Integrand size = 24, antiderivative size = 119 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {1314}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:
4644/5929/(1-2*x)^(1/2)-340/77/(1-2*x)^(1/2)/(3+5*x)+3/7/(1-2*x)^(1/2)/(2+ 3*x)/(3+5*x)-1314/343*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))+3150/13 31*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {-21955+9696 x+69660 x^2}{5929 \sqrt {1-2 x} \left (6+19 x+15 x^2\right )}-\frac {1314}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:
Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^2),x]
Output:
(-21955 + 9696*x + 69660*x^2)/(5929*Sqrt[1 - 2*x]*(6 + 19*x + 15*x^2)) - ( 1314*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (3150*Sqrt[5/11]*Arc Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {114, 168, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{7} \int \frac {23-75 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}dx+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (-\frac {1}{11} \int \frac {9 (41-340 x)}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {340}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {9}{11} \int \frac {41-340 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx-\frac {340}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{7} \left (-\frac {9}{11} \left (-\frac {2}{77} \int -\frac {6253-3870 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {516}{77 \sqrt {1-2 x}}\right )-\frac {340}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {9}{11} \left (\frac {1}{77} \int \frac {6253-3870 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {516}{77 \sqrt {1-2 x}}\right )-\frac {340}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{7} \left (-\frac {9}{11} \left (\frac {1}{77} \left (42875 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-26499 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {516}{77 \sqrt {1-2 x}}\right )-\frac {340}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{7} \left (-\frac {9}{11} \left (\frac {1}{77} \left (26499 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-42875 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {516}{77 \sqrt {1-2 x}}\right )-\frac {340}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{7} \left (-\frac {9}{11} \left (\frac {1}{77} \left (17666 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-17150 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {516}{77 \sqrt {1-2 x}}\right )-\frac {340}{11 \sqrt {1-2 x} (5 x+3)}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}\) |
Input:
Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^2),x]
Output:
3/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) + (-340/(11*Sqrt[1 - 2*x]*(3 + 5*x )) - (9*(-516/(77*Sqrt[1 - 2*x]) + (17666*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt [1 - 2*x]] - 17150*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77))/11)/ 7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]
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]
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]
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])
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {69660 x^{2}+9696 x -21955}{5929 \left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}-\frac {1314 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}+\frac {3150 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}\) | \(69\) |
| derivativedivides | \(\frac {18 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}-\frac {1314 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}+\frac {50 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {3150 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}+\frac {16}{5929 \sqrt {1-2 x}}\) | \(79\) |
| default | \(\frac {18 \sqrt {1-2 x}}{49 \left (-\frac {4}{3}-2 x \right )}-\frac {1314 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{343}+\frac {50 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {3150 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{1331}+\frac {16}{5929 \sqrt {1-2 x}}\) | \(79\) |
| pseudoelliptic | \(\frac {-1748934 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (15 x^{2}+19 x +6\right ) \sqrt {21}+1080450 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (15 x^{2}+19 x +6\right ) \sqrt {55}+5363820 x^{2}+746592 x -1690535}{\sqrt {1-2 x}\, \left (6847995 x^{2}+8674127 x +2739198\right )}\) | \(102\) |
| trager | \(-\frac {\left (69660 x^{2}+9696 x -21955\right ) \sqrt {1-2 x}}{5929 \left (30 x^{3}+23 x^{2}-7 x -6\right )}+\frac {1575 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}+\frac {657 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{343}\) | \(126\) |
Input:
int(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^2,x,method=_RETURNVERBOSE)
Output:
1/5929*(69660*x^2+9696*x-21955)/(15*x^2+19*x+6)/(1-2*x)^(1/2)-1314/343*21^ (1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))+3150/1331*55^(1/2)*arctanh(1/11* 55^(1/2)*(1-2*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {77175 \, \sqrt {\frac {5}{11}} {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (\frac {5 \, x - 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 79497 \, \sqrt {\frac {3}{7}} {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (\frac {3 \, x + 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - {\left (69660 \, x^{2} + 9696 \, x - 21955\right )} \sqrt {-2 \, x + 1}}{5929 \, {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^2,x, algorithm="fricas")
Output:
1/5929*(77175*sqrt(5/11)*(30*x^3 + 23*x^2 - 7*x - 6)*log((5*x - 11*sqrt(5/ 11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 79497*sqrt(3/7)*(30*x^3 + 23*x^2 - 7* x - 6)*log((3*x + 7*sqrt(3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - (69660*x^2 + 9696*x - 21955)*sqrt(-2*x + 1))/(30*x^3 + 23*x^2 - 7*x - 6)
Result contains complex when optimal does not.
