\(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx\) [770]

Optimal result

Integrand size = 24, antiderivative size = 112 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx=-\frac {2525}{3773 \sqrt {1-2 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {225}{98 \sqrt {1-2 x} (2+3 x)}+\frac {8025}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:

-2525/3773/(1-2*x)^(1/2)+3/14/(1-2*x)^(1/2)/(2+3*x)^2+225/98/(1-2*x)^(1/2) 
/(2+3*x)+8025/2401*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-250/121*55 
^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {16067-8625 x-45450 x^2}{7546 \sqrt {1-2 x} (2+3 x)^2}+\frac {8025}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:

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

Output:

(16067 - 8625*x - 45450*x^2)/(7546*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (8025*Sqrt 
[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt 
[5/11]*Sqrt[1 - 2*x]])/11
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {114, 27, 168, 169, 27, 174, 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.

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{14} \int \frac {25 (1-3 x)}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}dx+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{14} \int \frac {1-3 x}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}dx+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {25}{14} \left (\frac {1}{7} \int \frac {17-135 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {9}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {25}{14} \left (\frac {1}{7} \left (-\frac {2}{77} \int -\frac {2521-1515 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {202}{77 \sqrt {1-2 x}}\right )+\frac {9}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{14} \left (\frac {1}{7} \left (\frac {1}{77} \int \frac {2521-1515 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {202}{77 \sqrt {1-2 x}}\right )+\frac {9}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {25}{14} \left (\frac {1}{7} \left (\frac {1}{77} \left (17150 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-10593 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {202}{77 \sqrt {1-2 x}}\right )+\frac {9}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {25}{14} \left (\frac {1}{7} \left (\frac {1}{77} \left (10593 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-17150 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {202}{77 \sqrt {1-2 x}}\right )+\frac {9}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {25}{14} \left (\frac {1}{7} \left (\frac {1}{77} \left (7062 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-6860 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {202}{77 \sqrt {1-2 x}}\right )+\frac {9}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2}\)

Input:

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

Output:

3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (25*(9/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (- 
202/(77*Sqrt[1 - 2*x]) + (7062*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] 
- 6860*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77)/7))/14
 

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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 174

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

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 = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.57

method	result	size
risch	\(-\frac {45450 x^{2}+8625 x -16067}{7546 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {8025 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2401}-\frac {250 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}\)	\(64\)
derivativedivides	\(\frac {16}{3773 \sqrt {1-2 x}}-\frac {250 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}-\frac {486 \left (\frac {77 \left (1-2 x \right )^{\frac {3}{2}}}{18}-\frac {553 \sqrt {1-2 x}}{54}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {8025 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2401}\)	\(75\)
default	\(\frac {16}{3773 \sqrt {1-2 x}}-\frac {250 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{121}-\frac {486 \left (\frac {77 \left (1-2 x \right )^{\frac {3}{2}}}{18}-\frac {553 \sqrt {1-2 x}}{54}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {8025 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2401}\)	\(75\)
pseudoelliptic	\(\frac {\frac {16067}{7546}+\frac {8025 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}}{2401}-\frac {250 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \sqrt {55}}{121}-\frac {22725 x^{2}}{3773}-\frac {8625 x}{7546}}{\sqrt {1-2 x}\, \left (2+3 x \right )^{2}}\)	\(91\)
trager	\(\frac {\left (45450 x^{2}+8625 x -16067\right ) \sqrt {1-2 x}}{7546 \left (2+3 x \right )^{2} \left (-1+2 x \right )}+\frac {125 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}+\frac {75 \operatorname {RootOf}\left (\textit {\_Z}^{2}-240429\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-240429\right ) x +2247 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-240429\right )}{2+3 x}\right )}{4802}\)	\(123\)

Input:

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

Output:

-1/7546*(45450*x^2+8625*x-16067)/(2+3*x)^2/(1-2*x)^(1/2)+8025/2401*21^(1/2 
)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-250/121*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 = 130, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {85750 \, \sqrt {\frac {5}{11}} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac {5 \, x + 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 88275 \, \sqrt {\frac {3}{7}} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac {3 \, x - 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (45450 \, x^{2} + 8625 \, x - 16067\right )} \sqrt {-2 \, x + 1}}{7546 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \] Input:

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

Output:

