Integrand size = 26, antiderivative size = 180 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {437 \sqrt {1-2 x} \sqrt {3+5 x}}{17640 (2+3 x)^4}-\frac {14831 \sqrt {1-2 x} \sqrt {3+5 x}}{105840 (2+3 x)^3}-\frac {12371 \sqrt {1-2 x} \sqrt {3+5 x}}{592704 (2+3 x)^2}+\frac {1948963 \sqrt {1-2 x} \sqrt {3+5 x}}{8297856 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{105 (2+3 x)^5}-\frac {933031 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{307328 \sqrt {7}} \]
-933031/2151296*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1/ 105*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^5+437/17640*(1-2*x)^(1/2)*(3+5*x)^ (1/2)/(2+3*x)^4-14831/105840*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3-12371/5 92704*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+1948963/8297856*(1-2*x)^(1/2)* (3+5*x)^(1/2)/(2+3*x)
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (14330592+93291272 x+222865988 x^2+231277650 x^3+87703335 x^4\right )}{(2+3 x)^5}-13995465 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{32269440} \]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(14330592 + 93291272*x + 222865988*x^2 + 2 31277650*x^3 + 87703335*x^4))/(2 + 3*x)^5 - 13995465*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/32269440
Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {109, 27, 166, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}-\frac {1}{105} \int -\frac {\sqrt {5 x+3} (1690 x+981)}{2 \sqrt {1-2 x} (3 x+2)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \int \frac {\sqrt {5 x+3} (1690 x+981)}{\sqrt {1-2 x} (3 x+2)^5}dx+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{84} \int \frac {446980 x+263381}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \int \frac {446980 x+263381}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \left (\frac {1}{21} \int \frac {35 (118648 x+74975)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \left (\frac {5}{6} \int \frac {118648 x+74975}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {247420 x+814601}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {12371 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {247420 x+814601}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {12371 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {25191837}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1948963 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {12371 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {25191837}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1948963 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {12371 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {25191837}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {1948963 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {12371 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{168} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1948963 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {25191837 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )-\frac {12371 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {14831 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {437 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}\) |
(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(105*(2 + 3*x)^5) + ((437*Sqrt[1 - 2*x]*Sq rt[3 + 5*x])/(84*(2 + 3*x)^4) + ((-14831*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*( 2 + 3*x)^3) + (5*((-12371*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((1948963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (25191837*ArcTan[Sq rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6)/168)/210
3.25.87.3.1 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[(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*((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 = 1.19 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (87703335 x^{4}+231277650 x^{3}+222865988 x^{2}+93291272 x +14330592\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4609920 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {933031 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4302592 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(134\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (3400897995 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+11336326650 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+15115102200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+1227846690 x^{4} \sqrt {-10 x^{2}-x +3}+10076734800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+3237887100 x^{3} \sqrt {-10 x^{2}-x +3}+3358911600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +3120123832 x^{2} \sqrt {-10 x^{2}-x +3}+447854880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1306077808 x \sqrt {-10 x^{2}-x +3}+200628288 \sqrt {-10 x^{2}-x +3}\right )}{64538880 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(298\) |
-1/4609920*(-1+2*x)*(3+5*x)^(1/2)*(87703335*x^4+231277650*x^3+222865988*x^ 2+93291272*x+14330592)/(2+3*x)^5/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x ))^(1/2)/(1-2*x)^(1/2)+933031/4302592*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 = 131, normalized size of antiderivative = 0.73 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=-\frac {13995465 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (87703335 \, x^{4} + 231277650 \, x^{3} + 222865988 \, x^{2} + 93291272 \, x + 14330592\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{64538880 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
-1/64538880*(13995465*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 24 0*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10 *x^2 + x - 3)) - 14*(87703335*x^4 + 231277650*x^3 + 222865988*x^2 + 932912 72*x + 14330592)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x ^3 + 720*x^2 + 240*x + 32)
\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{6}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.02 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {933031}{4302592} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{315 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {239 \, \sqrt {-10 \, x^{2} - x + 3}}{5880 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {14831 \, \sqrt {-10 \, x^{2} - x + 3}}{105840 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {12371 \, \sqrt {-10 \, x^{2} - x + 3}}{592704 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1948963 \, \sqrt {-10 \, x^{2} - x + 3}}{8297856 \, {\left (3 \, x + 2\right )}} \]
933031/4302592*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/315*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240 *x + 32) + 239/5880*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96 *x + 16) - 14831/105840*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) - 12371/592704*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 1948963/8297856 *sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (141) = 282\).
Time = 0.61 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {933031}{43025920} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (2103 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 2747920 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 1406935040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 74141312000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {10228753920000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {40915015680000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{460992 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
933031/43025920*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)))) - 1331/460992*sqrt(10)*(2103*((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 + 2747920*((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 + 1406935040 *((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)))^5 - 74141312000*((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 - 10228753920000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5* x + 3) + 40915015680000*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)^5
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^6} \,d x \]