Integrand size = 26, antiderivative size = 151 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {16925425 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \] Output:
1/4*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+81/56*(1-2*x)^(1/2)*(3+5*x)^(1/2 )/(2+3*x)^3+14145/1568*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+1479375*(1-2* x)^(1/2)*(3+5*x)^(1/2)/(43904+65856*x)-16925425/153664*7^(1/2)*arctan(1/7* (1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
Time = 2.70 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {25 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (12696112+55729116 x+81668520 x^2+39943125 x^3\right )}{25 (2+3 x)^4}+677017 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+677017 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{153664} \] Input:
Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^5*Sqrt[3 + 5*x]),x]
Output:
(25*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12696112 + 55729116*x + 81668520*x^2 + 39943125*x^3))/(25*(2 + 3*x)^4) + 677017*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sq rt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 677017*Sqrt[7]*A rcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))] ))/153664
Time = 0.25 (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 = {110, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x}}{(3 x+2)^5 \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}-\frac {1}{4} \int -\frac {41-60 x}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \frac {41-60 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{21} \int \frac {15 (511-648 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {5}{14} \int \frac {511-648 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{8} \left (\frac {5}{14} \left (\frac {1}{14} \int \frac {5 (12181-11316 x)}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {2829 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {5}{14} \left (\frac {5}{28} \int \frac {12181-11316 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {2829 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{8} \left (\frac {5}{14} \left (\frac {5}{28} \left (\frac {1}{7} \int \frac {677017}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {59175 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {2829 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {5}{14} \left (\frac {5}{28} \left (\frac {677017}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {59175 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {2829 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{8} \left (\frac {5}{14} \left (\frac {5}{28} \left (\frac {677017}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {59175 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {2829 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{8} \left (\frac {5}{14} \left (\frac {5}{28} \left (\frac {59175 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {677017 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {2829 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4}\) |
Input:
Int[Sqrt[1 - 2*x]/((2 + 3*x)^5*Sqrt[3 + 5*x]),x]
Output:
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4*(2 + 3*x)^4) + ((81*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (5*((2829*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + (5*((59175*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (677017* ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])))/28))/14)/8
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])
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (39943125 x^{3}+81668520 x^{2}+55729116 x +12696112\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{21952 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {16925425 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(129\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1370959425 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+3655891800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+3655891800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+559203750 x^{3} \sqrt {-10 x^{2}-x +3}+1624840800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1143359280 x^{2} \sqrt {-10 x^{2}-x +3}+270806800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+780207624 x \sqrt {-10 x^{2}-x +3}+177745568 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(250\) |
Input:
int((1-2*x)^(1/2)/(2+3*x)^5/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/21952*(-1+2*x)*(3+5*x)^(1/2)*(39943125*x^3+81668520*x^2+55729116*x+1269 6112)/(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)+16925425/307328*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 = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=-\frac {16925425 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (39943125 \, x^{3} + 81668520 \, x^{2} + 55729116 \, x + 12696112\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:
integrate((1-2*x)^(1/2)/(2+3*x)^5/(3+5*x)^(1/2),x, algorithm="fricas")
Output:
-1/307328*(16925425*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arcta n(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(39943125*x^3 + 81668520*x^2 + 55729116*x + 12696112)*sqrt(5*x + 3)*s qrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
\[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\int \frac {\sqrt {1 - 2 x}}{\left (3 x + 2\right )^{5} \sqrt {5 x + 3}}\, dx \] Input:
integrate((1-2*x)**(1/2)/(2+3*x)**5/(3+5*x)**(1/2),x)
Output:
Integral(sqrt(1 - 2*x)/((3*x + 2)**5*sqrt(5*x + 3)), x)
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {16925425}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{4 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {81 \, \sqrt {-10 \, x^{2} - x + 3}}{56 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {14145 \, \sqrt {-10 \, x^{2} - x + 3}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1479375 \, \sqrt {-10 \, x^{2} - x + 3}}{21952 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(1/2)/(2+3*x)^5/(3+5*x)^(1/2),x, algorithm="maxima")
Output:
16925425/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/4*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 81/ 56*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 14145/1568*sqrt(-1 0*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 1479375/21952*sqrt(-10*x^2 - x + 3)/(3 *x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (118) = 236\).
