Integrand size = 26, antiderivative size = 108 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {845 \sqrt {1-2 x}}{231 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {84235 \sqrt {1-2 x}}{2541 \sqrt {3+5 x}}-\frac {1593 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}} \]
-1593/49*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-845/231*( 1-2*x)^(1/2)/(3+5*x)^(3/2)+3/7*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(3/2)+84235/2 541*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=\frac {7 \sqrt {1-2 x} \left (487909+1572580 x+1263525 x^2\right )-578259 \sqrt {7} \sqrt {3+5 x} \left (6+19 x+15 x^2\right ) \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{17787 (2+3 x) (3+5 x)^{3/2}} \]
(7*Sqrt[1 - 2*x]*(487909 + 1572580*x + 1263525*x^2) - 578259*Sqrt[7]*Sqrt[ 3 + 5*x]*(6 + 19*x + 15*x^2)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])] )/(17787*(2 + 3*x)*(3 + 5*x)^(3/2))
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {114, 27, 169, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{7} \int \frac {97-120 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {97-120 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (-\frac {2}{33} \int \frac {10763-10140 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {1690 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (-\frac {1}{33} \int \frac {10763-10140 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {1690 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {578259}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {168470 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {1690 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{33} \left (52569 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {168470 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {1690 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{33} \left (105138 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {168470 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {1690 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{33} \left (\frac {168470 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {105138 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {1690 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}\right )+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\) |
(3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^(3/2)) + ((-1690*Sqrt[1 - 2*x])/( 33*(3 + 5*x)^(3/2)) + ((168470*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) - (105138 *ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/33)/14
3.26.14.3.1 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[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[((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. \(2(81)=162\).
Time = 3.96 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.87
| method | result | size |
| default | \(\frac {\left (43369425 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+80956260 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+50308533 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +17689350 x^{2} \sqrt {-10 x^{2}-x +3}+10408662 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+22016120 x \sqrt {-10 x^{2}-x +3}+6830726 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{35574 \left (2+3 x \right ) \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(202\) |
1/35574*(43369425*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))*x^3+80956260*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x^2+50308533*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x +17689350*x^2*(-10*x^2-x+3)^(1/2)+10408662*7^(1/2)*arctan(1/14*(37*x+20)*7 ^(1/2)/(-10*x^2-x+3)^(1/2))+22016120*x*(-10*x^2-x+3)^(1/2)+6830726*(-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 = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {578259 \, \sqrt {7} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1263525 \, x^{2} + 1572580 \, x + 487909\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{35574 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]
-1/35574*(578259*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)) - 14*(1263525 *x^2 + 1572580*x + 487909)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 18)
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{2} \sqrt {-2 \, x + 1}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (81) = 162\).
Time = 0.36 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.90 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=-\frac {5}{5808} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1593}{980} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {160}{121} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {594 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{7 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
-5/5808*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 1593/980*sqrt(70)*s qrt(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)))) + 160/121*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))) + 594/7*sqrt(10)*((sqr t(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*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)
Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2}} \,d x \]