Integrand size = 26, antiderivative size = 209 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{18 (2+3 x)^6}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{1260 (2+3 x)^5}+\frac {10921 \sqrt {1-2 x} \sqrt {3+5 x}}{70560 (2+3 x)^4}+\frac {126799 \sqrt {1-2 x} \sqrt {3+5 x}}{141120 (2+3 x)^3}+\frac {4429459 \sqrt {1-2 x} \sqrt {3+5 x}}{790272 (2+3 x)^2}+\frac {463266973 \sqrt {1-2 x} \sqrt {3+5 x}}{11063808 (2+3 x)}-\frac {588912203 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312 \sqrt {7}} \] Output:
-1/18*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^6+37/1260*(1-2*x)^(1/2)*(3+5*x)^ (1/2)/(2+3*x)^5+10921/70560*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+126799/1 41120*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+4429459/790272*(1-2*x)^(1/2)*( 3+5*x)^(1/2)/(2+3*x)^2+463266973*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(22127616+331 91424*x)-588912203/8605184*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x )^(1/2))
Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\frac {121 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (8835086144+65287037520 x+193055073632 x^2+285550790544 x^3+211260697020 x^4+62541041355 x^5\right )}{121 (2+3 x)^6}-24335215 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{43025920} \] Input:
Integrate[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^7,x]
Output:
(121*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8835086144 + 65287037520*x + 1930550 73632*x^2 + 285550790544*x^3 + 211260697020*x^4 + 62541041355*x^5))/(121*( 2 + 3*x)^6) - 24335215*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x] )]))/43025920
Time = 0.30 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {108, 27, 168, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{(3 x+2)^7} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{18} \int -\frac {20 x+1}{2 \sqrt {1-2 x} (3 x+2)^6 \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{36} \int \frac {20 x+1}{\sqrt {1-2 x} (3 x+2)^6 \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}-\frac {1}{35} \int -\frac {1667-2960 x}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \int \frac {1667-2960 x}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {1}{28} \int \frac {3 (150251-218420 x)}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \int \frac {150251-218420 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \left (\frac {1}{21} \int \frac {35 (800225-1014392 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {126799 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \left (\frac {5}{6} \int \frac {800225-1014392 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {126799 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {95362871-88589180 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {4429459 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {126799 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {95362871-88589180 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {4429459 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {126799 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {5300209827}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {463266973 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {4429459 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {126799 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {5300209827}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {463266973 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {4429459 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {126799 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {5300209827}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {463266973 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {4429459 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {126799 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{36} \left (\frac {1}{70} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {463266973 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {5300209827 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {4429459 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {126799 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {10921 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^5}\right )-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{18 (3 x+2)^6}\) |
Input:
Int[(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^7,x]
Output:
-1/18*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^6 + ((37*Sqrt[1 - 2*x]*Sqrt[ 3 + 5*x])/(35*(2 + 3*x)^5) + ((10921*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (3*((126799*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5*( (4429459*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((463266973*Sqrt[ 1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (5300209827*ArcTan[Sqrt[1 - 2*x]/( Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6))/56)/70)/36
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 + 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 = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (62541041355 x^{5}+211260697020 x^{4}+285550790544 x^{3}+193055073632 x^{2}+65287037520 x +8835086144\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6146560 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {588912203 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{17210368 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (2146584979935 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+8586339919740 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+14310566532900 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+875574578970 x^{5} \sqrt {-10 x^{2}-x +3}+12720503584800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+2957649758280 x^{4} \sqrt {-10 x^{2}-x +3}+6360251792400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+3997711067616 x^{3} \sqrt {-10 x^{2}-x +3}+1696067144640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +2702771030848 x^{2} \sqrt {-10 x^{2}-x +3}+188451904960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+914018525280 x \sqrt {-10 x^{2}-x +3}+123691206016 \sqrt {-10 x^{2}-x +3}\right )}{86051840 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
Input:
int((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^7,x,method=_RETURNVERBOSE)
Output:
-1/6146560*(-1+2*x)*(3+5*x)^(1/2)*(62541041355*x^5+211260697020*x^4+285550 790544*x^3+193055073632*x^2+65287037520*x+8835086144)/(2+3*x)^6/(-(-1+2*x) *(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+588912203/17210368*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 = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=-\frac {2944561015 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (62541041355 \, x^{5} + 211260697020 \, x^{4} + 285550790544 \, x^{3} + 193055073632 \, x^{2} + 65287037520 \, x + 8835086144\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{86051840 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^7,x, algorithm="fricas")
Output:
-1/86051840*(2944561015*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqr t(-2*x + 1)/(10*x^2 + x - 3)) - 14*(62541041355*x^5 + 211260697020*x^4 + 2 85550790544*x^3 + 193055073632*x^2 + 65287037520*x + 8835086144)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{7}}\, dx \] Input:
integrate((1-2*x)**(1/2)*(3+5*x)**(1/2)/(2+3*x)**7,x)
Output:
Integral(sqrt(1 - 2*x)*sqrt(5*x + 3)/(3*x + 2)**7, x)
Time = 0.11 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\frac {588912203}{17210368} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {24335215}{921984} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{14 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {333 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{980 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {11721 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{7840 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {137455 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {14601129 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {180080591 \, \sqrt {-10 \, x^{2} - x + 3}}{3687936 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^7,x, algorithm="maxima")
Output:
588912203/17210368*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2 )) + 24335215/921984*sqrt(-10*x^2 - x + 3) + 1/14*(-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) + 333/9 80*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 11721/7840*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 137455/21952*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 14601129/614656*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 180080 591/3687936*sqrt(-10*x^2 - x + 3)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).
