Integrand size = 26, antiderivative size = 173 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=-\frac {107245 \sqrt {3+5 x}}{230496 \sqrt {1-2 x}}-\frac {\sqrt {3+5 x}}{28 \sqrt {1-2 x} (2+3 x)^4}+\frac {17 \sqrt {3+5 x}}{1176 \sqrt {1-2 x} (2+3 x)^3}+\frac {313 \sqrt {3+5 x}}{1568 \sqrt {1-2 x} (2+3 x)^2}+\frac {34635 \sqrt {3+5 x}}{21952 \sqrt {1-2 x} (2+3 x)}-\frac {1244755 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{153664 \sqrt {7}} \] Output:
-107245/230496*(3+5*x)^(1/2)/(1-2*x)^(1/2)-1/28*(3+5*x)^(1/2)/(1-2*x)^(1/2 )/(2+3*x)^4+17/1176*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^3+313/1568*(3+5*x) ^(1/2)/(1-2*x)^(1/2)/(2+3*x)^2+34635/21952*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+ 3*x)-1244755/1075648*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2 ))
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {-7 \sqrt {3+5 x} \left (-917264-2239092 x+2075184 x^2+8897265 x^3+5791230 x^4\right )-1244755 \sqrt {7-14 x} (2+3 x)^4 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1075648 \sqrt {1-2 x} (2+3 x)^4} \] Input:
Integrate[Sqrt[3 + 5*x]/((1 - 2*x)^(3/2)*(2 + 3*x)^5),x]
Output:
(-7*Sqrt[3 + 5*x]*(-917264 - 2239092*x + 2075184*x^2 + 8897265*x^3 + 57912 30*x^4) - 1244755*Sqrt[7 - 14*x]*(2 + 3*x)^4*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7] *Sqrt[3 + 5*x])])/(1075648*Sqrt[1 - 2*x]*(2 + 3*x)^4)
Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {110, 27, 168, 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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{(1-2 x)^{3/2} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {2}{7} \int -\frac {120 x+71}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {120 x+71}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{28} \int \frac {1620 x+989}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \int \frac {1620 x+989}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \left (\frac {1}{21} \int \frac {105 (104 x+125)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \left (\frac {5}{2} \int \frac {104 x+125}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{14} \int \frac {4923-3340 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {167 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \int \frac {4923-3340 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {167 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {248951}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {21449 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {167 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {248951}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {21449 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {167 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {248951}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {21449 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {167 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{56} \left (\frac {5}{2} \left (\frac {1}{28} \left (\frac {21449 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {248951 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {167 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^3}\right )-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}\) |
Input:
Int[Sqrt[3 + 5*x]/((1 - 2*x)^(3/2)*(2 + 3*x)^5),x]
Output:
(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^4) + ((-27*Sqrt[1 - 2*x]*Sqrt [3 + 5*x])/(28*(2 + 3*x)^4) + ((-13*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x) ^3 + (5*((167*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((21449*Sqrt [1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (248951*ArcTan[Sqrt[1 - 2*x]/(Sqr t[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/2)/56)/7
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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (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. \(297\) vs. \(2(134)=268\).
