Integrand size = 24, antiderivative size = 97 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx=-\frac {5 \sqrt {1-2 x}}{22 (3+5 x)^2}+\frac {315 \sqrt {1-2 x}}{242 (3+5 x)}+18 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {2115}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
18/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2115/1331*arctanh(1/11*5 5^(1/2)*(1-2*x)^(1/2))*55^(1/2)-5/22*(1-2*x)^(1/2)/(3+5*x)^2+315/242*(1-2* x)^(1/2)/(3+5*x)
Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx=18 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {5 \left (\frac {11 \sqrt {1-2 x} (178+315 x)}{(3+5 x)^2}-846 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )}{2662} \]
18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (5*((11*Sqrt[1 - 2*x]*(178 + 315*x))/(3 + 5*x)^2 - 846*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/ 2662
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {114, 27, 168, 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.
\(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2) (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {1}{22} \int \frac {9 (4-5 x)}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {5 \sqrt {1-2 x}}{22 (5 x+3)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {9}{22} \int \frac {4-5 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^2}dx-\frac {5 \sqrt {1-2 x}}{22 (5 x+3)^2}\) |
\(\Big \downarrow \) 168 |
\(\displaystyle -\frac {9}{22} \left (-\frac {1}{11} \int \frac {172-105 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx-\frac {35 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {5 \sqrt {1-2 x}}{22 (5 x+3)^2}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle -\frac {9}{22} \left (\frac {1}{11} \left (726 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-1175 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )-\frac {35 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {5 \sqrt {1-2 x}}{22 (5 x+3)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {9}{22} \left (\frac {1}{11} \left (1175 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-726 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {35 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {5 \sqrt {1-2 x}}{22 (5 x+3)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {9}{22} \left (\frac {1}{11} \left (470 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-484 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )-\frac {35 \sqrt {1-2 x}}{11 (5 x+3)}\right )-\frac {5 \sqrt {1-2 x}}{22 (5 x+3)^2}\) |
(-5*Sqrt[1 - 2*x])/(22*(3 + 5*x)^2) - (9*((-35*Sqrt[1 - 2*x])/(11*(3 + 5*x )) + (-484*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 470*Sqrt[5/11]*Arc Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11))/22
3.21.65.3.1 Defintions of rubi rules used
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[(((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 = 1.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {5 \left (630 x^{2}+41 x -178\right )}{242 \left (3+5 x \right )^{2} \sqrt {1-2 x}}+\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}-\frac {2115 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}\) | \(64\) |
derivativedivides | \(\frac {-\frac {1575 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {305 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {2115 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(66\) |
default | \(\frac {-\frac {1575 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {305 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {2115 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {18 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\) | \(66\) |
pseudoelliptic | \(\frac {47916 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (3+5 x \right )^{2} \sqrt {21}-29610 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right )^{2} \sqrt {55}+385 \sqrt {1-2 x}\, \left (315 x +178\right )}{18634 \left (3+5 x \right )^{2}}\) | \(75\) |
trager | \(\frac {5 \left (315 x +178\right ) \sqrt {1-2 x}}{242 \left (3+5 x \right )^{2}}+\frac {2115 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{2662}-\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{7}\) | \(111\) |
-5/242*(630*x^2+41*x-178)/(3+5*x)^2/(1-2*x)^(1/2)+18/7*arctanh(1/7*21^(1/2 )*(1-2*x)^(1/2))*21^(1/2)-2115/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*5 5^(1/2)
Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx=\frac {14805 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 23958 \, \sqrt {7} \sqrt {3} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 385 \, {\left (315 \, x + 178\right )} \sqrt {-2 \, x + 1}}{18634 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
1/18634*(14805*sqrt(11)*sqrt(5)*(25*x^2 + 30*x + 9)*log((sqrt(11)*sqrt(5)* sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 23958*sqrt(7)*sqrt(3)*(25*x^2 + 30* x + 9)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 385*(3 15*x + 178)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)
Result contains complex when optimal does not.
Time = 8.17 (sec) , antiderivative size = 1953, normalized size of antiderivative = 20.13 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx=\text {Too large to display} \]
24255000*sqrt(2)*I*(x - 1/2)**(11/2)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*( x - 1/2)**2) + 79194500*sqrt(2)*I*(x - 1/2)**(9/2)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 86182250*sqrt(2)*I*(x - 1/2)**(7/2)/(93170000 *(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 49603708 0*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 31258535*sqrt(2)*I*(x - 1/2)**( 5/2)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2) **4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 4270000*sqrt(55)* I*(x - 1/2)**6*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) - 143780000*sqrt(55)*I*(x - 1/2)**6*atan(sqrt(11 0)*sqrt(x - 1/2)/11)/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676 414200*(x - 1/2)**4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 2 39580000*sqrt(21)*I*(x - 1/2)**6*atan(sqrt(42)*sqrt(x - 1/2)/7)/(93170000* (x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2)**4 + 496037080 *(x - 1/2)**3 + 136410197*(x - 1/2)**2) - 119790000*sqrt(21)*I*pi*(x - 1/2 )**6/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 676414200*(x - 1/2) **4 + 496037080*(x - 1/2)**3 + 136410197*(x - 1/2)**2) + 71890000*sqrt(55) *I*pi*(x - 1/2)**6/(93170000*(x - 1/2)**6 + 409948000*(x - 1/2)**5 + 67...
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx=\frac {2115}{2662} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9}{7} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {5 \, {\left (315 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 671 \, \sqrt {-2 \, x + 1}\right )}}{121 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
2115/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(- 2*x + 1))) - 9/7*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) - 5/121*(315*(-2*x + 1)^(3/2) - 671*sqrt(-2*x + 1))/(25* (2*x - 1)^2 + 220*x + 11)
Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx=\frac {2115}{2662} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {9}{7} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {5 \, {\left (315 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 671 \, \sqrt {-2 \, x + 1}\right )}}{484 \, {\left (5 \, x + 3\right )}^{2}} \]
2115/2662*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/7*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5/484*(315*(-2*x + 1)^(3/2) - 671*s qrt(-2*x + 1))/(5*x + 3)^2
Time = 1.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx=\frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {2115\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {61\,\sqrt {1-2\,x}}{55}-\frac {63\,{\left (1-2\,x\right )}^{3/2}}{121}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \]