Integrand size = 26, antiderivative size = 166 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}-\frac {4246733 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}} \] Output:
-204595/168*(1-2*x)^(1/2)/(3+5*x)^(3/2)+7/9*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x )^(3/2)+301/36*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3/2)+24469/168*(1-2*x)^(1/ 2)/(2+3*x)/(3+5*x)^(3/2)+618645/56*(1-2*x)^(1/2)/(3+5*x)^(1/2)-4246733/392 *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 = 84, normalized size of antiderivative = 0.51 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\frac {\sqrt {1-2 x} \left (43006496+267610802 x+623901861 x^2+645909120 x^3+250551225 x^4\right )}{168 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {4246733 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}} \] Input:
Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)^(5/2)),x]
Output:
(Sqrt[1 - 2*x]*(43006496 + 267610802*x + 623901861*x^2 + 645909120*x^3 + 2 50551225*x^4))/(168*(2 + 3*x)^3*(3 + 5*x)^(3/2)) - (4246733*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(56*Sqrt[7])
Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {109, 27, 168, 27, 168, 27, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^4 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{9} \int \frac {345-536 x}{2 \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{5/2}}dx+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \int \frac {345-536 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{5/2}}dx+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{18} \left (\frac {1}{14} \int \frac {21 (4143-6020 x)}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \int \frac {4143-6020 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \left (\frac {1}{7} \int \frac {763071-978760 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {24469 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \left (\frac {1}{14} \int \frac {763071-978760 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {24469 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \left (\frac {1}{14} \left (-\frac {2}{33} \int \frac {99 (869991-818380 x)}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {409190 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24469 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \left (\frac {1}{14} \left (-3 \int \frac {869991-818380 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {409190 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24469 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \left (\frac {1}{14} \left (-3 \left (-\frac {2}{11} \int \frac {46714063}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {1237290 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {409190 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24469 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \left (\frac {1}{14} \left (-3 \left (-4246733 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {1237290 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {409190 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24469 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \left (\frac {1}{14} \left (-3 \left (-8493466 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {1237290 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {409190 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24469 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{18} \left (\frac {3}{4} \left (\frac {1}{14} \left (-3 \left (\frac {8493466 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}-\frac {1237290 \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {409190 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24469 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {301 \sqrt {1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}\) |
Input:
Int[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)^(5/2)),x]
Output:
(7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + ((301*Sqrt[1 - 2*x])/( 2*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (3*((24469*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^(3/2)) + ((-409190*Sqrt[1 - 2*x])/(3 + 5*x)^(3/2) - 3*((-1237290*S qrt[1 - 2*x])/Sqrt[3 + 5*x] + (8493466*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[ 3 + 5*x])])/Sqrt[7]))/14))/4)/18
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. \(297\) vs. \(2(127)=254\).
Time = 0.22 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.80
| method | result | size |
| default | \(\frac {\left (8599634325 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+27518829840 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+35201169837 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+3507717150 x^{4} \sqrt {-10 x^{2}-x +3}+22499191434 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+9042727680 x^{3} \sqrt {-10 x^{2}-x +3}+7185472236 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +8734626054 x^{2} \sqrt {-10 x^{2}-x +3}+917294328 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3746551228 x \sqrt {-10 x^{2}-x +3}+602090944 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{2352 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(298\) |
Input:
int((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/2352*(8599634325*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 2))*x^5+27518829840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1 /2))*x^4+35201169837*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^( 1/2))*x^3+3507717150*x^4*(-10*x^2-x+3)^(1/2)+22499191434*7^(1/2)*arctan(1/ 14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+9042727680*x^3*(-10*x^2-x+3) ^(1/2)+7185472236*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2 ))*x+8734626054*x^2*(-10*x^2-x+3)^(1/2)+917294328*7^(1/2)*arctan(1/14*(37* x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+3746551228*x*(-10*x^2-x+3)^(1/2)+602090 944*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5* x)^(3/2)
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {12740199 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (250551225 \, x^{4} + 645909120 \, x^{3} + 623901861 \, x^{2} + 267610802 \, x + 43006496\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2352 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-1/2352*(12740199*sqrt(7)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564* x + 72)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x ^2 + x - 3)) - 14*(250551225*x^4 + 645909120*x^3 + 623901861*x^2 + 2676108 02*x + 43006496)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763* x^3 + 1766*x^2 + 564*x + 72)
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**(5/2),x)
Output:
Integral((1 - 2*x)**(3/2)/((3*x + 2)**4*(5*x + 3)**(5/2)), x)
Time = 0.12 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.45 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\frac {4246733}{784} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {618645 \, x}{28 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1937773}{168 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {199895 \, x}{36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {343}{81 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {4655}{108 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {165739}{216 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {1943461}{648 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
4246733/784*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 61 8645/28*x/sqrt(-10*x^2 - x + 3) + 1937773/168/sqrt(-10*x^2 - x + 3) + 1998 95/36*x/(-10*x^2 - x + 3)^(3/2) + 343/81/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^ 2 - x + 3)^(3/2)) + 4655/108/(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)) + 165739/216/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 1943461/648/(-10*x^2 - x + 3) ^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (127) = 254\).
Time = 0.32 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.59 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=-\frac {5}{48} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {4246733}{7840} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + 335 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {99 \, \sqrt {10} {\left (21713 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 10391360 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1283172800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5132691200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{28 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \] Input:
integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x, algorithm="giac")
Output:
-5/48*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 4246733/7840*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)))) + 335*sqrt(10)*((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))) + 99/28*sqrt(10)*(21713* ((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr t(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 10391360*((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 + 1283172800*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 5132691200*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)^3
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int((1 - 2*x)^(3/2)/((3*x + 2)^4*(5*x + 3)^(5/2)),x)
Output:
int((1 - 2*x)^(3/2)/((3*x + 2)^4*(5*x + 3)^(5/2)), x)
Time = 0.29 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.00 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x)
Output:
(1719926865*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt ( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 4471809849*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s qrt(11))/2))/sqrt(2))*x**3 + 4357148058*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(3 3) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x **2 + 1885549452*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin( (sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 305764776*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/ sqrt(11))/2))/sqrt(2)) - 1719926865*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 4471809849*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqr t( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 - 4357148058*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/ sqrt(11))/2))/sqrt(2))*x**2 - 1885549452*sqrt(5*x + 3)*sqrt(7)*atan((sqrt( 33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))* x - 305764776*sqrt(5*x + 3)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sq rt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 1753858575*sqrt( - 2*x + 1)*x**4 + 4521363840*sqrt( - 2*x + 1)*x**3 + 4367313027*sqrt( - 2*x + 1)*x **2 + 1873275614*sqrt( - 2*x + 1)*x + 301045472*sqrt( - 2*x + 1))/(1176*sq rt(5*x + 3)*(135*x**4 + 351*x**3 + 342*x**2 + 148*x + 24))