Integrand size = 26, antiderivative size = 151 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=-\frac {3993 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}-\frac {121 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac {11 \sqrt {1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}-\frac {43923 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \] Output:
-3993*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(6272+9408*x)-121/224*(1-2*x)^(1/2)*(3+5 *x)^(3/2)/(2+3*x)^2+1/4*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4+11/8*(1-2*x) ^(1/2)*(3+5*x)^(5/2)/(2+3*x)^3-43923/21952*7^(1/2)*arctan(1/7*(1-2*x)^(1/2 )*7^(1/2)/(3+5*x)^(1/2))
Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (32400+145940 x+213240 x^2+100159 x^3\right )}{(2+3 x)^4}-43923 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952} \] Input:
Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^5,x]
Output:
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(32400 + 145940*x + 213240*x^2 + 100159*x^ 3))/(2 + 3*x)^4 - 43923*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x ])])/21952
Time = 0.23 (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 = {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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {33}{8} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^4}dx+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {33}{8} \left (\frac {11}{6} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {33}{8} \left (\frac {11}{6} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}\) |
Input:
Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^5,x]
Output:
((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(4*(2 + 3*x)^4) + (33*((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) - (11*Ar cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])))/28))/6))/8
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_)^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 = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (100159 x^{3}+213240 x^{2}+145940 x +32400\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3136 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {43923 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{43904 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (3557763 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+9487368 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+9487368 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1402226 x^{3} \sqrt {-10 x^{2}-x +3}+4216608 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2985360 x^{2} \sqrt {-10 x^{2}-x +3}+702768 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2043160 x \sqrt {-10 x^{2}-x +3}+453600 \sqrt {-10 x^{2}-x +3}\right )}{43904 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
Input:
int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^5,x,method=_RETURNVERBOSE)
Output:
-1/3136*(-1+2*x)*(3+5*x)^(1/2)*(100159*x^3+213240*x^2+145940*x+32400)/(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)+4392 3/43904*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 = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=-\frac {43923 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (100159 \, x^{3} + 213240 \, x^{2} + 145940 \, x + 32400\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43904 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^5,x, algorithm="fricas")
Output:
-1/43904*(43923*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 *(100159*x^3 + 213240*x^2 + 145940*x + 32400)*sqrt(5*x + 3)*sqrt(-2*x + 1) )/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{5}}\, dx \] Input:
integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**5,x)
Output:
Integral((1 - 2*x)**(3/2)*(5*x + 3)**(3/2)/(3*x + 2)**5, x)
Time = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.23 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {8245}{16464} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{392 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {4947 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {67155}{10976} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {43923}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {59169}{21952} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {19573 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{65856 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^5,x, algorithm="maxima")
Output:
8245/16464*(-10*x^2 - x + 3)^(3/2) + 3/28*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 111/392*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 4947/10976*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 67155/10976*sqrt(-10*x^2 - x + 3)*x + 43923/43904*sqrt(7)*arcsin(37/ 11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 59169/21952*sqrt(-10*x^2 - x + 3 ) + 19573/65856*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).
Time = 0.32 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\frac {43923}{439040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {14641 \, \sqrt {10} {\left (3 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 3080 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 862400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {65856000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {263424000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1568 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^5,x, algorithm="giac")
Output:
43923/439040*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)))) - 14641/1568*sqrt(10)*(3*((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 + 3080*((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)))^5 - 862400*((sqrt(2)*sqr t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10 *x + 5) - sqrt(22)))^3 - 65856000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr t(5*x + 3) + 263424000*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)/(s qrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^5} \,d x \] Input:
int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^5,x)
Output:
int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^5, x)
Time = 0.24 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.93 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^5,x)
Output:
(3557763*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt (5))/sqrt(11))/2))/sqrt(2))*x**4 + 9487368*sqrt(7)*atan((sqrt(33) - sqrt(3 5)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 94873 68*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s qrt(11))/2))/sqrt(2))*x**2 + 4216608*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan (asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 702768*sqrt(7) *atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2 ))/sqrt(2)) - 3557763*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 9487368*sqrt(7)*atan((sqrt (33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) *x**3 - 9487368*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 4216608*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 7 02768*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5) )/sqrt(11))/2))/sqrt(2)) + 701113*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 14 92680*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 1021580*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 226800*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(21952*(81*x**4 + 216* x**3 + 216*x**2 + 96*x + 16))