\(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx\) [834]

Optimal result

Integrand size = 24, antiderivative size = 159 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {15185}{2541 (1-2 x)^{3/2}}+\frac {172105}{65219 \sqrt {1-2 x}}-\frac {745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {4455}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {117500 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \] Output:

15185/2541/(1-2*x)^(3/2)+172105/65219/(1-2*x)^(1/2)-745/22/(1-2*x)^(3/2)/( 
3+5*x)+3/14/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)+24/7/(1-2*x)^(3/2)/(2+3*x)/(3+ 
5*x)-4455/343*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))+117500/14641*55 
^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {9784671-9008764 x-58371045 x^2+27977220 x^3+92936700 x^4}{391314 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}-\frac {4455}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {117500 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \] Input:

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

Output:

-1/391314*(9784671 - 9008764*x - 58371045*x^2 + 27977220*x^3 + 92936700*x^ 
4)/((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)) - (4455*Sqrt[3/7]*ArcTanh[Sqrt[ 
3/7]*Sqrt[1 - 2*x]])/49 + (117500*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2 
*x]])/1331
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {114, 168, 27, 168, 25, 169, 27, 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.

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{14} \int \frac {22-135 x}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^2}dx+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{14} \left (\frac {1}{7} \int \frac {245 (1-48 x)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (35 \int \frac {1-48 x}{(1-2 x)^{5/2} (3 x+2) (5 x+3)^2}dx+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{14} \left (35 \left (-\frac {1}{11} \int -\frac {2235 x+401}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{14} \left (35 \left (\frac {1}{11} \int \frac {2235 x+401}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{14} \left (35 \left (\frac {1}{11} \left (\frac {6074}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int \frac {3 (5567-45555 x)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (35 \left (\frac {1}{11} \left (\frac {6074}{231 (1-2 x)^{3/2}}-\frac {1}{77} \int \frac {5567-45555 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{14} \left (35 \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {2}{77} \int -\frac {841711-516315 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {68842}{77 \sqrt {1-2 x}}\right )+\frac {6074}{231 (1-2 x)^{3/2}}\right )-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (35 \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {68842}{77 \sqrt {1-2 x}}-\frac {1}{77} \int \frac {841711-516315 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx\right )+\frac {6074}{231 (1-2 x)^{3/2}}\right )-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{14} \left (35 \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (3557763 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-5757500 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {68842}{77 \sqrt {1-2 x}}\right )+\frac {6074}{231 (1-2 x)^{3/2}}\right )-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{14} \left (35 \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (5757500 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-3557763 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {68842}{77 \sqrt {1-2 x}}\right )+\frac {6074}{231 (1-2 x)^{3/2}}\right )-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{14} \left (35 \left (\frac {1}{11} \left (\frac {1}{77} \left (\frac {1}{77} \left (2303000 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-2371842 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {68842}{77 \sqrt {1-2 x}}\right )+\frac {6074}{231 (1-2 x)^{3/2}}\right )-\frac {149}{11 (1-2 x)^{3/2} (5 x+3)}\right )+\frac {48}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}\right )+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}\)

Input:

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

Output:

3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)) + (48/((1 - 2*x)^(3/2)*(2 + 3 
*x)*(3 + 5*x)) + 35*(-149/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) + (6074/(231*(1 - 
 2*x)^(3/2)) + (68842/(77*Sqrt[1 - 2*x]) + (-2371842*Sqrt[3/7]*ArcTanh[Sqr 
t[3/7]*Sqrt[1 - 2*x]] + 2303000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x 
]])/77)/77)/11))/14
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.55

