Integrand size = 26, antiderivative size = 224 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=-\frac {754386765 \sqrt {1-2 x}}{6272 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 (3+5 x)^{3/2}}+\frac {1001 \sqrt {1-2 x}}{120 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {53009 \sqrt {1-2 x}}{720 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {3329689 \sqrt {1-2 x}}{4032 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {270667969 \sqrt {1-2 x}}{18816 (2+3 x) (3+5 x)^{3/2}}+\frac {20529722435 \sqrt {1-2 x}}{18816 \sqrt {3+5 x}}-\frac {46975917593 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}} \]
7/15*(1-2*x)^(3/2)/(2+3*x)^5/(3+5*x)^(3/2)-46975917593/43904*arctan(1/7*(1 -2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-754386765/6272*(1-2*x)^(1/2)/(3 +5*x)^(3/2)+1001/120*(1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(3/2)+53009/720*(1-2* x)^(1/2)/(2+3*x)^3/(3+5*x)^(3/2)+3329689/4032*(1-2*x)^(1/2)/(2+3*x)^2/(3+5 *x)^(3/2)+270667969/18816*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^(3/2)+20529722435/ 18816*(1-2*x)^(1/2)/(3+5*x)^(1/2)
Time = 8.21 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=\frac {395136 (1-2 x)^{7/2}+3252816 (1-2 x)^{7/2} (2+3 x)+(2+3 x)^2 \left (29407896 (1-2 x)^{7/2}+(2+3 x) \left (324091386 (1-2 x)^{7/2}+4270537963 (2+3 x) \left (3 (1-2 x)^{5/2}-55 (2+3 x) \left (-\sqrt {1-2 x} (62+107 x)+21 \sqrt {7} (3+5 x)^{3/2} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )\right )\right )\right )}{4609920 (2+3 x)^5 (3+5 x)^{3/2}} \]
(395136*(1 - 2*x)^(7/2) + 3252816*(1 - 2*x)^(7/2)*(2 + 3*x) + (2 + 3*x)^2* (29407896*(1 - 2*x)^(7/2) + (2 + 3*x)*(324091386*(1 - 2*x)^(7/2) + 4270537 963*(2 + 3*x)*(3*(1 - 2*x)^(5/2) - 55*(2 + 3*x)*(-(Sqrt[1 - 2*x]*(62 + 107 *x)) + 21*Sqrt[7]*(3 + 5*x)^(3/2)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5 *x])])))))/(4609920*(2 + 3*x)^5*(3 + 5*x)^(3/2))
Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {109, 27, 166, 27, 168, 27, 168, 27, 168, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^6 (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {1}{15} \int \frac {33 (17-20 x) \sqrt {1-2 x}}{2 (3 x+2)^5 (5 x+3)^{5/2}}dx+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{10} \int \frac {(17-20 x) \sqrt {1-2 x}}{(3 x+2)^5 (5 x+3)^{5/2}}dx+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {11}{10} \left (\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}-\frac {1}{12} \int -\frac {5391-8780 x}{2 \sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{5/2}}dx\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \int \frac {5391-8780 x}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{5/2}}dx+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {1}{21} \int \frac {35 (49497-77104 x)}{2 \sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{5/2}}dx+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \int \frac {49497-77104 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)^{5/2}}dx+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {3 (4166073-6053980 x)}{2 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \int \frac {4166073-6053980 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}}dx+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \left (\frac {1}{7} \int \frac {767347881-984247160 x}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {24606179 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \left (\frac {1}{14} \int \frac {767347881-984247160 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{5/2}}dx+\frac {24606179 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \left (\frac {1}{14} \left (-\frac {2}{33} \int \frac {33 (2624603203-2468902140 x)}{2 \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {411483690 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24606179 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \left (\frac {1}{14} \left (-\int \frac {2624603203-2468902140 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2}}dx-\frac {411483690 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24606179 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \left (\frac {1}{14} \left (\frac {2}{11} \int \frac {140927752779}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {41059444870 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {411483690 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24606179 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \left (\frac {1}{14} \left (12811613889 \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {41059444870 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {411483690 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24606179 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \left (\frac {1}{14} \left (25623227778 \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {41059444870 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {411483690 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24606179 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11}{10} \left (\frac {1}{24} \left (\frac {5}{6} \left (\frac {3}{28} \left (\frac {1}{14} \left (-\frac {25623227778 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}}+\frac {41059444870 \sqrt {1-2 x}}{11 \sqrt {5 x+3}}-\frac {411483690 \sqrt {1-2 x}}{(5 x+3)^{3/2}}\right )+\frac {24606179 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}\right )+\frac {302699 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}\right )+\frac {4819 \sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}\right )+\frac {91 \sqrt {1-2 x}}{12 (3 x+2)^4 (5 x+3)^{3/2}}\right )+\frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)^{3/2}}\) |
(7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5*(3 + 5*x)^(3/2)) + (11*((91*Sqrt[1 - 2 *x])/(12*(2 + 3*x)^4*(3 + 5*x)^(3/2)) + ((4819*Sqrt[1 - 2*x])/(3*(2 + 3*x) ^3*(3 + 5*x)^(3/2)) + (5*((302699*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x) ^(3/2)) + (3*((24606179*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^(3/2)) + ((- 411483690*Sqrt[1 - 2*x])/(3 + 5*x)^(3/2) + (41059444870*Sqrt[1 - 2*x])/(11 *Sqrt[3 + 5*x]) - (25623227778*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x] )])/Sqrt[7])/14))/28))/6)/24))/10
3.25.57.3.1 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_))^(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] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(393\) vs. \(2(173)=346\).
