Integrand size = 26, antiderivative size = 115 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {2615 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 \sqrt {1-2 x}}{28 (2+3 x) \sqrt {3+5 x}}+\frac {17951 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}} \]
17951/196*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-2615/28* (1-2*x)^(1/2)/(3+5*x)^(1/2)+1/2*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(1/2)+173/ 28*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(1/2)
Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\frac {1}{196} \left (-\frac {7 \sqrt {1-2 x} \left (10100+30861 x+23535 x^2\right )}{(2+3 x)^2 \sqrt {3+5 x}}+17951 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \]
((-7*Sqrt[1 - 2*x]*(10100 + 30861*x + 23535*x^2))/((2 + 3*x)^2*Sqrt[3 + 5* x]) + 17951*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/196
Time = 0.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {110, 27, 168, 27, 169, 27, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}-\frac {1}{2} \int -\frac {31-40 x}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {31-40 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2}}dx+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{7} \int \frac {3677-3460 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {173 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \int \frac {3677-3460 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx+\frac {173 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (-\frac {2}{11} \int \frac {197461}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {5230 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {173 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (-17951 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {5230 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {173 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (-35902 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {5230 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {173 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (\frac {35902 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {5230 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {173 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}\) |
Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*Sqrt[3 + 5*x]) + ((173*Sqrt[1 - 2*x])/(7*(2 + 3*x)*Sqrt[3 + 5*x]) + ((-5230*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + (35902*ArcTa n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/14)/4
3.24.16.3.1 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_))/((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[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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[((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[((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. \(2(88)=176\).
Time = 1.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(-\frac {\left (807795 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+1561737 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1005256 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +329490 x^{2} \sqrt {-10 x^{2}-x +3}+215412 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+432054 x \sqrt {-10 x^{2}-x +3}+141400 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{392 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(202\) |
-1/392*(807795*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))* x^3+1561737*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2 +1005256*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+3294 90*x^2*(-10*x^2-x+3)^(1/2)+215412*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(- 10*x^2-x+3)^(1/2))+432054*x*(-10*x^2-x+3)^(1/2)+141400*(-10*x^2-x+3)^(1/2) )*(1-2*x)^(1/2)/(2+3*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\frac {17951 \, \sqrt {7} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (23535 \, x^{2} + 30861 \, x + 10100\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{392 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]
1/392*(17951*sqrt(7)*(45*x^3 + 87*x^2 + 56*x + 12)*arctan(1/14*sqrt(7)*(37 *x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(23535*x^2 + 30861*x + 10100)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 1 2)
\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {1 - 2 x}}{\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {17951}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2615 \, x}{14 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {8191}{84 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {7}{6 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {169}{12 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
-17951/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 261 5/14*x/sqrt(-10*x^2 - x + 3) - 8191/84/sqrt(-10*x^2 - x + 3) + 7/6/(9*sqrt (-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 169/12/(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. 316 vs. \(2 (88) = 176\).
Time = 0.44 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=-\frac {1}{3920} \, \sqrt {5} {\left (17951 \, \sqrt {70} \sqrt {2} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + 9800 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {9240 \, \sqrt {2} {\left (313 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {69160 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {276640 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}}\right )} \]
-1/3920*sqrt(5)*(17951*sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqr t(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2) *sqrt(-10*x + 5) - sqrt(22)))) + 9800*sqrt(2)*((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))) + 9240*sqrt(2)*(313*((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 + 69160*(sqr t(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 276640*sqrt(5*x + 3)/(sqr t(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 2 80)^2)
Timed out. \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}} \,d x \]