method	result	size
risch	\(\frac {92936700 x^{4}[phone] x^{3}-58371045 x^{2}-9008764 x +9784671}{391314 \left (2+3 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right ) \left (3+5 x \right )}-\frac {4455 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}+\frac {117500 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{14641}\)	\(88\)
derivativedivides	\(\frac {1250 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {117500 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{14641}+\frac {\frac {36693 \left (1-2 x \right )^{\frac {3}{2}}}{2401}-\frac {12393 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {4455 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}+\frac {32}{124509 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5408}{3195731 \sqrt {1-2 x}}\)	\(100\)
default	\(\frac {1250 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {117500 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{14641}+\frac {\frac {36693 \left (1-2 x \right )^{\frac {3}{2}}}{2401}-\frac {12393 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {4455 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}+\frac {32}{124509 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5408}{3195731 \sqrt {1-2 x}}\)	\(100\)
pseudoelliptic	\(-\frac {1057500 \left (\frac {35877127}{103635000}-\frac {1449459 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{2} \sqrt {21}}{8060500}+\frac {\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{2} \sqrt {55}}{9}+\frac {378631 x^{4}}{115150}+\frac {36377 x^{3}}{36750}-\frac {42805433 x^{2}}{20727000}-\frac {24774101 x}{77726250}\right )}{14641 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{2} \left (3+5 x \right )}\)	\(124\)
trager	\(-\frac {\left (92936700 x^{4}[phone] x^{3}-58371045 x^{2}-9008764 x +9784671\right ) \sqrt {1-2 x}}{391314 \left (6 x^{2}+x -2\right )^{2} \left (3+5 x \right )}-\frac {58750 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{14641}-\frac {4455 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )-21 \sqrt {1-2 x}}{2+3 x}\right )}{686}\)	\(137\)

Input:

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

Output:

1/391314*(92936700*x^4+27977220*x^3-58371045*x^2-9008764*x+9784671)/(2+3*x 
)^2/(1-2*x)^(1/2)/(-1+2*x)/(3+5*x)-4455/343*21^(1/2)*arctanh(1/7*21^(1/2)* 
(1-2*x)^(1/2))+117500/14641*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 = 171, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {17272500 \, \sqrt {\frac {5}{11}} {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (\frac {5 \, x - 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 17788815 \, \sqrt {\frac {3}{7}} {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (\frac {3 \, x + 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - {\left (92936700 \, x^{4} + 27977220 \, x^{3} - 58371045 \, x^{2} - 9008764 \, x + 9784671\right )} \sqrt {-2 \, x + 1}}{391314 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \] Input:

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

Output:

1/391314*(17272500*sqrt(5/11)*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 
 12)*log((5*x - 11*sqrt(5/11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 17788815*sq 
rt(3/7)*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log((3*x + 7*sqrt 
(3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - (92936700*x^4 + 27977220*x^3 - 5837 
1045*x^2 - 9008764*x + 9784671)*sqrt(-2*x + 1))/(180*x^5 + 168*x^4 - 79*x^ 
3 - 89*x^2 + 8*x + 12)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.84 (sec) , antiderivative size = 2966, normalized size of antiderivative = 18.65 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-8587351080000*sqrt(2)*I*(x - 1/2)**(17/2)/(6508334448000*(x - 1/2)**9 + 4 
4256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 18938578305292 
8*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)** 
4 + 13755877085874*(x - 1/2)**3) - 48670545924000*sqrt(2)*I*(x - 1/2)**(15 
/2)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 1253722159 
16640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1 
/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) - 1103 
21398202400*sqrt(2)*I*(x - 1/2)**(13/2)/(6508334448000*(x - 1/2)**9 + 4425 
6674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*( 
x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 
 13755877085874*(x - 1/2)**3) - 125018036238480*sqrt(2)*I*(x - 1/2)**(11/2 
)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916 
640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2 
)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) - 708383 
64022580*sqrt(2)*I*(x - 1/2)**(9/2)/(6508334448000*(x - 1/2)**9 + 44256674 
246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 
1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 137 
55877085874*(x - 1/2)**3) - 16066680171234*sqrt(2)*I*(x - 1/2)**(7/2)/(650 
8334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x 
 - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**...
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {58750}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4455}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {23234175 \, {\left (2 \, x - 1\right )}^{4} + 106925310 \, {\left (2 \, x - 1\right )}^{3} + 122999835 \, {\left (2 \, x - 1\right )}^{2} + 285824 \, x - 170016}{195657 \, {\left (45 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 309 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 707 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 539 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \] Input:

