Integrand size = 24, antiderivative size = 105 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {720}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
-190/1617/(1-2*x)^(3/2)+3/7/(1-2*x)^(3/2)/(2+3*x)+720/2401*arctanh(1/7*21^ (1/2)*(1-2*x)^(1/2))*21^(1/2)-250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2) )*55^(1/2)-1370/41503/(1-2*x)^(1/2)
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {15881-39780 x+24660 x^2}{124509 (1-2 x)^{3/2} (2+3 x)}+\frac {720}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
(15881 - 39780*x + 24660*x^2)/(124509*(1 - 2*x)^(3/2)*(2 + 3*x)) + (720*Sq rt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sq rt[5/11]*Sqrt[1 - 2*x]])/121
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {114, 27, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^2 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{7} \int -\frac {5 (15 x+2)}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx+\frac {3}{7 (1-2 x)^{3/2} (3 x+2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \int \frac {15 x+2}{(1-2 x)^{5/2} (3 x+2) (5 x+3)}dx\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {38}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int \frac {3 (74-285 x)}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {38}{231 (1-2 x)^{3/2}}-\frac {1}{77} \int \frac {74-285 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)}dx\right )\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {2}{77} \int -\frac {7342-2055 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {274}{77 \sqrt {1-2 x}}\right )+\frac {38}{231 (1-2 x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {274}{77 \sqrt {1-2 x}}-\frac {1}{77} \int \frac {7342-2055 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx\right )+\frac {38}{231 (1-2 x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (26136 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-42875 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx\right )+\frac {274}{77 \sqrt {1-2 x}}\right )+\frac {38}{231 (1-2 x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (42875 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}-26136 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {274}{77 \sqrt {1-2 x}}\right )+\frac {38}{231 (1-2 x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{7} \left (\frac {1}{77} \left (\frac {1}{77} \left (17150 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-17424 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {274}{77 \sqrt {1-2 x}}\right )+\frac {38}{231 (1-2 x)^{3/2}}\right )\) |
3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)) - (5*(38/(231*(1 - 2*x)^(3/2)) + (274/(77* Sqrt[1 - 2*x]) + (-17424*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 1715 0*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/77)/77))/7
3.22.77.3.1 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] && 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 = 1.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.69
| method | result | size |
| derivativedivides | \(-\frac {250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {18 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}+\frac {720 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {8}{1617 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {808}{41503 \sqrt {1-2 x}}\) | \(72\) |
| default | \(-\frac {250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {18 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}+\frac {720 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {8}{1617 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {808}{41503 \sqrt {1-2 x}}\) | \(72\) |
| pseudoelliptic | \(\frac {-2874960 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (6 x^{2}+x -2\right ) \sqrt {21}+1800750 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (6 x^{2}+x -2\right ) \sqrt {55}+1898820 x^{2}-3063060 x +1222837}{\left (1-2 x \right )^{\frac {3}{2}} \left (19174386+28761579 x \right )}\) | \(93\) |
| trager | \(\frac {\left (24660 x^{2}-39780 x +15881\right ) \sqrt {1-2 x}}{124509 \left (-1+2 x \right )^{2} \left (2+3 x \right )}+\frac {360 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2401}-\frac {125 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}\) | \(123\) |
-250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-18/343*(1-2*x)^(1/ 2)/(-4/3-2*x)+720/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+8/1617 /(1-2*x)^(3/2)+808/41503/(1-2*x)^(1/2)
Time = 0.25 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {900375 \, \sqrt {11} \sqrt {5} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 1437480 \, \sqrt {7} \sqrt {3} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (24660 \, x^{2} - 39780 \, x + 15881\right )} \sqrt {-2 \, x + 1}}{9587193 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
1/9587193*(900375*sqrt(11)*sqrt(5)*(12*x^3 - 4*x^2 - 5*x + 2)*log((sqrt(11 )*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 1437480*sqrt(7)*sqrt(3)*( 12*x^3 - 4*x^2 - 5*x + 2)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/ (3*x + 2)) + 77*(24660*x^2 - 39780*x + 15881)*sqrt(-2*x + 1))/(12*x^3 - 4* x^2 - 5*x + 2)
Result contains complex when optimal does not.
Time = 7.10 (sec) , antiderivative size = 1352, normalized size of antiderivative = 12.88 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx=\text {Too large to display} \]
-388962000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/( 2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) + 620991360*sqrt(21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*( x - 1/2)**(5/2)) - 310495680*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(2070833688*( x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/ 2) + 3288407199*(x - 1/2)**(5/2)) + 194481000*sqrt(55)*I*pi*(x - 1/2)**(11 /2)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 84559042 26*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) - 1361367000*sqrt(55)*I *(x - 1/2)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(2070833688*(x - 1/2)** (11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 32884 07199*(x - 1/2)**(5/2)) + 2173469760*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqrt (42)*sqrt(x - 1/2)/7)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2) **(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) - 108 6734880*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(2070833688*(x - 1/2)**(11/2) + 724 7917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1 /2)**(5/2)) + 680683500*sqrt(55)*I*pi*(x - 1/2)**(9/2)/(2070833688*(x - 1/ 2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3 288407199*(x - 1/2)**(5/2)) - 1588261500*sqrt(55)*I*(x - 1/2)**(7/2)*at...
Time = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {125}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {360}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (6165 \, {\left (2 \, x - 1\right )}^{2} - 15120 \, x + 9716\right )}}{124509 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
125/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) - 360/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/124509*(6165*(2*x - 1)^2 - 15120*x + 9716)/(3*(- 2*x + 1)^(5/2) - 7*(-2*x + 1)^(3/2))
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {125}{1331} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {360}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (303 \, x - 190\right )}}{124509 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {27 \, \sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} \]
125/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 360/2401*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(- 2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/124509*(303*x - 190)/((2*x - 1)*sqrt(-2*x + 1)) + 27/343*sqrt(-2*x + 1)/(3*x + 2)
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx=\frac {\frac {1370\,{\left (2\,x-1\right )}^2}{41503}-\frac {480\,x}{5929}+\frac {2776}{53361}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}+\frac {720\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \]