Integrand size = 26, antiderivative size = 238 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=-\frac {37 (1-2 x)^{3/2} \sqrt {3+5 x}}{1764 (2+3 x)^6}+\frac {2309 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}+\frac {341917 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^4}+\frac {4014523 \sqrt {1-2 x} \sqrt {3+5 x}}{5927040 (2+3 x)^3}+\frac {140331343 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)^2}+\frac {14677525921 \sqrt {1-2 x} \sqrt {3+5 x}}{464679936 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}-\frac {6219452877 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{17210368 \sqrt {7}} \] Output:
-37/1764*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6+2309/52920*(1-2*x)^(1/2)*(3 +5*x)^(1/2)/(2+3*x)^5+341917/2963520*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4 +4014523/5927040*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+140331343/33191424* (1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+14677525921*(1-2*x)^(1/2)*(3+5*x)^(1 /2)/(929359872+1394039808*x)-1/21*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^7-62 19452877/120472576*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
Time = 0.42 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.40 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\frac {14641 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (186609267072+1658923773088 x+6146173476816 x^2+12147806104256 x^3+13509190228248 x^4+8014272743430 x^5+1981465999335 x^6\right )}{14641 (2+3 x)^7}-2123985 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{602362880} \] Input:
Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^8,x]
Output:
(14641*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(186609267072 + 1658923773088*x + 6 146173476816*x^2 + 12147806104256*x^3 + 13509190228248*x^4 + 8014272743430 *x^5 + 1981465999335*x^6))/(14641*(2 + 3*x)^7) - 2123985*Sqrt[7]*ArcTan[Sq rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/602362880
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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^8} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{21} \int -\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{2 (3 x+2)^7}dx-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{14} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{(3 x+2)^7}dx-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{18} \int \frac {(1647-2480 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^6}dx+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \int \frac {(1647-2480 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^6}dx+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{105} \int \frac {63359-75920 x}{2 \sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \int \frac {63359-75920 x}{\sqrt {1-2 x} (3 x+2)^5 \sqrt {5 x+3}}dx-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {1}{28} \int \frac {3 (4808327-6838340 x)}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \int \frac {4808327-6838340 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {1}{21} \int \frac {35 (25366325-32116184 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \int \frac {25366325-32116184 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{14} \int \frac {3021424067-2806626860 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {140331343 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \int \frac {3021424067-2806626860 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {140331343 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {167925227679}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {14677525921 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {140331343 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {167925227679}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {14677525921 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {140331343 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {167925227679}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {14677525921 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {140331343 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (\frac {1}{36} \left (\frac {1}{210} \left (\frac {3}{56} \left (\frac {5}{6} \left (\frac {1}{28} \left (\frac {14677525921 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {167925227679 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {140331343 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}\right )+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}\right )-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^5}\right )+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}\) |
Input:
Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^8,x]
Output:
-1/21*((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^7 + ((37*Sqrt[1 - 2*x]*( 3 + 5*x)^(3/2))/(18*(2 + 3*x)^6) + ((-9901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1 05*(2 + 3*x)^5) + ((341917*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (3*((4014523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (5*((14033134 3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + ((14677525921*Sqrt[1 - 2 *x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (167925227679*ArcTan[Sqrt[1 - 2*x]/(Sqr t[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/6))/56)/210)/36)/14
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 = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (1981465999335 x^{6}+8014272743430 x^{5}+13509190228248 x^{4}+12147806104256 x^{3}+6146173476816 x^{2}+1658923773088 x +186609267072\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{86051840 \left (2+3 x \right )^{7} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {6219452877 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{240945152 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(144\) |
| default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (68009717209995 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}+317378680313310 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+634757360626620 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+27740523990690 \sqrt {-10 x^{2}-x +3}\, x^{6}+705285956251800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+112199818408020 x^{5} \sqrt {-10 x^{2}-x +3}+470190637501200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+189128663195472 x^{4} \sqrt {-10 x^{2}-x +3}+188076255000480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+170069285459584 x^{3} \sqrt {-10 x^{2}-x +3}+41794723333440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +86046428675424 x^{2} \sqrt {-10 x^{2}-x +3}+3980449841280 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+23224932823232 x \sqrt {-10 x^{2}-x +3}+2612529739008 \sqrt {-10 x^{2}-x +3}\right )}{1204725760 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{7}}\) | \(394\) |
Input:
