Integrand size = 26, antiderivative size = 151 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=-\frac {13915 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {1265 \sqrt {1-2 x} (3+5 x)^{3/2}}{4704 (2+3 x)^2}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{5/2}}{28 (2+3 x)^4}+\frac {115 \sqrt {1-2 x} (3+5 x)^{5/2}}{168 (2+3 x)^3}-\frac {153065 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \]
3/28*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4-153065/153664*arctan(1/7*(1-2*x )^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1265/4704*(3+5*x)^(3/2)*(1-2*x)^(1/ 2)/(2+3*x)^2+115/168*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3-13915/21952*(1- 2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (328464+1512052 x+2269240 x^2+1104135 x^3\right )}{(2+3 x)^4}-459195 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{460992} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(328464 + 1512052*x + 2269240*x^2 + 110413 5*x^3))/(2 + 3*x)^4 - 459195*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/460992
Time = 0.22 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {107, 105, 105, 105, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {115}{56} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^4}dx+\frac {3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {115}{56} \left (\frac {11}{6} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {115}{56} \left (\frac {11}{6} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {115}{56} \left (\frac {11}{6} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {115}{56} \left (\frac {11}{6} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {115}{56} \left (\frac {11}{6} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{28 (3 x+2)^4}\) |
(3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(28*(2 + 3*x)^4) + (115*((Sqrt[1 - 2*x ]*(3 + 5*x)^(5/2))/(3*(2 + 3*x)^3) + (11*(-1/14*(Sqrt[1 - 2*x]*(3 + 5*x)^( 3/2))/(2 + 3*x)^2 + (33*(-1/7*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x) - (1 1*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])))/28))/6))/56
3.23.83.3.1 Defintions of rubi rules used
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[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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])
Time = 3.64 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (1104135 x^{3}+2269240 x^{2}+1512052 x +328464\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{65856 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {153065 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (37194795 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+99186120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+99186120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+15457890 x^{3} \sqrt {-10 x^{2}-x +3}+44082720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +31769360 x^{2} \sqrt {-10 x^{2}-x +3}+7347120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+21168728 x \sqrt {-10 x^{2}-x +3}+4598496 \sqrt {-10 x^{2}-x +3}\right )}{921984 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
-1/65856*(-1+2*x)*(3+5*x)^(1/2)*(1104135*x^3+2269240*x^2+1512052*x+328464) /(2+3*x)^4/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2) +153065/307328*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67 +111*x)^(1/2))*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=-\frac {459195 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1104135 \, x^{3} + 2269240 \, x^{2} + 1512052 \, x + 328464\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{921984 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
-1/921984*(459195*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan( 1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1104135*x^3 + 2269240*x^2 + 1512052*x + 328464)*sqrt(5*x + 3)*sqrt(-2* x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{5}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {153065}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {6325}{16464} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {95 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1176 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {3795 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {46805 \, \sqrt {-10 \, x^{2} - x + 3}}{65856 \, {\left (3 \, x + 2\right )}} \]
153065/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 6325/16464*sqrt(-10*x^2 - x + 3) - 1/28*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 95/1176*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 3795/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4 ) - 46805/65856*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).
Time = 0.47 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {30613}{614656} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {6655 \, \sqrt {10} {\left (69 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 70840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 15821120 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {1514688000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {6058752000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{32928 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
30613/614656*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)))) - 6655/32928*sqrt(10)*(69*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr t(22)))^7 + 70840*((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)))^5 - 15821120*((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 - 1514688000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 ))/sqrt(5*x + 3) + 6058752000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr t(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)^4
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^5} \,d x \]