Integrand size = 26, antiderivative size = 151 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {65 \sqrt {1-2 x} \sqrt {3+5 x}}{1372 (2+3 x)}+\frac {65 \sqrt {1-2 x} (3+5 x)^{3/2}}{3234 (2+3 x)^2}+\frac {26 (3+5 x)^{5/2}}{231 \sqrt {1-2 x} (2+3 x)^2}+\frac {4 (3+5 x)^{7/2}}{231 (1-2 x)^{3/2} (2+3 x)^2}+\frac {715 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \]
4/231*(3+5*x)^(7/2)/(1-2*x)^(3/2)/(2+3*x)^2+715/9604*arctan(1/7*(1-2*x)^(1 /2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+26/231*(3+5*x)^(5/2)/(2+3*x)^2/(1-2*x)^ (1/2)+65/3234*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2+65/1372*(1-2*x)^(1/2)* (3+5*x)^(1/2)/(2+3*x)
Time = 2.24 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.99 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {5 \left (-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (-6732-13627 x+1620 x^2+10260 x^3\right )}{5 \left (-2+x+6 x^2\right )^2}-429 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-429 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{28812} \]
(5*((-7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-6732 - 13627*x + 1620*x^2 + 10260*x^ 3))/(5*(-2 + x + 6*x^2)^2) - 429*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])] *Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] - 429*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]))/28812
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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{5/2} (3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {13}{33} \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{3/2} (3 x+2)^3}dx+\frac {4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {13}{33} \left (\frac {2 (5 x+3)^{5/2}}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx\right )+\frac {4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {13}{33} \left (\frac {2 (5 x+3)^{5/2}}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )\right )+\frac {4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {13}{33} \left (\frac {2 (5 x+3)^{5/2}}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )\right )+\frac {4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {13}{33} \left (\frac {2 (5 x+3)^{5/2}}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )\right )+\frac {4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {13}{33} \left (\frac {2 (5 x+3)^{5/2}}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )\right )+\frac {4 (5 x+3)^{7/2}}{231 (1-2 x)^{3/2} (3 x+2)^2}\) |
(4*(3 + 5*x)^(7/2))/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (13*((2*(3 + 5*x)^ (5/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*(-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))/7))/3 3
3.27.5.3.1 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])
Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(118)=236\).
Time = 1.24 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(-\frac {\left (77220 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+25740 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-49335 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+143640 x^{3} \sqrt {-10 x^{2}-x +3}-8580 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +22680 x^{2} \sqrt {-10 x^{2}-x +3}+8580 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-190778 x \sqrt {-10 x^{2}-x +3}-94248 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{57624 \left (2+3 x \right )^{2} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(257\) |
-1/57624*(77220*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x^4+25740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3- 49335*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+14364 0*x^3*(-10*x^2-x+3)^(1/2)-8580*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10* x^2-x+3)^(1/2))*x+22680*x^2*(-10*x^2-x+3)^(1/2)+8580*7^(1/2)*arctan(1/14*( 37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-190778*x*(-10*x^2-x+3)^(1/2)-94248*( -10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2/(-1+2*x)^2/(-10* x^2-x+3)^(1/2)
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {2145 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (10260 \, x^{3} + 1620 \, x^{2} - 13627 \, x - 6732\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{57624 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
1/57624*(2145*sqrt(7)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*arctan(1/14*sqr t(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1026 0*x^3 + 1620*x^2 - 13627*x - 6732)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.14 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {715}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {475 \, x}{686 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {215}{4116 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {17375 \, x}{2646 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{1134 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1}{36 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {60695}{15876 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
-715/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 475 /686*x/sqrt(-10*x^2 - x + 3) - 215/4116/sqrt(-10*x^2 - x + 3) + 17375/2646 *x/(-10*x^2 - x + 3)^(3/2) - 1/1134/(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)) + 1/36/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 60695/15876/(-10*x^2 - x + 3 )^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (118) = 236\).
Time = 0.53 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.93 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {143}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {22 \, {\left (104 \, \sqrt {5} {\left (5 \, x + 3\right )} - 957 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{180075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {11 \, \sqrt {10} {\left (223 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {80920 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {323680 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4802 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
-143/38416*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)))) - 22/180075*(104*sqrt(5)*(5*x + 3) - 957*sqrt(5))*sqrt (5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 11/4802*sqrt(10)*(223*((sqrt(2)*sq rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 0*x + 5) - sqrt(22)))^3 + 80920*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt( 5*x + 3) - 323680*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((s qrt(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. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3} \,d x \]