Integrand size = 26, antiderivative size = 151 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}-\frac {145 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}-\frac {415 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}-\frac {2805 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \] Output:
11/7*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^3-2/3*(1-2*x)^(1/2)*(3+5*x)^(1/2) /(2+3*x)^3-145/588*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2-415*(1-2*x)^(1/2) *(3+5*x)^(1/2)/(16464+24696*x)-2805/19208*7^(1/2)*arctan(1/7*(1-2*x)^(1/2) *7^(1/2)/(3+5*x)^(1/2))
Time = 2.35 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.97 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {5 \left (\frac {7 \sqrt {3+5 x} \left (576+3782 x+6135 x^2+2490 x^3\right )}{5 \sqrt {1-2 x} (2+3 x)^3}+561 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+561 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{19208} \] Input:
Integrate[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]
Output:
(5*((7*Sqrt[3 + 5*x]*(576 + 3782*x + 6135*x^2 + 2490*x^3))/(5*Sqrt[1 - 2*x ]*(2 + 3*x)^3) + 561*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5* x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 561*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqr t[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]))/19208
Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.10, 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, 168, 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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{(1-2 x)^{3/2} (3 x+2)^4} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}-\frac {1}{7} \int -\frac {815 x+478}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \int \frac {815 x+478}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{21} \int \frac {35 (112 x+65)}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {28 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{3} \int \frac {112 x+65}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {28 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{3} \left (\frac {1}{14} \int \frac {580 x+359}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {29 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {28 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{3} \left (\frac {1}{28} \int \frac {580 x+359}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {29 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {28 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{3} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {1683}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {83 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {29 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {28 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{3} \left (\frac {1}{28} \left (\frac {1683}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {83 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {29 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {28 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{3} \left (\frac {1}{28} \left (\frac {1683}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {83 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {29 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {28 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {5}{3} \left (\frac {1}{28} \left (-\frac {1683 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {83 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {29 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {28 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\) |
Input:
Int[(3 + 5*x)^(3/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]
Output:
(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + ((-28*Sqrt[1 - 2*x]*Sqr t[3 + 5*x])/(3*(2 + 3*x)^3) + (5*((-29*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((-83*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (1683*ArcT an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/3)/14
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_)^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. \(249\) vs. \(2(118)=236\).
Time = 0.24 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.66
| method | result | size |
| default | \(-\frac {\left (151470 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+227205 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+50490 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-34860 x^{3} \sqrt {-10 x^{2}-x +3}-56100 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -85890 x^{2} \sqrt {-10 x^{2}-x +3}-22440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-52948 x \sqrt {-10 x^{2}-x +3}-8064 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{38416 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \sqrt {-10 x^{2}-x +3}}\) | \(250\) |
Input:
int((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^4,x,method=_RETURNVERBOSE)
Output:
-1/38416*(151470*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2) )*x^4+227205*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^ 3+50490*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-348 60*x^3*(-10*x^2-x+3)^(1/2)-56100*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1 0*x^2-x+3)^(1/2))*x-85890*x^2*(-10*x^2-x+3)^(1/2)-22440*7^(1/2)*arctan(1/1 4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-52948*x*(-10*x^2-x+3)^(1/2)-8064* (-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2)/(-10*x^2-x+3)^( 1/2)
Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=-\frac {2805 \, \sqrt {7} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (2490 \, x^{3} + 6135 \, x^{2} + 3782 \, x + 576\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38416 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="fricas")
Output:
-1/38416*(2805*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 90*x^3 + 6135*x^2 + 3782*x + 576)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)
\[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{4}}\, dx \] Input:
integrate((3+5*x)**(3/2)/(1-2*x)**(3/2)/(2+3*x)**4,x)
Output:
Integral((5*x + 3)**(3/2)/((1 - 2*x)**(3/2)*(3*x + 2)**4), x)
Time = 0.16 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.40 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {2805}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2075 \, x}{12348 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4415}{24696 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1}{189 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {53}{756 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {275}{1176 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="maxima")
Output:
2805/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 207 5/12348*x/sqrt(-10*x^2 - x + 3) + 4415/24696/sqrt(-10*x^2 - x + 3) - 1/189 /(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-1 0*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 53/756/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 275/1176 /(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 = 0.37 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.23 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {561}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {88 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{12005 \, {\left (2 \, x - 1\right )}} - \frac {11 \, \sqrt {10} {\left (1849 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1386560 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {15601600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {62406400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9604 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \] Input:
integrate((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")
Output:
561/76832*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)))) - 88/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 11/9604*sqrt(10)*(1849*((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 + 1386560 *((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 + 15601600*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 62406400*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*sqr t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^4} \,d x \] Input:
int((5*x + 3)^(3/2)/((1 - 2*x)^(3/2)*(3*x + 2)^4),x)
Output:
int((5*x + 3)^(3/2)/((1 - 2*x)^(3/2)*(3*x + 2)^4), x)
Time = 0.27 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.63 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {75735 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{3}+151470 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}+100980 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +22440 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-75735 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{3}-151470 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}-100980 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -22440 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )+17430 \sqrt {5 x +3}\, x^{3}+42945 \sqrt {5 x +3}\, x^{2}+26474 \sqrt {5 x +3}\, x +4032 \sqrt {5 x +3}}{19208 \sqrt {-2 x +1}\, \left (27 x^{3}+54 x^{2}+36 x +8\right )} \] Input:
int((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^4,x)
Output:
(75735*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 151470*sqrt( - 2*x + 1)* sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt (11))/2))/sqrt(2))*x**2 + 100980*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 2 2440*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 75735*sqrt( - 2*x + 1)*sqrt(7)* atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2) )/sqrt(2))*x**3 - 151470*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35 )*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 100980 *sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 22440*sqrt( - 2*x + 1)*sqrt(7)*at an((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/ sqrt(2)) + 17430*sqrt(5*x + 3)*x**3 + 42945*sqrt(5*x + 3)*x**2 + 26474*sqr t(5*x + 3)*x + 4032*sqrt(5*x + 3))/(19208*sqrt( - 2*x + 1)*(27*x**3 + 54*x **2 + 36*x + 8))