Integrand size = 24, antiderivative size = 97 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2}+\frac {219 \sqrt {1-2 x}}{98 (2+3 x)}+\frac {2523}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-50 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
2523/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50/11*arctanh(1/11*5 5^(1/2)*(1-2*x)^(1/2))*55^(1/2)+3/14*(1-2*x)^(1/2)/(2+3*x)^2+219/98*(1-2*x )^(1/2)/(2+3*x)
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx=\frac {9 \sqrt {1-2 x} (51+73 x)}{98 (2+3 x)^2}+\frac {2523}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-50 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(9*Sqrt[1 - 2*x]*(51 + 73*x))/(98*(2 + 3*x)^2) + (2523*Sqrt[3/7]*ArcTanh[S qrt[3/7]*Sqrt[1 - 2*x]])/49 - 50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2* x]]
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {114, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{14} \int \frac {43-45 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \int \frac {1793-1095 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {219 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (12250 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-7569 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {219 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (7569 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-12250 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {219 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{7} \left (5046 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-4900 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {219 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2}\) |
(3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2) + ((219*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + (5046*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 4900*Sqrt[5/11]*ArcTanh [Sqrt[5/11]*Sqrt[1 - 2*x]])/7)/14
3.21.44.3.1 Defintions of rubi rules used
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 = 3.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(-\frac {9 \left (146 x^{2}+29 x -51\right )}{98 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {2523 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}\) | \(64\) |
| derivativedivides | \(-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {162 \left (\frac {73 \left (1-2 x \right )^{\frac {3}{2}}}{882}-\frac {25 \sqrt {1-2 x}}{126}\right )}{\left (-4-6 x \right )^{2}}+\frac {2523 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(66\) |
| default | \(-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {162 \left (\frac {73 \left (1-2 x \right )^{\frac {3}{2}}}{882}-\frac {25 \sqrt {1-2 x}}{126}\right )}{\left (-4-6 x \right )^{2}}+\frac {2523 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\) | \(66\) |
| pseudoelliptic | \(\frac {55506 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \sqrt {21}-34300 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \sqrt {55}+693 \sqrt {1-2 x}\, \left (73 x +51\right )}{7546 \left (2+3 x \right )^{2}}\) | \(75\) |
| trager | \(\frac {9 \left (73 x +51\right ) \sqrt {1-2 x}}{98 \left (2+3 x \right )^{2}}-\frac {2523 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{686}+\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{11}\) | \(111\) |
-9/98*(146*x^2+29*x-51)/(2+3*x)^2/(1-2*x)^(1/2)+2523/343*arctanh(1/7*21^(1 /2)*(1-2*x)^(1/2))*21^(1/2)-50/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^ (1/2)
Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx=\frac {17150 \, \sqrt {11} \sqrt {5} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 27753 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 693 \, {\left (73 \, x + 51\right )} \sqrt {-2 \, x + 1}}{7546 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
1/7546*(17150*sqrt(11)*sqrt(5)*(9*x^2 + 12*x + 4)*log((sqrt(11)*sqrt(5)*sq rt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 27753*sqrt(7)*sqrt(3)*(9*x^2 + 12*x + 4)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 693*(73*x + 51)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
Result contains complex when optimal does not.
Time = 8.99 (sec) , antiderivative size = 1953, normalized size of antiderivative = 20.13 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx=\text {Too large to display} \]
3642408*sqrt(2)*I*(x - 1/2)**(11/2)/(4889808*(x - 1/2)**6 + 22819104*(x - 1/2)**5 + 39933432*(x - 1/2)**4 + 31059336*(x - 1/2)**3 + 9058973*(x - 1/2 )**2) + 12864852*sqrt(2)*I*(x - 1/2)**(9/2)/(4889808*(x - 1/2)**6 + 228191 04*(x - 1/2)**5 + 39933432*(x - 1/2)**4 + 31059336*(x - 1/2)**3 + 9058973* (x - 1/2)**2) + 15144822*sqrt(2)*I*(x - 1/2)**(7/2)/(4889808*(x - 1/2)**6 + 22819104*(x - 1/2)**5 + 39933432*(x - 1/2)**4 + 31059336*(x - 1/2)**3 + 9058973*(x - 1/2)**2) + 5942475*sqrt(2)*I*(x - 1/2)**(5/2)/(4889808*(x - 1 /2)**6 + 22819104*(x - 1/2)**5 + 39933432*(x - 1/2)**4 + 31059336*(x - 1/2 )**3 + 9058973*(x - 1/2)**2) - 22226400*sqrt(55)*I*(x - 1/2)**6*atan(sqrt( 110)*sqrt(x - 1/2)/11)/(4889808*(x - 1/2)**6 + 22819104*(x - 1/2)**5 + 399 33432*(x - 1/2)**4 + 31059336*(x - 1/2)**3 + 9058973*(x - 1/2)**2) + 10692 00*sqrt(21)*I*(x - 1/2)**6*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(4889808*(x - 1/2)**6 + 22819104*(x - 1/2)**5 + 39933432*(x - 1/2)**4 + 31059336*(x - 1/ 2)**3 + 9058973*(x - 1/2)**2) + 37037088*sqrt(21)*I*(x - 1/2)**6*atan(sqrt (42)*sqrt(x - 1/2)/7)/(4889808*(x - 1/2)**6 + 22819104*(x - 1/2)**5 + 3993 3432*(x - 1/2)**4 + 31059336*(x - 1/2)**3 + 9058973*(x - 1/2)**2) - 185185 44*sqrt(21)*I*pi*(x - 1/2)**6/(4889808*(x - 1/2)**6 + 22819104*(x - 1/2)** 5 + 39933432*(x - 1/2)**4 + 31059336*(x - 1/2)**3 + 9058973*(x - 1/2)**2) + 11113200*sqrt(55)*I*pi*(x - 1/2)**6/(4889808*(x - 1/2)**6 + 22819104*(x - 1/2)**5 + 39933432*(x - 1/2)**4 + 31059336*(x - 1/2)**3 + 9058973*(x ...
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx=\frac {25}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2523}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9 \, {\left (73 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 175 \, \sqrt {-2 \, x + 1}\right )}}{49 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
25/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2523/686*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 9/49*(73*(-2*x + 1)^(3/2) - 175*sqrt(-2*x + 1))/(9*(2 *x - 1)^2 + 84*x + 7)
Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx=\frac {25}{11} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2523}{686} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {9 \, {\left (73 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 175 \, \sqrt {-2 \, x + 1}\right )}}{196 \, {\left (3 \, x + 2\right )}^{2}} \]
25/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5* sqrt(-2*x + 1))) - 2523/686*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 9/196*(73*(-2*x + 1)^(3/2) - 175*s qrt(-2*x + 1))/(3*x + 2)^2
Time = 1.58 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx=\frac {2523\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {25\,\sqrt {1-2\,x}}{7}-\frac {73\,{\left (1-2\,x\right )}^{3/2}}{49}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \]