Integrand size = 26, antiderivative size = 238 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}-\frac {3735929329 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{120472576 \sqrt {7}} \]
-3735929329/843308032*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1 /2)-59/1764*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^6-1/21*(3+5*x)^(5/2)*(1-2* x)^(1/2)/(2+3*x)^7-6577/370440*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^5+36940 9/20744640*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+2524471/41489280*(1-2*x)^ (1/2)*(3+5*x)^(1/2)/(2+3*x)^3+84539611/232339968*(1-2*x)^(1/2)*(3+5*x)^(1/ 2)/(2+3*x)^2+8818415317/3252759552*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
Time = 0.46 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {14641 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (335335888512+2987299350368 x+11077661454896 x^2+21898948566336 x^3+24351227238888 x^4+14445612678330 x^5+3571458203385 x^6\right )}{14641 (2+3 x)^7}-3827535 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{12649620480} \]
(14641*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(335335888512 + 2987299350368*x + 1 1077661454896*x^2 + 21898948566336*x^3 + 24351227238888*x^4 + 144456126783 30*x^5 + 3571458203385*x^6))/(14641*(2 + 3*x)^7) - 3827535*Sqrt[7]*ArcTan[ Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/12649620480
Time = 0.32 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.13, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {108, 27, 166, 27, 166, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{(3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{21} \int \frac {(19-60 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^7}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \int \frac {(19-60 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^7}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{126} \int -\frac {3 \sqrt {5 x+3} (3680 x+261)}{2 \sqrt {1-2 x} (3 x+2)^6}dx-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (-\frac {1}{84} \int \frac {\sqrt {5 x+3} (3680 x+261)}{\sqrt {1-2 x} (3 x+2)^6}dx-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (-\frac {1}{105} \int \frac {761840 x+384757}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (-\frac {1}{210} \int \frac {761840 x+384757}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}-\frac {1}{28} \int -\frac {3 (964979-7388180 x)}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \int \frac {964979-7388180 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {1}{21} \int \frac {35 (14716025-20195768 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \int \frac {14716025-20195768 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {1812276959-1690792220 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {1812276959-1690792220 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {100870091883}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {8818415317 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {100870091883}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {8818415317 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {100870091883}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {8818415317 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{42} \left (\frac {1}{84} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {8818415317 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {100870091883 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^6}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}\) |
-1/21*(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^7 + ((-59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(42*(2 + 3*x)^6) + ((-6577*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10 5*(2 + 3*x)^5) + ((369409*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (3*((2524471*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5*((84539611* Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((8818415317*Sqrt[1 - 2*x] *Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (100870091883*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7 ]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6))/56)/210)/84)/42
3.23.98.3.1 Defintions of rubi rules used
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/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
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 = 1.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (3571458203385 x^{6}+14445612678330 x^{5}+24351227238888 x^{4}+21898948566336 x^{3}+11077661454896 x^{2}+2987299350368 x +335335888512\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1807088640 \left (2+3 x \right )^{7} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {3735929329 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1686616064 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(144\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (122557161637845 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}+571933420976610 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+1143866841953220 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+50000414847390 \sqrt {-10 x^{2}-x +3}\, x^{6}+1270963157725800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+202238577496620 x^{5} \sqrt {-10 x^{2}-x +3}+847308771817200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+340917181344432 x^{4} \sqrt {-10 x^{2}-x +3}+338923508726880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+306585279928704 x^{3} \sqrt {-10 x^{2}-x +3}+75316335272640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +155087260368544 x^{2} \sqrt {-10 x^{2}-x +3}+7172984311680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+41822190905152 x \sqrt {-10 x^{2}-x +3}+4694702439168 \sqrt {-10 x^{2}-x +3}\right )}{25299240960 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{7}}\) | \(394\) |
-1/1807088640*(-1+2*x)*(3+5*x)^(1/2)*(3571458203385*x^6+14445612678330*x^5 +24351227238888*x^4+21898948566336*x^3+11077661454896*x^2+2987299350368*x+ 335335888512)/(2+3*x)^7/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/ (1-2*x)^(1/2)+3735929329/1686616064*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 = 0.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=-\frac {56038939935 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (3571458203385 \, x^{6} + 14445612678330 \, x^{5} + 24351227238888 \, x^{4} + 21898948566336 \, x^{3} + 11077661454896 \, x^{2} + 2987299350368 \, x + 335335888512\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{25299240960 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
-1/25299240960*(56038939935*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22 680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(3571458203385*x^ 6 + 14445612678330*x^5 + 24351227238888*x^4 + 21898948566336*x^3 + 1107766 1454896*x^2 + 2987299350368*x + 335335888512)*sqrt(5*x + 3)*sqrt(-2*x + 1) )/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1 344*x + 128)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {3735929329}{1686616064} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {154377245}{90354432} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{147 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} - \frac {191 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4116 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {919 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{96040 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {72203 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{768320 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2612695 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6453888 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {92626347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{60236288 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1142391613 \, \sqrt {-10 \, x^{2} - x + 3}}{361417728 \, {\left (3 \, x + 2\right )}} \]
3735929329/1686616064*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 154377245/90354432*sqrt(-10*x^2 - x + 3) + 1/147*(-10*x^2 - x + 3) ^(3/2)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^ 2 + 1344*x + 128) - 191/4116*(-10*x^2 - x + 3)^(3/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 919/96040*(-10*x^2 - x + 3 )^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 72203/7683 20*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 2612 695/6453888*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 9262634 7/60236288*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1142391613/3614177 28*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (187) = 374\).
Time = 0.81 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {3735929329}{16866160640} \, \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 (765507 \, {\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 )}^{13} + 1428946400 \, {\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 )}^{11} + 1132297127360 \, {\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 )}^{9} - 334448649830400 \, {\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} - 85378328229376000 \, {\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} - 8754907317452800000 \, {\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 {368890400944128000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1475561603776512000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{180708864 \, {\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 )}^{7}} \]
3735929329/16866160640*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sq rt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2 )*sqrt(-10*x + 5) - sqrt(22)))) - 14641/180708864*sqrt(10)*(765507*((sqrt( 2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq rt(-10*x + 5) - sqrt(22)))^13 + 1428946400*((sqrt(2)*sqrt(-10*x + 5) - sqr t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22) ))^11 + 1132297127360*((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)))^9 - 33444864983040 0*((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)))^7 - 85378328229376000*((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 - 8754907317452800000*((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 - 368890400944128000000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1475561603776512000000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr t(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)^7
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^8} \,d x \]