Integrand size = 26, antiderivative size = 151 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=-\frac {165 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {5 \sqrt {1-2 x} (3+5 x)^{3/2}}{196 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{77 (2+3 x)^3}+\frac {4 (3+5 x)^{7/2}}{77 \sqrt {1-2 x} (2+3 x)^3}-\frac {1815 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]
-1815/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+4/77*( 3+5*x)^(7/2)/(2+3*x)^3/(1-2*x)^(1/2)-5/196*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+ 3*x)^2-1/77*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3-165/2744*(1-2*x)^(1/2)*( 3+5*x)^(1/2)/(2+3*x)
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.60 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {7 \sqrt {3+5 x} \left (2448+17666 x+37405 x^2+24670 x^3\right )-1815 \sqrt {7-14 x} (2+3 x)^3 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {1-2 x} (2+3 x)^3} \]
(7*Sqrt[3 + 5*x]*(2448 + 17666*x + 37405*x^2 + 24670*x^3) - 1815*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sq rt[1 - 2*x]*(2 + 3*x)^3)
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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{3/2} (3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {3}{11} \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {4 (5 x+3)^{7/2}}{77 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {3}{11} \left (\frac {55}{42} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {4 (5 x+3)^{7/2}}{77 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {3}{11} \left (\frac {55}{42} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {4 (5 x+3)^{7/2}}{77 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {3}{11} \left (\frac {55}{42} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {4 (5 x+3)^{7/2}}{77 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {3}{11} \left (\frac {55}{42} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {4 (5 x+3)^{7/2}}{77 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{11} \left (\frac {55}{42} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {4 (5 x+3)^{7/2}}{77 \sqrt {1-2 x} (3 x+2)^3}\) |
(4*(3 + 5*x)^(7/2))/(77*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (3*(-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))/11
3.26.47.3.1 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 = 1.20 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(\frac {\left (98010 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+147015 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+32670 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-345380 x^{3} \sqrt {-10 x^{2}-x +3}-36300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -523670 x^{2} \sqrt {-10 x^{2}-x +3}-14520 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-247324 x \sqrt {-10 x^{2}-x +3}-34272 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{38416 \left (2+3 x \right )^{3} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(257\) |
1/38416*(98010*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))* x^4+147015*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+ 32670*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-34538 0*x^3*(-10*x^2-x+3)^(1/2)-36300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10 *x^2-x+3)^(1/2))*x-523670*x^2*(-10*x^2-x+3)^(1/2)-14520*7^(1/2)*arctan(1/1 4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-247324*x*(-10*x^2-x+3)^(1/2)-3427 2*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)/(-10 *x^2-x+3)^(1/2)
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=-\frac {1815 \, \sqrt {7} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (24670 \, x^{3} + 37405 \, x^{2} + 17666 \, x + 2448\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38416 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
-1/38416*(1815*sqrt(7)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*arctan(1/14*s qrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(24 670*x^3 + 37405*x^2 + 17666*x + 2448)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(54*x^ 4 + 81*x^3 + 18*x^2 - 20*x - 8)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{4}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.40 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {1815}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {61675 \, x}{37044 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {14335}{74088 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{567 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {83}{2268 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {3175}{10584 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
1815/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 616 75/37044*x/sqrt(-10*x^2 - x + 3) + 14335/74088/sqrt(-10*x^2 - x + 3) + 1/5 67/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt( -10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) - 83/2268/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 3175/ 10584/(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. 336 vs. \(2 (118) = 236\).
Time = 0.58 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.23 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {363}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {484 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{12005 \, {\left (2 \, x - 1\right )}} - \frac {121 \, \sqrt {10} {\left (137 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 105280 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {25636800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {102547200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9604 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
363/76832*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)))) - 484/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 121/9604*sqrt(10)*(137*((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 + 105280 *((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sq rt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 25636800*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 102547200*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*sq rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^4} \,d x \]