5/2}}{54}+\frac {125\,{\left (1-2\,x\right )}^{7/2}}{126}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,8{}\mathrm {i}}{27} \] Input:

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

Output:

(21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*8i)/27 + (14*(1 - 2*x)^(1/ 
2))/(729*(2*x + 4/3)) + (214*(1 - 2*x)^(1/2))/243 + (10*(1 - 2*x)^(3/2))/8 
1 - (145*(1 - 2*x)^(5/2))/54 + (125*(1 - 2*x)^(7/2))/126
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.26 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx=\frac {-4500 \sqrt {-2 x +1}\, x^{4}-2340 \sqrt {-2 x +1}\, x^{3}+3015 \sqrt {-2 x +1}\, x^{2}+1326 \sqrt {-2 x +1}\, x -255 \sqrt {-2 x +1}+84 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +56 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-84 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -56 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{567 x +378} \] Input:

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

Output:

( - 4500*sqrt( - 2*x + 1)*x**4 - 2340*sqrt( - 2*x + 1)*x**3 + 3015*sqrt( - 
 2*x + 1)*x**2 + 1326*sqrt( - 2*x + 1)*x - 255*sqrt( - 2*x + 1) + 84*sqrt( 
21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 56*sqrt(21)*log(3*sqrt( - 2*x + 
 1) - sqrt(21)) - 84*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x - 56*sq 
rt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(189*(3*x + 2))