Integrand size = 26, antiderivative size = 137 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {655 \sqrt {1-2 x}}{4 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {235 \sqrt {1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac {17825 \sqrt {1-2 x}}{12 \sqrt {3+5 x}}-\frac {40787 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}} \]
-40787/28*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-655/4*(1 -2*x)^(1/2)/(3+5*x)^(3/2)+7/6*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3/2)+235/12 *(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(3/2)+17825/12*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (206524+975325 x+1533090 x^2+802125 x^3\right )}{12 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {40787 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4 \sqrt {7}} \]
(Sqrt[1 - 2*x]*(206524 + 975325*x + 1533090*x^2 + 802125*x^3))/(12*(2 + 3* x)^2*(3 + 5*x)^(3/2)) - (40787*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x] )])/(4*Sqrt[7])
Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {109, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^3 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{6} \int \frac {279-404 x}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \int \frac {279-404 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{7} \int \frac {7 (7329-9400 x)}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {235 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \int \frac {7329-9400 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {235 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (-\frac {2}{33} \int \frac {33 (25067-23580 x)}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {3930 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {235 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (-\int \frac {25067-23580 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {3930 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {235 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (\frac {2}{11} \int \frac {1345971}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {35650 \sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {3930 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {235 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (122361 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {35650 \sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {3930 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {235 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (244722 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {35650 \sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {3930 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {235 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{12} \left (\frac {1}{2} \left (-\frac {244722 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}+\frac {35650 \sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {3930 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {235 \sqrt {1-2 x}}{(3 x+2) (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}\) |
(7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + ((235*Sqrt[1 - 2*x])/( (2 + 3*x)*(3 + 5*x)^(3/2)) + ((-3930*Sqrt[1 - 2*x])/(3 + 5*x)^(3/2) + (356 50*Sqrt[1 - 2*x])/Sqrt[3 + 5*x] - (244722*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sq rt[3 + 5*x])])/Sqrt[7])/2)/12
3.24.86.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 = 1.14 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\frac {\left (27531225 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+69745770 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+66197301 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+11229750 x^{3} \sqrt {-10 x^{2}-x +3}+27898308 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +21463260 x^{2} \sqrt {-10 x^{2}-x +3}+4404996 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13654550 x \sqrt {-10 x^{2}-x +3}+2891336 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{168 \left (2+3 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(250\) |
1/168*(27531225*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x^4+69745770*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x ^3+66197301*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2 +11229750*x^3*(-10*x^2-x+3)^(1/2)+27898308*7^(1/2)*arctan(1/14*(37*x+20)*7 ^(1/2)/(-10*x^2-x+3)^(1/2))*x+21463260*x^2*(-10*x^2-x+3)^(1/2)+4404996*7^( 1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+13654550*x*(-10*x^ 2-x+3)^(1/2)+2891336*(-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 = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {122361 \, \sqrt {7} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (802125 \, x^{3} + 1533090 \, x^{2} + 975325 \, x + 206524\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{168 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
-1/168*(122361*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arctan(1 /14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 1 4*(802125*x^3 + 1533090*x^2 + 975325*x + 206524)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.26 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\frac {40787}{56} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {17825 \, x}{6 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {18611}{12 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {13439 \, x}{18 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {343}{54 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {11123}{108 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {1613}{4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
40787/56*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 17825 /6*x/sqrt(-10*x^2 - x + 3) + 18611/12/sqrt(-10*x^2 - x + 3) + 13439/18*x/( -10*x^2 - x + 3)^(3/2) + 343/54/(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)) + 11123/108/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 1613/4/(-10*x^2 - x + 3)^(3 /2)
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (104) = 208\).
Time = 0.42 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.72 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=-\frac {1}{48} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {40787}{560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {101}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {165 \, \sqrt {10} {\left (89 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {21224 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {84896 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{2 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
-1/48*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 40787/560*sqrt(70)*sq rt(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)))) + 101/2*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 165/2*sqrt(10)*(89*((sq rt(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 + 21224*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 ))/sqrt(5*x + 3) - 84896*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. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}} \,d x \]