Integrand size = 26, antiderivative size = 137 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {6095 \sqrt {1-2 x}}{84 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {243 \sqrt {1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac {608185 \sqrt {1-2 x}}{924 \sqrt {3+5 x}}-\frac {126513 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}} \]
-126513/196*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-6095/8 4*(1-2*x)^(1/2)/(3+5*x)^(3/2)+1/2*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3/2)+24 3/28*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(3/2)+608185/924*(1-2*x)^(1/2)/(3+5*x)^ (1/2)
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (7046540+33277877 x+52308690 x^2+27368325 x^3\right )}{924 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {126513 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}} \]
(Sqrt[1 - 2*x]*(7046540 + 33277877*x + 52308690*x^2 + 27368325*x^3))/(924* (2 + 3*x)^2*(3 + 5*x)^(3/2)) - (126513*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[ 3 + 5*x])])/(28*Sqrt[7])
Time = 0.22 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {110, 27, 168, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^3 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {1}{2} \int -\frac {41-60 x}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {41-60 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{7} \int \frac {7577-9720 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {243 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \int \frac {7577-9720 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {243 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (-\frac {2}{33} \int \frac {11 (77753-73140 x)}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {12190 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {243 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (-\frac {1}{3} \int \frac {77753-73140 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {12190 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {243 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {2}{11} \int \frac {4174929}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1216370 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {12190 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {243 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (\frac {1}{3} \left (379539 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1216370 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {12190 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {243 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (\frac {1}{3} \left (759078 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {1216370 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {12190 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {243 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (\frac {1}{3} \left (\frac {1216370 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {759078 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {12190 \sqrt {1-2 x}}{3 (5 x+3)^{3/2}}\right )+\frac {243 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\) |
Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + ((243*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^(3/2)) + ((-12190*Sqrt[1 - 2*x])/(3*(3 + 5*x)^(3/2)) + ( (1216370*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) - (759078*ArcTan[Sqrt[1 - 2*x]/ (Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/3)/14)/4
3.24.25.3.1 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. \(249\) vs. \(2(104)=208\).
Time = 1.12 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\frac {\left (939359025 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+2379709530 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+2258636589 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+383156550 x^{3} \sqrt {-10 x^{2}-x +3}+951883812 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +732321660 x^{2} \sqrt {-10 x^{2}-x +3}+150297444 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+465890278 x \sqrt {-10 x^{2}-x +3}+98651560 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{12936 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(250\) |
1/12936*(939359025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 2))*x^4+2379709530*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 2))*x^3+2258636589*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 2))*x^2+383156550*x^3*(-10*x^2-x+3)^(1/2)+951883812*7^(1/2)*arctan(1/14*(3 7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+732321660*x^2*(-10*x^2-x+3)^(1/2)+1 50297444*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+465890 278*x*(-10*x^2-x+3)^(1/2)+98651560*(-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)^(3/2)
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {4174929 \, \sqrt {7} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (27368325 \, x^{3} + 52308690 \, x^{2} + 33277877 \, x + 7046540\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12936 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
-1/12936*(4174929*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arcta n(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(27368325*x^3 + 52308690*x^2 + 33277877*x + 7046540)*sqrt(5*x + 3)*sq rt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1 - 2 x}}{\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {126513}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {608185 \, x}{462 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {635003}{924 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1985 \, x}{6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {49}{18 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1645}{36 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {6433}{36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
126513/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 608 185/462*x/sqrt(-10*x^2 - x + 3) + 635003/924/sqrt(-10*x^2 - x + 3) + 1985/ 6*x/(-10*x^2 - x + 3)^(3/2) + 49/18/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(- 10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 1645/36/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 6433/36/(-10*x^2 - x + 3) ^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (104) = 208\).
Time = 0.44 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {1}{129360} \, \sqrt {5} {\left (1225 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - 4174929 \, \sqrt {70} \sqrt {2} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - 2910600 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {2744280 \, \sqrt {2} {\left (151 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {36120 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {144480 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}}\right )} \]
-1/129360*sqrt(5)*(1225*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)))^3 - 4174 929*sqrt(70)*sqrt(2)*(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)))) - 2910600*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))) - 2744280* sqrt(2)*(151*((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 + 36120*(sqrt(2)*sqrt(-10 *x + 5) - sqrt(22))/sqrt(5*x + 3) - 144480*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)^2)
Timed out. \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}} \,d x \]