Integrand size = 26, antiderivative size = 101 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac {385 \sqrt {1-2 x}}{\sqrt {3+5 x}}-385 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
-55/3*(1-2*x)^(3/2)/(3+5*x)^(3/2)+(1-2*x)^(5/2)/(2+3*x)/(3+5*x)^(3/2)-385* arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+385*(1-2*x)^(1/2)/ (3+5*x)^(1/2)
Time = 1.62 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.37 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (6823+21988 x+17667 x^2\right )}{3 (2+3 x) (3+5 x)^{3/2}}+385 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+385 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right ) \]
(Sqrt[1 - 2*x]*(6823 + 21988*x + 17667*x^2))/(3*(2 + 3*x)*(3 + 5*x)^(3/2)) + 385*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 385*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[11 55]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {105, 105, 105, 104, 217}
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.
\(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^2 (5 x+3)^{5/2}} \, dx\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {55}{2} \int \frac {(1-2 x)^{3/2}}{(3 x+2) (5 x+3)^{5/2}}dx+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {55}{2} \left (-7 \int \frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}\right )+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {55}{2} \left (-7 \left (-7 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}\right )+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {55}{2} \left (-7 \left (-14 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}\right )+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {55}{2} \left (-7 \left (2 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}\right )+\frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}\) |
(1 - 2*x)^(5/2)/((2 + 3*x)*(3 + 5*x)^(3/2)) + (55*((-2*(1 - 2*x)^(3/2))/(3 *(3 + 5*x)^(3/2)) - 7*((-2*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + 2*Sqrt[7]*ArcTan [Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])))/2
3.25.53.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(80)=160\).
Time = 1.17 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.00
method | result | size |
default | \(\frac {\left (86625 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+161700 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+100485 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +35334 x^{2} \sqrt {-10 x^{2}-x +3}+20790 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+43976 x \sqrt {-10 x^{2}-x +3}+13646 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{6 \left (2+3 x \right ) \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(202\) |
1/6*(86625*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+ 161700*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1004 85*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+35334*x^2* (-10*x^2-x+3)^(1/2)+20790*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x +3)^(1/2))+43976*x*(-10*x^2-x+3)^(1/2)+13646*(-10*x^2-x+3)^(1/2))*(1-2*x)^ (1/2)/(2+3*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {1155 \, \sqrt {7} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 2 \, {\left (17667 \, x^{2} + 21988 \, x + 6823\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]
-1/6*(1155*sqrt(7)*(75*x^3 + 140*x^2 + 87*x + 18)*arctan(1/14*sqrt(7)*(37* x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 2*(17667*x^2 + 21 988*x + 6823)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 18)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.37 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx=\frac {385}{2} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {3926 \, x}{5 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {16 \, x^{2}}{45 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {30743}{75 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {133642 \, x}{675 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{81 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {217433}{2025 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
385/2*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 3926/5*x /sqrt(-10*x^2 - x + 3) - 16/45*x^2/(-10*x^2 - x + 3)^(3/2) + 30743/75/sqrt (-10*x^2 - x + 3) + 133642/675*x/(-10*x^2 - x + 3)^(3/2) + 2401/81/(3*(-10 *x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 217433/2025/(-10*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (80) = 160\).
Time = 0.42 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.10 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {11}{1200} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {77}{4} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {77}{5} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280} \]
-11/1200*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4* sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 77/4*sqrt(70)*sqrt (10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5 ) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 77 /5*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5 *x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 1078*sqrt(10)*((sqrt(2)*sq rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 0*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2}} \,d x \]