Integrand size = 26, antiderivative size = 180 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^5}-\frac {367 \sqrt {1-2 x} \sqrt {3+5 x}}{5880 (2+3 x)^4}-\frac {73 \sqrt {1-2 x} \sqrt {3+5 x}}{11760 (2+3 x)^3}+\frac {6107 \sqrt {1-2 x} \sqrt {3+5 x}}{65856 (2+3 x)^2}+\frac {694229 \sqrt {1-2 x} \sqrt {3+5 x}}{921984 (2+3 x)}-\frac {2664057 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{307328 \sqrt {7}} \]
-2664057/2151296*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1 /105*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^5-367/5880*(1-2*x)^(1/2)*(3+5*x)^ (1/2)/(2+3*x)^4-73/11760*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+6107/65856* (1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+694229/921984*(1-2*x)^(1/2)*(3+5*x)^ (1/2)/(2+3*x)
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.48 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {121 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (19437408+115804328 x+257531412 x^2+253769850 x^3+93720915 x^4\right )}{121 (2+3 x)^5}-110085 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{10756480} \]
(121*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(19437408 + 115804328*x + 257531412*x ^2 + 253769850*x^3 + 93720915*x^4))/(121*(2 + 3*x)^5) - 110085*Sqrt[7]*Arc Tan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/10756480
Time = 0.27 (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, 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.
| \(\displaystyle \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}-\frac {1}{105} \int -\frac {1670 x+991}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \int \frac {1670 x+991}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{210} \left (\frac {1}{28} \int \frac {3 (7340 x+4723)}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \int \frac {7340 x+4723}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \left (\frac {1}{21} \int \frac {35 (584 x+2425)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \int \frac {584 x+2425}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {149983-122140 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {6107 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {149983-122140 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {6107 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {7992171}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {694229 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {6107 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {7992171}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {694229 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {6107 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {7992171}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {694229 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {6107 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {694229 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {7992171 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {6107 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )-\frac {73 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )-\frac {367 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\) |
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^5) + ((-367*Sqrt[1 - 2*x]*Sqr t[3 + 5*x])/(28*(2 + 3*x)^4) + (3*((-73*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5*((6107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((6 94229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (7992171*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6))/56)/210
3.25.77.3.1 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 + 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 = 1.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (93720915 x^{4}+253769850 x^{3}+257531412 x^{2}+115804328 x +19437408\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1536640 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {2664057 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4302592 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(134\) |
| default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (3236829255 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+10789430850 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+14385907800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+1312092810 x^{4} \sqrt {-10 x^{2}-x +3}+9590605200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+3552777900 x^{3} \sqrt {-10 x^{2}-x +3}+3196868400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +3605439768 x^{2} \sqrt {-10 x^{2}-x +3}+426249120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1621260592 x \sqrt {-10 x^{2}-x +3}+272123712 \sqrt {-10 x^{2}-x +3}\right )}{21512960 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(298\) |
-1/1536640*(-1+2*x)*(3+5*x)^(1/2)*(93720915*x^4+253769850*x^3+257531412*x^ 2+115804328*x+19437408)/(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)+2664057/4302592*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 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=-\frac {13320285 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (93720915 \, x^{4} + 253769850 \, x^{3} + 257531412 \, x^{2} + 115804328 \, x + 19437408\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{21512960 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
-1/21512960*(13320285*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*(93720915*x^4 + 253769850*x^3 + 257531412*x^2 + 115804 328*x + 19437408)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080* x^3 + 720*x^2 + 240*x + 32)
\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {3}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{6}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.02 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {2664057}{4302592} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{105 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac {367 \, \sqrt {-10 \, x^{2} - x + 3}}{5880 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {73 \, \sqrt {-10 \, x^{2} - x + 3}}{11760 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {6107 \, \sqrt {-10 \, x^{2} - x + 3}}{65856 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {694229 \, \sqrt {-10 \, x^{2} - x + 3}}{921984 \, {\left (3 \, x + 2\right )}} \]
2664057/4302592*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/105*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 24 0*x + 32) - 367/5880*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 9 6*x + 16) - 73/11760*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 6107/65856*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 694229/921984*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 = 0.63 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {2664057}{43025920} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left (22017 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 28768880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 9856573440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 2123818368000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {133530503680000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {534122014720000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{153664 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
2664057/43025920*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)))) - 121/153664*sqrt(10)*(22017*((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 + 28768880*((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 - 98565734 40*((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 - 2123818368000*((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 - 133530503680000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq rt(5*x + 3) + 534122014720000*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)^5
Timed out. \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^6} \,d x \]