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

Output:

-58750/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr 
t(-2*x + 1))) + 4455/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt 
(21) + 3*sqrt(-2*x + 1))) + 1/195657*(23234175*(2*x - 1)^4 + 106925310*(2* 
x - 1)^3 + 122999835*(2*x - 1)^2 + 285824*x - 170016)/(45*(-2*x + 1)^(9/2) 
 - 309*(-2*x + 1)^(7/2) + 707*(-2*x + 1)^(5/2) - 539*(-2*x + 1)^(3/2))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {58750}{14641} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4455}{686} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {64 \, {\left (507 \, x - 292\right )}}{9587193 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {3125 \, \sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} + \frac {243 \, {\left (151 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 357 \, \sqrt {-2 \, x + 1}\right )}}{9604 \, {\left (3 \, x + 2\right )}^{2}} \] Input:

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

Output:

-58750/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(5 
5) + 5*sqrt(-2*x + 1))) + 4455/686*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sq 
rt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/9587193*(507*x - 292)/(( 
2*x - 1)*sqrt(-2*x + 1)) - 3125/1331*sqrt(-2*x + 1)/(5*x + 3) + 243/9604*( 
151*(-2*x + 1)^(3/2) - 357*sqrt(-2*x + 1))/(3*x + 2)^2
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {117500\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {4455\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {3712\,x}{114345}+\frac {1171427\,{\left (2\,x-1\right )}^2}{83853}+\frac {2376118\,{\left (2\,x-1\right )}^3}{195657}+\frac {172105\,{\left (2\,x-1\right )}^4}{65219}-\frac {736}{38115}}{\frac {539\,{\left (1-2\,x\right )}^{3/2}}{45}-\frac {707\,{\left (1-2\,x\right )}^{5/2}}{45}+\frac {103\,{\left (1-2\,x\right )}^{7/2}}{15}-{\left (1-2\,x\right )}^{9/2}} \] Input:

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

Output:

(117500*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641 - (4455*21^(1 
/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - ((3712*x)/114345 + (1171427 
*(2*x - 1)^2)/83853 + (2376118*(2*x - 1)^3)/195657 + (172105*(2*x - 1)^4)/ 
65219 - 736/38115)/((539*(1 - 2*x)^(3/2))/45 - (707*(1 - 2*x)^(5/2))/45 + 
(103*(1 - 2*x)^(7/2))/15 - (1 - 2*x)^(9/2))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.47 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 10881675000*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55 
))*x**4 - 15597067500*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - s 
qrt(55))*x**3 - 3022687500*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1 
) - sqrt(55))*x**2 + 3869040000*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2* 
x + 1) - sqrt(55))*x + 1450890000*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 
2*x + 1) - sqrt(55)) + 10881675000*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 
 2*x + 1) + sqrt(55))*x**4 + 15597067500*sqrt( - 2*x + 1)*sqrt(55)*log(5*s 
qrt( - 2*x + 1) + sqrt(55))*x**3 + 3022687500*sqrt( - 2*x + 1)*sqrt(55)*lo 
g(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 3869040000*sqrt( - 2*x + 1)*sqrt(5 
5)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x - 1450890000*sqrt( - 2*x + 1)*sqrt 
(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) + 17610926850*sqrt( - 2*x + 1)*sqr 
t(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**4 + 25242328485*sqrt( - 2*x + 
1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**3 + 4891924125*sqrt( - 2 
*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 - 6261662880*sqrt 
( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x - 2348123580*sq 
rt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 17610926850*s 
qrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**4 - 2524232 
8485*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x**3 - 4 
891924125*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x** 
2 + 6261662880*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(...