Integrand size = 24, antiderivative size = 112 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx=-\frac {2049}{9317 \sqrt {1-2 x}}-\frac {5}{22 \sqrt {1-2 x} (3+5 x)^2}+\frac {305}{242 \sqrt {1-2 x} (3+5 x)}+\frac {54}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {9975 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \] Output:
-2049/9317/(1-2*x)^(1/2)-5/22/(1-2*x)^(1/2)/(3+5*x)^2+305/242/(1-2*x)^(1/2 )/(3+5*x)+54/49*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-9975/14641*55 ^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx=\frac {54}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {-\frac {11 \left (-29338+5515 x+102450 x^2\right )}{\sqrt {1-2 x} (3+5 x)^2}-139650 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{204974} \] Input:
Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^3),x]
Output:
(54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + ((-11*(-29338 + 5515*x + 102450*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 139650*Sqrt[55]*ArcTanh[Sqrt [5/11]*Sqrt[1 - 2*x]])/204974
Time = 0.25 (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, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {1}{22} \int \frac {16-75 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^2}dx-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \int \frac {3 (116-915 x)}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {305}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{22} \left (\frac {3}{11} \int \frac {116-915 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx+\frac {305}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{22} \left (\frac {3}{11} \left (-\frac {2}{77} \int -\frac {17128-10245 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {1366}{77 \sqrt {1-2 x}}\right )+\frac {305}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{22} \left (\frac {3}{11} \left (\frac {1}{77} \int \frac {17128-10245 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {1366}{77 \sqrt {1-2 x}}\right )+\frac {305}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{22} \left (\frac {3}{11} \left (\frac {1}{77} \left (116375 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-71874 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )-\frac {1366}{77 \sqrt {1-2 x}}\right )+\frac {305}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{22} \left (\frac {3}{11} \left (\frac {1}{77} \left (71874 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-116375 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1366}{77 \sqrt {1-2 x}}\right )+\frac {305}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{22} \left (\frac {3}{11} \left (\frac {1}{77} \left (47916 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-46550 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {1366}{77 \sqrt {1-2 x}}\right )+\frac {305}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}\) |
Input:
Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^3),x]
Output:
-5/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) + (305/(11*Sqrt[1 - 2*x]*(3 + 5*x)) + (3 *(-1366/(77*Sqrt[1 - 2*x]) + (47916*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2 *x]] - 46550*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77))/11)/22
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 = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {102450 x^{2}+5515 x -29338}{18634 \left (3+5 x \right )^{2} \sqrt {1-2 x}}+\frac {54 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}-\frac {9975 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{14641}\) | \(64\) |
derivativedivides | \(\frac {-\frac {7375 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {1425 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {9975 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{14641}+\frac {54 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}+\frac {16}{9317 \sqrt {1-2 x}}\) | \(75\) |
default | \(\frac {-\frac {7375 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {1425 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {9975 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{14641}+\frac {54 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{49}+\frac {16}{9317 \sqrt {1-2 x}}\) | \(75\) |
pseudoelliptic | \(-\frac {9975 \left (-\frac {161359}{69825}-\frac {263538 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right )^{2} \sqrt {21}}{162925}+\sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}+\frac {7513 x^{2}}{931}+\frac {12133 x}{27930}\right )}{14641 \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}\) | \(90\) |
trager | \(\frac {\left (102450 x^{2}+5515 x -29338\right ) \sqrt {1-2 x}}{18634 \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{49}+\frac {9975 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{29282}\) | \(123\) |
Input:
int(1/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/18634*(102450*x^2+5515*x-29338)/(3+5*x)^2/(1-2*x)^(1/2)+54/49*21^(1/2)* arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-9975/14641*55^(1/2)*arctanh(1/11*55^(1 /2)*(1-2*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.16 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx=\frac {69825 \, \sqrt {\frac {5}{11}} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 71874 \, \sqrt {\frac {3}{7}} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {3 \, x - 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + {\left (102450 \, x^{2} + 5515 \, x - 29338\right )} \sqrt {-2 \, x + 1}}{18634 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^3,x, algorithm="fricas")
Output:
1/18634*(69825*sqrt(5/11)*(50*x^3 + 35*x^2 - 12*x - 9)*log((5*x + 11*sqrt( 5/11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 71874*sqrt(3/7)*(50*x^3 + 35*x^2 - 12*x - 9)*log((3*x - 7*sqrt(3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + (102450* x^2 + 5515*x - 29338)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*x - 9)
Result contains complex when optimal does not.
Time = 7.86 (sec) , antiderivative size = 1875, normalized size of antiderivative = 16.74 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-2*x)**(3/2)/(2+3*x)/(3+5*x)**3,x)
Output:
-9775500000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/ (14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 1041677868 00*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2) **(3/2)) + 15812280000*sqrt(21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 1 04167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338* (x - 1/2)**(3/2)) - 7906140000*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(1434818000 0*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2 )**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 4887750000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 763897103 20*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 43012200000*sqrt(55) *I*(x - 1/2)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(14348180000*(x - 1/2 )**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 6957403200 0*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(14348180000* (x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)* *(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 34 787016000*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710...
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx=\frac {9975}{29282} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {27}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {51225 \, {\left (2 \, x - 1\right )}^{2} + 215930 \, x - 109901}{9317 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^3,x, algorithm="maxima")
Output:
9975/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) - 27/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/9317*(51225*(2*x - 1)^2 + 215930*x - 109901)/(25* (-2*x + 1)^(5/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx=\frac {9975}{29282} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {27}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16}{9317 \, \sqrt {-2 \, x + 1}} - \frac {25 \, {\left (295 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 627 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^3,x, algorithm="giac")
Output:
9975/29282*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/49*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2 *x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/9317/sqrt(-2*x + 1) - 25/5324 *(295*(-2*x + 1)^(3/2) - 627*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 1.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx=\frac {54\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {9975\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {\frac {3926\,x}{4235}+\frac {2049\,{\left (2\,x-1\right )}^2}{9317}-\frac {9991}{21175}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}} \] Input:
int(1/((1 - 2*x)^(3/2)*(3*x + 2)*(5*x + 3)^3),x)
Output:
(54*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (9975*55^(1/2)*atan h((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641 - ((3926*x)/4235 + (2049*(2*x - 1) ^2)/9317 - 9991/21175)/((121*(1 - 2*x)^(1/2))/25 - (22*(1 - 2*x)^(3/2))/5 + (1 - 2*x)^(5/2))
Time = 0.15 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.89 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx=\frac {12219375 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+14663250 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +4398975 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-12219375 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-14663250 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -4398975 \sqrt {-2 x +1}\, \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-19765350 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-23718420 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -7115526 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+19765350 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+23718420 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +7115526 \sqrt {-2 x +1}\, \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )-7888650 x^{2}-424655 x +2259026}{1434818 \sqrt {-2 x +1}\, \left (25 x^{2}+30 x +9\right )} \] Input:
int(1/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^3,x)
Output:
(12219375*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x** 2 + 14663250*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))* x + 4398975*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 12219375*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x** 2 - 14663250*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))* x - 4398975*sqrt( - 2*x + 1)*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) - 19765350*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x** 2 - 23718420*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))* x - 7115526*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 19765350*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x** 2 + 23718420*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))* x + 7115526*sqrt( - 2*x + 1)*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)) - 7888650*x**2 - 424655*x + 2259026)/(1434818*sqrt( - 2*x + 1)*(25*x**2 + 3 0*x + 9))