int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^8,x,method=_RETURNVERBOSE)
Output:
-1/86051840*(-1+2*x)*(3+5*x)^(1/2)*(1981465999335*x^6+8014272743430*x^5+13 509190228248*x^4+12147806104256*x^3+6146173476816*x^2+1658923773088*x+1866 09267072)/(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)+6219452877/240945152*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 = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=-\frac {31097264385 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1981465999335 \, x^{6} + 8014272743430 \, x^{5} + 13509190228248 \, x^{4} + 12147806104256 \, x^{3} + 6146173476816 \, x^{2} + 1658923773088 \, x + 186609267072\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1204725760 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^8,x, algorithm="fricas")
Output:
-1/1204725760*(31097264385*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 226 80*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*(1981465999335*x^6 + 8014272743430*x^5 + 13509190228248*x^4 + 12147806104256*x^3 + 614617347 6816*x^2 + 1658923773088*x + 186609267072)*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 + 1344 *x + 128)
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{8}}\, dx \] Input:
integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**8,x)
Output:
Integral((1 - 2*x)**(3/2)*(5*x + 3)**(3/2)/(3*x + 2)**8, x)
Time = 0.13 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.36 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\frac {1167483755}{90354432} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{49 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {333 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1372 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {11841 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{13720 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {424797 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{153664 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {15717489 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2151296 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {700490253 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{60236288 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {9509080845}{60236288} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {6219452877}{240945152} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {8378271231}{120472576} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {2771517227 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{361417728 \, {\left (3 \, x + 2\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^8,x, algorithm="maxima")
Output:
1167483755/90354432*(-10*x^2 - x + 3)^(3/2) + 3/49*(-10*x^2 - x + 3)^(5/2) /(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 13 44*x + 128) + 333/1372*(-10*x^2 - x + 3)^(5/2)/(729*x^6 + 2916*x^5 + 4860* x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 11841/13720*(-10*x^2 - x + 3)^(5 /2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 424797/153664* (-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1571748 9/2151296*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 700490253 /60236288*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 9509080845/60236288 *sqrt(-10*x^2 - x + 3)*x + 6219452877/240945152*sqrt(7)*arcsin(37/11*x/abs (3*x + 2) + 20/11/abs(3*x + 2)) - 8378271231/120472576*sqrt(-10*x^2 - x + 3) + 2771517227/361417728*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (187) = 374\).
Time = 0.58 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.28 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\frac {6219452877}{2409451520} \, \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 (424797 \, {\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} + 792954400 \, {\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} - 748492373440 \, {\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} - 270037116518400 \, {\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} - 49241484970496000 \, {\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} - 4873941796864000000 \, {\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 {204705555468288000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {818822221873152000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{8605184 \, {\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}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^8,x, algorithm="giac")
Output:
6219452877/2409451520*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqr t(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2) *sqrt(-10*x + 5) - sqrt(22)))) - 14641/8605184*sqrt(10)*(424797*((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)))^13 + 792954400*((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)))^1 1 - 748492373440*((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 - 270037116518400*((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)))^7 - 49241484970496000*((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 - 4873941796864000000*((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 - 204705555468288000000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 818822221873152000000*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)^7
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^8} \,d x \] Input:
int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^8,x)
Output:
int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^8, x)
Time = 0.29 (sec) , antiderivative size = 727, normalized size of antiderivative = 3.05 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^8,x)
Output:
(68009717209995*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**7 + 317378680313310*sqrt(7)*atan((sq rt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2 ))*x**6 + 634757360626620*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt ( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 + 705285956251800*sqrt(7 )*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/ 2))/sqrt(2))*x**4 + 470190637501200*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan( asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 188076255000 480*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/ sqrt(11))/2))/sqrt(2))*x**2 + 41794723333440*sqrt(7)*atan((sqrt(33) - sqrt (35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 398044 9841280*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt( 5))/sqrt(11))/2))/sqrt(2)) - 68009717209995*sqrt(7)*atan((sqrt(33) + sqrt( 35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**7 - 3173 78680313310*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*s qrt(5))/sqrt(11))/2))/sqrt(2))*x**6 - 634757360626620*sqrt(7)*atan((sqrt(3 3) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x **5 - 705285956251800*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 470190637501200*sqrt(7)*at an((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2...