Time = 0.44 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\frac {588912203}{172103680} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left (4867043 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} - 12766158440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 6076175020160 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 1409555377484800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 169516778170880000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {8376360110182400000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {33505440440729600000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{614656 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \] Input:
integrate((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^7,x, algorithm="giac")
Output:
588912203/172103680*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)*s qrt(-10*x + 5) - sqrt(22)))) - 121/614656*sqrt(10)*(4867043*((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)))^11 - 12766158440*((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 - 6076175020160*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 1409555377484800*((sq rt(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 - 169516778170880000*((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 - 8376360110182400000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/ sqrt(5*x + 3) + 33505440440729600000*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*sq rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^6
Time = 16.42 (sec) , antiderivative size = 1981, normalized size of antiderivative = 9.48 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx=\text {Too large to display} \] Input:
int(((1 - 2*x)^(1/2)*(5*x + 3)^(1/2))/(3*x + 2)^7,x)
Output:
((16944332789147*((1 - 2*x)^(1/2) - 1)^7)/(234472656250*(3^(1/2) - (5*x + 3)^(1/2))^7) - (166614000612*((1 - 2*x)^(1/2) - 1)^3)/(117236328125*(3^(1/ 2) - (5*x + 3)^(1/2))^3) - (3113868401706*((1 - 2*x)^(1/2) - 1)^5)/(837402 34375*(3^(1/2) - (5*x + 3)^(1/2))^5) - (471037564*((1 - 2*x)^(1/2) - 1))/( 117236328125*(3^(1/2) - (5*x + 3)^(1/2))) + (788747583108129*((1 - 2*x)^(1 /2) - 1)^9)/(2344726562500*(3^(1/2) - (5*x + 3)^(1/2))^9) - (2357601925166 3873*((1 - 2*x)^(1/2) - 1)^11)/(11723632812500*(3^(1/2) - (5*x + 3)^(1/2)) ^11) + (23576019251663873*((1 - 2*x)^(1/2) - 1)^13)/(4689453125000*(3^(1/2 ) - (5*x + 3)^(1/2))^13) - (788747583108129*((1 - 2*x)^(1/2) - 1)^15)/(150 062500000*(3^(1/2) - (5*x + 3)^(1/2))^15) - (16944332789147*((1 - 2*x)^(1/ 2) - 1)^17)/(2401000000*(3^(1/2) - (5*x + 3)^(1/2))^17) + (1556934200853*( (1 - 2*x)^(1/2) - 1)^19)/(68600000*(3^(1/2) - (5*x + 3)^(1/2))^19) + (4165 3500153*((1 - 2*x)^(1/2) - 1)^21)/(7683200*(3^(1/2) - (5*x + 3)^(1/2))^21) + (117759391*((1 - 2*x)^(1/2) - 1)^23)/(1229312*(3^(1/2) - (5*x + 3)^(1/2 ))^23) + (38864901622*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(586181640625*(3^(1 /2) - (5*x + 3)^(1/2))^2) + (655789856256*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4) /(117236328125*(3^(1/2) - (5*x + 3)^(1/2))^4) + (8465281916419*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(234472656250*(3^(1/2) - (5*x + 3)^(1/2))^6) - (1626 52045859857*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(586181640625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (21120963153193039*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/...
Time = 0.29 (sec) , antiderivative size = 632, normalized size of antiderivative = 3.02 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^7,x)
Output:
(2146584979935*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1 )*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**6 + 8586339919740*sqrt(7)*atan((sqrt( 33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))* x**5 + 14310566532900*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 + 12720503584800*sqrt(7)*ata n((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/s qrt(2))*x**3 + 6360251792400*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((s qrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 + 1696067144640*sqrt( 7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) /2))/sqrt(2))*x + 188451904960*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin( (sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 2146584979935*sqrt(7)* atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2) )/sqrt(2))*x**6 - 8586339919740*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin ((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 - 14310566532900*s qrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt( 11))/2))/sqrt(2))*x**4 - 12720503584800*sqrt(7)*atan((sqrt(33) + sqrt(35)* tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 - 63602517 92400*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5) )/sqrt(11))/2))/sqrt(2))*x**2 - 1696067144640*sqrt(7)*atan((sqrt(33) + sqr t(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 18...