Time = 0.28 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.47 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\frac {55}{614656} \, \sqrt {5} {\left (61547 \, \sqrt {70} \sqrt {2} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {280 \, \sqrt {2} {\left (157973 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 83743800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 17691640512 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1351079744000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5404318976000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}}\right )} \] Input:
integrate((1-2*x)^(1/2)/(2+3*x)^5/(3+5*x)^(1/2),x, algorithm="giac")
Output:
55/614656*sqrt(5)*(61547*sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*s qrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt( 2)*sqrt(-10*x + 5) - sqrt(22)))) + 280*sqrt(2)*(157973*((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 + 83743800*((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 + 17691640 512*((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 + 1351079744000*(sqrt(2)*sqrt(-10* x + 5) - sqrt(22))/sqrt(5*x + 3) - 5404318976000*sqrt(5*x + 3)/(sqrt(2)*sq rt(-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)
Time = 11.54 (sec) , antiderivative size = 1509, normalized size of antiderivative = 9.99 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx=\text {Too large to display} \] Input:
int((1 - 2*x)^(1/2)/((3*x + 2)^5*(5*x + 3)^(1/2)),x)
Output:
((51771207139*((1 - 2*x)^(1/2) - 1)^7)/(76562500*(3^(1/2) - (5*x + 3)^(1/2 ))^7) - (90265424*((1 - 2*x)^(1/2) - 1)^3)/(2734375*(3^(1/2) - (5*x + 3)^( 1/2))^3) - (622608669*((1 - 2*x)^(1/2) - 1)^5)/(7656250*(3^(1/2) - (5*x + 3)^(1/2))^5) - (33845362*((1 - 2*x)^(1/2) - 1))/(133984375*(3^(1/2) - (5*x + 3)^(1/2))) - (51771207139*((1 - 2*x)^(1/2) - 1)^9)/(30625000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (622608669*((1 - 2*x)^(1/2) - 1)^11)/(490000*(3^(1/ 2) - (5*x + 3)^(1/2))^11) + (5641589*((1 - 2*x)^(1/2) - 1)^13)/(1750*(3^(1 /2) - (5*x + 3)^(1/2))^13) + (16922681*((1 - 2*x)^(1/2) - 1)^15)/(109760*( 3^(1/2) - (5*x + 3)^(1/2))^15) + (10154863*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2 )/(3828125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (169254138*3^(1/2)*((1 - 2*x)^ (1/2) - 1)^4)/(2734375*(3^(1/2) - (5*x + 3)^(1/2))^4) - (8594094207*3^(1/2 )*((1 - 2*x)^(1/2) - 1)^6)/(38281250*(3^(1/2) - (5*x + 3)^(1/2))^6) + (143 880176831*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(267968750*(3^(1/2) - (5*x + 3) ^(1/2))^8) - (8594094207*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(6125000*(3^(1/ 2) - (5*x + 3)^(1/2))^10) + (84627069*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(3 5000*(3^(1/2) - (5*x + 3)^(1/2))^12) + (10154863*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(15680*(3^(1/2) - (5*x + 3)^(1/2))^14))/((45056*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (294784*((1 - 2*x)^(1/2) - 1)^4)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^4) - (1921024*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (5828656*((1 - 2*x)^(...
Time = 0.26 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.93 \[ \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(1/2)/(2+3*x)^5/(3+5*x)^(1/2),x)
Output:
(1370959425*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*s qrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 3655891800*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 3655891800*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*s qrt(5))/sqrt(11))/2))/sqrt(2))*x**2 + 1624840800*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 27 0806800*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt( 5))/sqrt(11))/2))/sqrt(2)) - 1370959425*sqrt(7)*atan((sqrt(33) + sqrt(35)* tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 36558918 00*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s qrt(11))/2))/sqrt(2))*x**3 - 3655891800*sqrt(7)*atan((sqrt(33) + sqrt(35)* tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 16248408 00*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s qrt(11))/2))/sqrt(2))*x - 270806800*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan( asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) + 279601875*sqrt(5* x + 3)*sqrt( - 2*x + 1)*x**3 + 571679640*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x* *2 + 390103812*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 88872784*sqrt(5*x + 3)*s qrt( - 2*x + 1))/(153664*(81*x**4 + 216*x**3 + 216*x**2 + 96*x + 16))