Time = 7.90 (sec) , antiderivative size = 818, normalized size of antiderivative = 6.87 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-2*x)**(3/2)/(2+3*x)**2/(3+5*x)**2,x)
Output:
64827000*sqrt(55)*I*(x - 1/2)**(7/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(273 91980*(x - 1/2)**(7/2) + 62088488*(x - 1/2)**(5/2) + 35153041*(x - 1/2)**( 3/2)) - 104936040*sqrt(21)*I*(x - 1/2)**(7/2)*atan(sqrt(42)*sqrt(x - 1/2)/ 7)/(27391980*(x - 1/2)**(7/2) + 62088488*(x - 1/2)**(5/2) + 35153041*(x - 1/2)**(3/2)) - 32413500*sqrt(55)*I*pi*(x - 1/2)**(7/2)/(27391980*(x - 1/2) **(7/2) + 62088488*(x - 1/2)**(5/2) + 35153041*(x - 1/2)**(3/2)) + 5246802 0*sqrt(21)*I*pi*(x - 1/2)**(7/2)/(27391980*(x - 1/2)**(7/2) + 62088488*(x - 1/2)**(5/2) + 35153041*(x - 1/2)**(3/2)) + 146941200*sqrt(55)*I*(x - 1/2 )**(5/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(27391980*(x - 1/2)**(7/2) + 620 88488*(x - 1/2)**(5/2) + 35153041*(x - 1/2)**(3/2)) - 237855024*sqrt(21)*I *(x - 1/2)**(5/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(27391980*(x - 1/2)**(7/2 ) + 62088488*(x - 1/2)**(5/2) + 35153041*(x - 1/2)**(3/2)) - 73470600*sqrt (55)*I*pi*(x - 1/2)**(5/2)/(27391980*(x - 1/2)**(7/2) + 62088488*(x - 1/2) **(5/2) + 35153041*(x - 1/2)**(3/2)) + 118927512*sqrt(21)*I*pi*(x - 1/2)** (5/2)/(27391980*(x - 1/2)**(7/2) + 62088488*(x - 1/2)**(5/2) + 35153041*(x - 1/2)**(3/2)) + 83194650*sqrt(55)*I*(x - 1/2)**(3/2)*atan(sqrt(110)*sqrt (x - 1/2)/11)/(27391980*(x - 1/2)**(7/2) + 62088488*(x - 1/2)**(5/2) + 351 53041*(x - 1/2)**(3/2)) - 134667918*sqrt(21)*I*(x - 1/2)**(3/2)*atan(sqrt( 42)*sqrt(x - 1/2)/7)/(27391980*(x - 1/2)**(7/2) + 62088488*(x - 1/2)**(5/2 ) + 35153041*(x - 1/2)**(3/2)) - 41597325*sqrt(55)*I*pi*(x - 1/2)**(3/2...
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {1575}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {657}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (17415 \, {\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 68 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 77 \, \sqrt {-2 \, x + 1}\right )}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^2,x, algorithm="maxima")
Output:
-1575/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) + 657/343*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) + 4/5929*(17415*(2*x - 1)^2 + 79356*x - 39370)/(15* (-2*x + 1)^(5/2) - 68*(-2*x + 1)^(3/2) + 77*sqrt(-2*x + 1))
Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx=-\frac {1575}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {657}{343} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (17415 \, {\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 68 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 77 \, \sqrt {-2 \, x + 1}\right )}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^2,x, algorithm="giac")
Output:
-1575/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 657/343*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt( -2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/5929*(17415*(2*x - 1)^2 + 79 356*x - 39370)/(15*(2*x - 1)^2*sqrt(-2*x + 1) - 68*(-2*x + 1)^(3/2) + 77*s qrt(-2*x + 1))
Time = 1.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {3150\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}-\frac {1314\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}+\frac {\frac {105808\,x}{29645}+\frac {4644\,{\left (2\,x-1\right )}^2}{5929}-\frac {31496}{17787}}{\frac {77\,\sqrt {1-2\,x}}{15}-\frac {68\,{\left (1-2\,x\right )}^{3/2}}{15}+{\left (1-2\,x\right )}^{5/2}} \] Input:
int(1/((1 - 2*x)^(3/2)*(3*x + 2)^2*(5*x + 3)^2),x)
Output:
(3150*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331 - (1314*21^(1/2) *atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 + ((105808*x)/29645 + (4644*(2*x - 1)^2)/5929 - 31496/17787)/((77*(1 - 2*x)^(1/2))/15 - (68*(1 - 2*x)^(3/2 ))/15 + (1 - 2*x)^(5/2))
Time = 0.16 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.72 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx=\frac {-8103375 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}-10264275 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x -3241350 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )+8103375 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}+10264275 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x +3241350 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )+13117005 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+16614873 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +5246802 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-13117005 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-16614873 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -5246802 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )+5363820 x^{2}+746592 x -1690535}{456533 \sqrt {-2 x +1}\, \left (15 x^{2}+19 x +6\right )} \] Input:
int(1/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^2,x)
Output:
( - 8103375*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x **2 - 10264275*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55) )*x - 3241350*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) + 8103375*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x* *2 + 10264275*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) *x + 3241350*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) + 13117005*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x* *2 + 16614873*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) *x + 5246802*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 13117005*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x* *2 - 16614873*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) *x - 5246802*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) + 5363820*x**2 + 746592*x - 1690535)/(456533*sqrt( - 2*x + 1)*(15*x**2 + 1 9*x + 6))