Time = 1.16 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(\frac {\left (4280680490662125 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}+19405751557668300 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+37689013564451865 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+1746052893096750 \sqrt {-10 x^{2}-x +3}\, x^{6}+40650610289102550 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+6829311689562600 x^{5} \sqrt {-10 x^{2}-x +3}+26297118668561400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+11125554365281230 x^{4} \sqrt {-10 x^{2}-x +3}+10203169301199600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+9662658051124260 x^{3} \sqrt {-10 x^{2}-x +3}+2198472943352400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +4718679545989416 x^{2} \sqrt {-10 x^{2}-x +3}+202935964001760 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1228469050319504 x \sqrt {-10 x^{2}-x +3}+133202515888064 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{1317120 \left (2+3 x \right )^{5} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(394\) |
1/1317120*(4280680490662125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2 -x+3)^(1/2))*x^7+19405751557668300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/( -10*x^2-x+3)^(1/2))*x^6+37689013564451865*7^(1/2)*arctan(1/14*(37*x+20)*7^ (1/2)/(-10*x^2-x+3)^(1/2))*x^5+1746052893096750*(-10*x^2-x+3)^(1/2)*x^6+40 650610289102550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) *x^4+6829311689562600*x^5*(-10*x^2-x+3)^(1/2)+26297118668561400*7^(1/2)*ar ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+11125554365281230*x^4 *(-10*x^2-x+3)^(1/2)+10203169301199600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/ 2)/(-10*x^2-x+3)^(1/2))*x^2+9662658051124260*x^3*(-10*x^2-x+3)^(1/2)+21984 72943352400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+4 718679545989416*x^2*(-10*x^2-x+3)^(1/2)+202935964001760*7^(1/2)*arctan(1/1 4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1228469050319504*x*(-10*x^2-x+3)^ (1/2)+133202515888064*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^5/(-10*x^ 2-x+3)^(1/2)/(3+5*x)^(3/2)
Time = 0.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=-\frac {704638763895 \, \sqrt {7} {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (124718063792625 \, x^{6} + 487807977825900 \, x^{5} + 794682454662945 \, x^{4} + 690189860794590 \, x^{3} + 337048538999244 \, x^{2} + 87747789308536 \, x + 9514465420576\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1317120 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]
-1/1317120*(704638763895*sqrt(7)*(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690 *x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*arctan(1/14*sqrt(7)*(37*x + 2 0)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(124718063792625*x^ 6 + 487807977825900*x^5 + 794682454662945*x^4 + 690189860794590*x^3 + 3370 48538999244*x^2 + 87747789308536*x + 9514465420576)*sqrt(5*x + 3)*sqrt(-2* x + 1))/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480* x^2 + 3120*x + 288)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (173) = 346\).
Time = 0.33 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.91 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=\frac {46975917593}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {20529722435 \, x}{9408 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {21434986553}{18816 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {2211170555 \, x}{4032 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{405 \, {\left (243 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} + 810 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + 1080 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 720 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 240 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 32 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {43561}{1080 \, {\left (81 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 96 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 16 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {2438681}{6480 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {110694619}{25920 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1309509421}{17280 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {21497905297}{72576 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
46975917593/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2) ) - 20529722435/9408*x/sqrt(-10*x^2 - x + 3) + 21434986553/18816/sqrt(-10* x^2 - x + 3) + 2211170555/4032*x/(-10*x^2 - x + 3)^(3/2) + 2401/405/(243*( -10*x^2 - x + 3)^(3/2)*x^5 + 810*(-10*x^2 - x + 3)^(3/2)*x^4 + 1080*(-10*x ^2 - x + 3)^(3/2)*x^3 + 720*(-10*x^2 - x + 3)^(3/2)*x^2 + 240*(-10*x^2 - x + 3)^(3/2)*x + 32*(-10*x^2 - x + 3)^(3/2)) + 43561/1080/(81*(-10*x^2 - x + 3)^(3/2)*x^4 + 216*(-10*x^2 - x + 3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^( 3/2)*x^2 + 96*(-10*x^2 - x + 3)^(3/2)*x + 16*(-10*x^2 - x + 3)^(3/2)) + 24 38681/6480/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^ 2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) + 110694619/ 25920/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(- 10*x^2 - x + 3)^(3/2)) + 1309509421/17280/(3*(-10*x^2 - x + 3)^(3/2)*x + 2 *(-10*x^2 - x + 3)^(3/2)) - 21497905297/72576/(-10*x^2 - x + 3)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (173) = 346\).
Time = 0.73 (sec) , antiderivative size = 546, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=-\frac {275}{48} \, \sqrt {10} {\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 {46975917593}{878080} \, \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 )} + 27775 \, \sqrt {10} {\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 )} + \frac {11 \, \sqrt {10} {\left (3277500437 \, {\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} + 3147123544880 \, {\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} + 1168996576419840 \, {\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} + 196941720284288000 \, {\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 {12621260024737280000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {50485040098949120000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3136 \, {\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}} \]
-275/48*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 46975917593/878080* sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*s qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqr t(22)))) + 27775*sqrt(10)*((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))) + 11/3136*sqrt (10)*(3277500437*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 3147123544880*((sqr t(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 + 1168996576419840*((sqrt(2)*sqrt(-10*x + 5 ) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - s qrt(22)))^5 + 196941720284288000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 126 21260024737280000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 504 85040098949120000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((s qrt(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. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^6\,{\left (5\,x+3\right )}^{5/2}} \,d x \]