Integrand size = 26, antiderivative size = 180 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=-\frac {22627 \sqrt {1-2 x} \sqrt {3+5 x}}{43904 (2+3 x)}-\frac {2057 \sqrt {1-2 x} (3+5 x)^{3/2}}{9408 (2+3 x)^2}-\frac {187 \sqrt {1-2 x} (3+5 x)^{5/2}}{1680 (2+3 x)^3}+\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2}}{35 (2+3 x)^5}+\frac {17 \sqrt {1-2 x} (3+5 x)^{7/2}}{40 (2+3 x)^4}-\frac {248897 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}} \]
3/35*(1-2*x)^(3/2)*(3+5*x)^(7/2)/(2+3*x)^5-248897/307328*arctan(1/7*(1-2*x )^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-2057/9408*(3+5*x)^(3/2)*(1-2*x)^(1/ 2)/(2+3*x)^2-187/1680*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3+17/40*(3+5*x)^ (7/2)*(1-2*x)^(1/2)/(2+3*x)^4-22627/43904*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3 *x)
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (5112864+32206264 x+74550556 x^2+74915550 x^3+27422145 x^4\right )}{(2+3 x)^5}-3733455 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4609920} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5112864 + 32206264*x + 74550556*x^2 + 749 15550*x^3 + 27422145*x^4))/(2 + 3*x)^5 - 3733455*Sqrt[7]*ArcTan[Sqrt[1 - 2 *x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4609920
Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {107, 105, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {17}{10} \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{(3 x+2)^5}dx+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {17}{10} \left (\frac {11}{8} \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {17}{10} \left (\frac {11}{8} \left (\frac {55}{42} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {17}{10} \left (\frac {11}{8} \left (\frac {55}{42} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {17}{10} \left (\frac {11}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {17}{10} \left (\frac {11}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {17}{10} \left (\frac {11}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{7/2}}{4 (3 x+2)^4}\right )+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{35 (3 x+2)^5}\) |
(3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(35*(2 + 3*x)^5) + (17*((Sqrt[1 - 2*x] *(3 + 5*x)^(7/2))/(4*(2 + 3*x)^4) + (11*(-1/21*(Sqrt[1 - 2*x]*(3 + 5*x)^(5 /2))/(2 + 3*x)^3 + (55*(-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) - (11*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])))/28))/42))/8))/10
3.23.96.3.1 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 = 1.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (27422145 x^{4}+74915550 x^{3}+74550556 x^{2}+32206264 x +5112864\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{658560 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {248897 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{614656 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(134\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (907229565 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+3024098550 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+4032131400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+383910030 x^{4} \sqrt {-10 x^{2}-x +3}+2688087600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1048817700 x^{3} \sqrt {-10 x^{2}-x +3}+896029200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1043707784 x^{2} \sqrt {-10 x^{2}-x +3}+119470560 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+450887696 x \sqrt {-10 x^{2}-x +3}+71580096 \sqrt {-10 x^{2}-x +3}\right )}{9219840 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(298\) |
-1/658560*(-1+2*x)*(3+5*x)^(1/2)*(27422145*x^4+74915550*x^3+74550556*x^2+3 2206264*x+5112864)/(2+3*x)^5/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^( 1/2)/(1-2*x)^(1/2)+248897/614656*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 = 131, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=-\frac {3733455 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (27422145 \, x^{4} + 74915550 \, x^{3} + 74550556 \, x^{2} + 32206264 \, x + 5112864\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9219840 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
-1/9219840*(3733455*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240* x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x ^2 + x - 3)) - 14*(27422145*x^4 + 74915550*x^3 + 74550556*x^2 + 32206264*x + 5112864)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{6}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {248897}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {10285}{32928} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{105 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{40 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {45 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{784 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {6171 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {76109 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \]
248897/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 10285/32928*sqrt(-10*x^2 - x + 3) + 1/105*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) - 3/40*(-10*x^2 - x + 3)^(3/ 2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 45/784*(-10*x^2 - x + 3)^(3/ 2)/(27*x^3 + 54*x^2 + 36*x + 8) + 6171/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^ 2 + 12*x + 4) - 76109/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (141) = 282\).
Time = 0.63 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {248897}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {14641 \, \sqrt {10} {\left (51 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 66640 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 34119680 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 3618944000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {313474560000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1253898240000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{65856 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
248897/6146560*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)))) - 14641/65856*sqrt(10)*(51*((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)))^9 + 66640*((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)))^7 + 34119680*((sqrt (2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s qrt(-10*x + 5) - sqrt(22)))^5 - 3618944000*((sqrt(2)*sqrt(-10*x + 5) - sqr t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22) ))^3 - 313474560000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1 253898240000*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)*sqr t(-10*x + 5) - sqrt(22)))^2 + 280)^5
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^6} \,d x \]