1/7546*(85750*sqrt(5/11)*(18*x^3 + 15*x^2 - 4*x - 4)*log((5*x + 11*sqrt(5/ 
11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 88275*sqrt(3/7)*(18*x^3 + 15*x^2 - 4* 
x - 4)*log((3*x - 7*sqrt(3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + (45450*x^2 
+ 8625*x - 16067)*sqrt(-2*x + 1))/(18*x^3 + 15*x^2 - 4*x - 4)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.06 (sec) , antiderivative size = 1875, normalized size of antiderivative = 16.74 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1555848000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/ 
(753030432*(x - 1/2)**(11/2) + 3514142016*(x - 1/2)**(9/2) + 6149748528*(x 
 - 1/2)**(7/2) + 4783137744*(x - 1/2)**(5/2) + 1395081842*(x - 1/2)**(3/2) 
) + 2516896800*sqrt(21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7) 
/(753030432*(x - 1/2)**(11/2) + 3514142016*(x - 1/2)**(9/2) + 6149748528*( 
x - 1/2)**(7/2) + 4783137744*(x - 1/2)**(5/2) + 1395081842*(x - 1/2)**(3/2 
)) - 1258448400*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(753030432*(x - 1/2)**(11/ 
2) + 3514142016*(x - 1/2)**(9/2) + 6149748528*(x - 1/2)**(7/2) + 478313774 
4*(x - 1/2)**(5/2) + 1395081842*(x - 1/2)**(3/2)) + 777924000*sqrt(55)*I*p 
i*(x - 1/2)**(11/2)/(753030432*(x - 1/2)**(11/2) + 3514142016*(x - 1/2)**( 
9/2) + 6149748528*(x - 1/2)**(7/2) + 4783137744*(x - 1/2)**(5/2) + 1395081 
842*(x - 1/2)**(3/2)) - 7260624000*sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(1 
10)*sqrt(x - 1/2)/11)/(753030432*(x - 1/2)**(11/2) + 3514142016*(x - 1/2)* 
*(9/2) + 6149748528*(x - 1/2)**(7/2) + 4783137744*(x - 1/2)**(5/2) + 13950 
81842*(x - 1/2)**(3/2)) + 11745518400*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqr 
t(42)*sqrt(x - 1/2)/7)/(753030432*(x - 1/2)**(11/2) + 3514142016*(x - 1/2) 
**(9/2) + 6149748528*(x - 1/2)**(7/2) + 4783137744*(x - 1/2)**(5/2) + 1395 
081842*(x - 1/2)**(3/2)) - 5872759200*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(7530 
30432*(x - 1/2)**(11/2) + 3514142016*(x - 1/2)**(9/2) + 6149748528*(x - 1/ 
2)**(7/2) + 4783137744*(x - 1/2)**(5/2) + 1395081842*(x - 1/2)**(3/2)) ...
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {125}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8025}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {22725 \, {\left (2 \, x - 1\right )}^{2} + 108150 \, x - 54859}{3773 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 49 \, \sqrt {-2 \, x + 1}\right )}} \] Input:

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

Output:

125/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2* 
x + 1))) - 8025/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) 
 + 3*sqrt(-2*x + 1))) - 1/3773*(22725*(2*x - 1)^2 + 108150*x - 54859)/(9*( 
-2*x + 1)^(5/2) - 42*(-2*x + 1)^(3/2) + 49*sqrt(-2*x + 1))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {125}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {8025}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16}{3773 \, \sqrt {-2 \, x + 1}} - \frac {9 \, {\left (33 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 79 \, \sqrt {-2 \, x + 1}\right )}}{196 \, {\left (3 \, x + 2\right )}^{2}} \] Input:

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

Output:

125/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 
5*sqrt(-2*x + 1))) - 8025/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(- 
2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/3773/sqrt(-2*x + 1) - 9/196* 
(33*(-2*x + 1)^(3/2) - 79*sqrt(-2*x + 1))/(3*x + 2)^2
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {8025\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {\frac {5150\,x}{1617}+\frac {2525\,{\left (2\,x-1\right )}^2}{3773}-\frac {7837}{4851}}{\frac {49\,\sqrt {1-2\,x}}{9}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{3}+{\left (1-2\,x\right )}^{5/2}} \] Input:

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

Output:

(8025*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (250*55^(1/2)*a 
tanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 - ((5150*x)/1617 + (2525*(2*x - 1 
)^2)/3773 - 7837/4851)/((49*(1 - 2*x)^(1/2))/9 - (14*(1 - 2*x)^(3/2))/3 + 
(1 - 2*x)^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.89 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx=\frac {5402250 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+7203000 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +2401000 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-5402250 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-7203000 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -2401000 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-8739225 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-11652300 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -3884100 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+8739225 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}[phone] \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +3884100 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )-3499650 x^{2}-664125 x +1237159}{581042 \sqrt {-2 x +1}\, \left (9 x^{2}+12 x +4\right )} \] Input:

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

Output:

(5402250*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**2 
 + 7203000*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x 
+ 2401000*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 5 
402250*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 
 7203000*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x - 
2401000*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) - 873 
9225*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 - 1 
1652300*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x - 3 
884100*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 8739 
225*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**2 + 11 
652300*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 38 
84100*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) - 34996 
50*x**2 - 664125*x + 1237159)/(581042*sqrt( - 2*x + 1)*(9*x**2 + 12*x + 4) 
)