Time = 0.23 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.72
| method | result | size |
| default | \(-\frac {\left (201650310 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+436909005 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+268867080 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+81077220 x^{4} \sqrt {-10 x^{2}-x +3}-29874120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+124561710 x^{3} \sqrt {-10 x^{2}-x +3}-79664320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +29052576 x^{2} \sqrt {-10 x^{2}-x +3}-19916080 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-31347288 x \sqrt {-10 x^{2}-x +3}-12841696 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{2151296 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \sqrt {-10 x^{2}-x +3}}\) | \(298\) |
Input:
int((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^5,x,method=_RETURNVERBOSE)
Output:
-1/2151296*(201650310*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^ (1/2))*x^5+436909005*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^( 1/2))*x^4+268867080*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1 /2))*x^3+81077220*x^4*(-10*x^2-x+3)^(1/2)-29874120*7^(1/2)*arctan(1/14*(37 *x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+124561710*x^3*(-10*x^2-x+3)^(1/2)- 79664320*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+2905 2576*x^2*(-10*x^2-x+3)^(1/2)-19916080*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2 )/(-10*x^2-x+3)^(1/2))-31347288*x*(-10*x^2-x+3)^(1/2)-12841696*(-10*x^2-x+ 3)^(1/2))*(3+5*x)^(1/2)/(2+3*x)^4/(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=-\frac {1244755 \, \sqrt {7} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (5791230 \, x^{4} + 8897265 \, x^{3} + 2075184 \, x^{2} - 2239092 \, x - 917264\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2151296 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \] Input:
integrate((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^5,x, algorithm="fricas")
Output:
-1/2151296*(1244755*sqrt(7)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(5791230*x^4 + 8897265*x^3 + 2075184*x^2 - 2239092*x - 9172 64)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)
\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{5}}\, dx \] Input:
integrate((3+5*x)**(1/2)/(1-2*x)**(3/2)/(2+3*x)**5,x)
Output:
Integral(sqrt(5*x + 3)/((1 - 2*x)**(3/2)*(3*x + 2)**5), x)
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (134) = 268\).
Time = 0.12 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {1244755}{2151296} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {536225 \, x}{230496 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {189585}{153664 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{84 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {227}{3528 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {599}{14112 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {12725}{65856 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \] Input:
integrate((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^5,x, algorithm="maxima")
Output:
1244755/2151296*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 536225/230496*x/sqrt(-10*x^2 - x + 3) + 189585/153664/sqrt(-10*x^2 - x + 3) + 1/84/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(-10*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10* x^2 - x + 3)) - 227/3528/(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)) - 599/ 14112/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(- 10*x^2 - x + 3)) - 12725/65856/(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. 394 vs. \(2 (134) = 268\).
Time = 0.49 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\frac {248951}{4302592} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{84035 \, {\left (2 \, x - 1\right )}} - \frac {33 \, \sqrt {10} {\left (264101 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 272107080 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 72200520000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {5707629760000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {22830519040000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{537824 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \] Input:
integrate((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^5,x, algorithm="giac")
Output:
248951/4302592*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)))) - 32/84035*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/( 2*x - 1) - 33/537824*sqrt(10)*(264101*((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 - 272107080*((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 - 72200520000*((sqrt(2)*sq rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1 0*x + 5) - sqrt(22)))^3 - 5707629760000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 ))/sqrt(5*x + 3) + 22830519040000*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)^4
Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^5} \,d x \] Input:
int((5*x + 3)^(1/2)/((1 - 2*x)^(3/2)*(3*x + 2)^5),x)
Output:
int((5*x + 3)^(1/2)/((1 - 2*x)^(3/2)*(3*x + 2)^5), x)
Time = 0.29 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.88 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx =\text {Too large to display} \] Input:
int((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^5,x)
Output:
(100825155*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sq rt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 268867080*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 + 268867080*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 + 119496480*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*t an(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 19916080*sqr t( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1) *sqrt(5))/sqrt(11))/2))/sqrt(2)) - 100825155*sqrt( - 2*x + 1)*sqrt(7)*atan ((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sq rt(2))*x**4 - 268867080*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 - 2688670 80*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 - 119496480*sqrt( - 2*x + 1)*s qrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt( 11))/2))/sqrt(2))*x - 19916080*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + s qrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 40538 610*sqrt(5*x + 3)*x**4 - 62280855*sqrt(5*x + 3)*x**3 - 14526288*sqrt(5*x + 3)*x**2 + 15673644*sqrt(5*x + 3)*x + 6420848*sqrt(5*x + 3))/(1075648*sqrt ( - 2*x + 1)*(81*x**4 + 216*x**3 + 216*x**2 + 96*x + 16))