Integrand size = 26, antiderivative size = 144 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {147015 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}+\frac {(1-2 x)^{7/2}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {81 (1-2 x)^{5/2}}{28 (2+3 x)^2 \sqrt {3+5 x}}+\frac {4455 (1-2 x)^{3/2}}{56 (2+3 x) \sqrt {3+5 x}}+\frac {147015 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{8 \sqrt {7}} \]
147015/56*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1/7*(1-2 *x)^(7/2)/(2+3*x)^3/(3+5*x)^(1/2)+81/28*(1-2*x)^(5/2)/(2+3*x)^2/(3+5*x)^(1 /2)+4455/56*(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^(1/2)-147015/56*(1-2*x)^(1/2)/(3 +5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {\sqrt {1-2 x} \left (165424+753654 x+1143741 x^2+578245 x^3\right )}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {147015 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{8 \sqrt {7}} \]
-1/8*(Sqrt[1 - 2*x]*(165424 + 753654*x + 1143741*x^2 + 578245*x^3))/((2 + 3*x)^3*Sqrt[3 + 5*x]) + (147015*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x ])])/(8*Sqrt[7])
Time = 0.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {107, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^4 (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {81}{14} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^3 (5 x+3)^{3/2}}dx+\frac {(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {81}{14} \left (\frac {55}{4} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {81}{14} \left (\frac {55}{4} \left (\frac {33}{2} \int \frac {\sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}dx+\frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {81}{14} \left (\frac {55}{4} \left (\frac {33}{2} \left (-7 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {81}{14} \left (\frac {55}{4} \left (\frac {33}{2} \left (-14 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {81}{14} \left (\frac {55}{4} \left (\frac {33}{2} \left (2 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {(1-2 x)^{3/2}}{(3 x+2) \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{5/2}}{2 (3 x+2)^2 \sqrt {5 x+3}}\right )+\frac {(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt {5 x+3}}\) |
(1 - 2*x)^(7/2)/(7*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (81*((1 - 2*x)^(5/2)/(2*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (55*((1 - 2*x)^(3/2)/((2 + 3*x)*Sqrt[3 + 5*x]) + (33*((-2*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + 2*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/( Sqrt[7]*Sqrt[3 + 5*x])]))/2))/4))/14
3.25.44.3.1 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_))^(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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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. \(249\) vs. \(2(111)=222\).
Time = 5.13 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(-\frac {\left (19847025 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+51602265 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+50279130 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+8095430 x^{3} \sqrt {-10 x^{2}-x +3}+21758220 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +16012374 x^{2} \sqrt {-10 x^{2}-x +3}+3528360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+10551156 x \sqrt {-10 x^{2}-x +3}+2315936 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{112 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(250\) |
-1/112*(19847025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2) )*x^4+51602265*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))* x^3+50279130*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^ 2+8095430*x^3*(-10*x^2-x+3)^(1/2)+21758220*7^(1/2)*arctan(1/14*(37*x+20)*7 ^(1/2)/(-10*x^2-x+3)^(1/2))*x+16012374*x^2*(-10*x^2-x+3)^(1/2)+3528360*7^( 1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+10551156*x*(-10*x^ 2-x+3)^(1/2)+2315936*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2 -x+3)^(1/2)/(3+5*x)^(1/2)
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=\frac {147015 \, \sqrt {7} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (578245 \, x^{3} + 1143741 \, x^{2} + 753654 \, x + 165424\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{112 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
1/112*(147015*sqrt(7)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*arctan(1/ 14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14 *(578245*x^3 + 1143741*x^2 + 753654*x + 165424)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.47 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {147015}{112} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {578245 \, x}{108 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {603743}{216 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{81 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {10339}{324 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {87199}{216 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
-147015/112*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 57 8245/108*x/sqrt(-10*x^2 - x + 3) - 603743/216/sqrt(-10*x^2 - x + 3) + 343/ 81/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt( -10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 10339/324/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 871 99/216/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (111) = 222\).
Time = 0.47 (sec) , antiderivative size = 369, normalized size of antiderivative = 2.56 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=-\frac {29403}{224} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {121 \, \sqrt {10} {\left (993 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 436800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {51352000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {205408000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
-29403/224*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)))) - 121/2*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))) - 1 21/4*sqrt(10)*(993*((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)))^5 + 436800*((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 + 51352000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/ sqrt(5*x + 3) - 205408000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*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)^3
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}} \,d x \]