Integrand size = 26, antiderivative size = 144 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {5 \sqrt {3+5 x}}{42 \sqrt {1-2 x}}+\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)}-\frac {5 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}} \]
-5/196*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+11/21*(3+5* x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^2+5/42*(3+5*x)^(1/2)/(1-2*x)^(1/2)-3/14*(3+ 5*x)^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)-5/28*(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(1/2 )
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {7 \sqrt {3+5 x} \left (-36-91 x+60 x^2+180 x^3\right )-15 \sqrt {7-14 x} (-1+2 x) (2+3 x)^2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{588 (1-2 x)^{3/2} (2+3 x)^2} \]
-1/588*(7*Sqrt[3 + 5*x]*(-36 - 91*x + 60*x^2 + 180*x^3) - 15*Sqrt[7 - 14*x ]*(-1 + 2*x)*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(( 1 - 2*x)^(3/2)*(2 + 3*x)^2)
Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {109, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{(1-2 x)^{5/2} (3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {1}{21} \int -\frac {465 x+268}{2 (1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \int \frac {465 x+268}{(1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{14} \int \frac {35 (72 x+41)}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {9 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {5}{2} \int \frac {72 x+41}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {9 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {5}{2} \left (\frac {1}{7} \int \frac {7 (60 x+47)}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}\right )-\frac {9 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {5}{2} \left (\frac {1}{2} \int \frac {60 x+47}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx-\frac {3 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}\right )-\frac {9 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{42} \left (\frac {5}{2} \left (\frac {1}{2} \left (\frac {4 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {2}{77} \int -\frac {231}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {3 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}\right )-\frac {9 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {5}{2} \left (\frac {1}{2} \left (3 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {4 \sqrt {5 x+3}}{\sqrt {1-2 x}}\right )-\frac {3 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}\right )-\frac {9 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{42} \left (\frac {5}{2} \left (\frac {1}{2} \left (6 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {4 \sqrt {5 x+3}}{\sqrt {1-2 x}}\right )-\frac {3 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}\right )-\frac {9 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{42} \left (\frac {5}{2} \left (\frac {1}{2} \left (\frac {4 \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {6 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}\right )-\frac {3 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)}\right )-\frac {9 \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}\right )+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}\) |
(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + ((-9*Sqrt[3 + 5*x])/ (Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*((-3*Sqrt[3 + 5*x])/(Sqrt[1 - 2*x]*(2 + 3 *x)) + ((4*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] - (6*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7] *Sqrt[3 + 5*x])])/Sqrt[7])/2))/2)/42
3.26.96.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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] && ILtQ[m, -1]
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. \(256\) vs. \(2(111)=222\).
Time = 1.22 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.78
| method | result | size |
| default | \(\frac {\left (540 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+180 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-345 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-2520 x^{3} \sqrt {-10 x^{2}-x +3}-60 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -840 x^{2} \sqrt {-10 x^{2}-x +3}+60 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1274 x \sqrt {-10 x^{2}-x +3}+504 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1176 \left (2+3 x \right )^{2} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(257\) |
1/1176*(540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4 +180*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-345*7^ (1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-2520*x^3*(-10 *x^2-x+3)^(1/2)-60*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 2))*x-840*x^2*(-10*x^2-x+3)^(1/2)+60*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2) /(-10*x^2-x+3)^(1/2))+1274*x*(-10*x^2-x+3)^(1/2)+504*(-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 = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.81 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {15 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (180 \, x^{3} + 60 \, x^{2} - 91 \, x - 36\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1176 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
-1/1176*(15*sqrt(7)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*arctan(1/14*sqrt( 7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(180*x^ 3 + 60*x^2 - 91*x - 36)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 2 3*x^2 - 4*x + 4)
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {5}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {25 \, x}{42 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {5}{84 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {125 \, x}{126 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1}{378 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {43}{756 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {205}{252 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
5/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 25/42*x/ sqrt(-10*x^2 - x + 3) + 5/84/sqrt(-10*x^2 - x + 3) + 125/126*x/(-10*x^2 - x + 3)^(3/2) + 1/378/(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)) - 43/756/(3*(-10*x^2 - x + 3)^(3/2)* x + 2*(-10*x^2 - x + 3)^(3/2)) + 205/252/(-10*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (111) = 222\).
Time = 0.52 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.02 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {1}{784} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {8 \, {\left (157 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1056 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{180075 \, {\left (2 \, x - 1\right )}^{2}} - \frac {33 \, \sqrt {10} {\left (83 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {41720 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {166880 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4802 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
1/784*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqr t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 8/180075*(157*sqrt(5)*(5*x + 3) - 1056*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 33/4802*sqrt(10)*(83*((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 + 41720*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 166880*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)*sqrt (-10*x + 5) - sqrt(22)))^2 + 280)^2
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3} \,d x \]