Integrand size = 26, antiderivative size = 108 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {58}{539 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {17735 \sqrt {1-2 x}}{5929 \sqrt {3+5 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {999 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}} \]
999/343*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-58/539/(1- 2*x)^(1/2)/(3+5*x)^(1/2)+3/7/(2+3*x)/(1-2*x)^(1/2)/(3+5*x)^(1/2)-17735/592 9*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {7 \sqrt {3+5 x} \left (-34205+15821 x+106410 x^2\right )+120879 \sqrt {7-14 x} \left (6+19 x+15 x^2\right ) \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{41503 \sqrt {1-2 x} (2+3 x) (3+5 x)} \]
(7*Sqrt[3 + 5*x]*(-34205 + 15821*x + 106410*x^2) + 120879*Sqrt[7 - 14*x]*( 6 + 19*x + 15*x^2)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(41503*S qrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x))
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.
\(\displaystyle \int \frac {1}{(1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {1}{7} \int \frac {31-120 x}{2 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}dx+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{14} \int \frac {31-120 x}{(1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}dx+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{14} \left (-\frac {2}{77} \int -\frac {2503-1740 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {116}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \int \frac {2503-1740 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {116}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (-\frac {2}{11} \int \frac {120879}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {35470 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {116}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (-10989 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {35470 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {116}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (-21978 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {35470 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {116}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{14} \left (\frac {1}{77} \left (\frac {21978 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {35470 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}\right )-\frac {116}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {3}{7 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}\) |
3/(7*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x]) + (-116/(77*Sqrt[1 - 2*x]*Sqrt [3 + 5*x]) + ((-35470*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x]) + (21978*ArcTan[Sq rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/Sqrt[7])/77)/14
3.26.68.3.1 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. \(208\) vs. \(2(81)=162\).
Time = 1.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \left (3626370 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+2780217 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-846153 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1489740 x^{2} \sqrt {-10 x^{2}-x +3}-725274 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+221494 x \sqrt {-10 x^{2}-x +3}-478870 \sqrt {-10 x^{2}-x +3}\right )}{83006 \left (2+3 x \right ) \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(209\) |
-1/83006*(1-2*x)^(1/2)*(3626370*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10 *x^2-x+3)^(1/2))*x^3+2780217*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^ 2-x+3)^(1/2))*x^2-846153*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+ 3)^(1/2))*x+1489740*x^2*(-10*x^2-x+3)^(1/2)-725274*7^(1/2)*arctan(1/14*(37 *x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+221494*x*(-10*x^2-x+3)^(1/2)-478870*(- 10*x^2-x+3)^(1/2))/(2+3*x)/(-1+2*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\frac {120879 \, \sqrt {7} {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (106410 \, x^{2} + 15821 \, x - 34205\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{83006 \, {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}} \]
1/83006*(120879*sqrt(7)*(30*x^3 + 23*x^2 - 7*x - 6)*arctan(1/14*sqrt(7)*(3 7*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(106410*x^2 + 15821*x - 34205)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(30*x^3 + 23*x^2 - 7*x - 6)
\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {999}{686} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {35470 \, x}{5929 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {18373}{5929 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3}{7 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
-999/686*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 35470 /5929*x/sqrt(-10*x^2 - x + 3) - 18373/5929/sqrt(-10*x^2 - x + 3) + 3/7/(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. 278 vs. \(2 (81) = 162\).
Time = 0.37 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.57 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=-\frac {999}{6860} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{242} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {16 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{29645 \, {\left (2 \, x - 1\right )}} - \frac {594 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{49 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
-999/6860*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)))) - 25/242*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 6/29645*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 594/49*sqrt(10)* ((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*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)
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,d x \]