Integrand size = 24, antiderivative size = 131 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {13 \sqrt {1-2 x} (3+5 x)^2}{56 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^3}-\frac {\sqrt {1-2 x} (18187+26775 x)}{1176 (2+3 x)}+\frac {13243 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{588 \sqrt {21}} \]
-1/12*(1-2*x)^(3/2)*(3+5*x)^3/(2+3*x)^4+13243/12348*arctanh(1/7*21^(1/2)*( 1-2*x)^(1/2))*21^(1/2)+13/56*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^2+(3+5*x)^3*( 1-2*x)^(1/2)/(2+3*x)^3-1/1176*(18187+26775*x)*(1-2*x)^(1/2)/(2+3*x)
Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (74810+401850 x+788415 x^2+661639 x^3+196000 x^4\right )}{(2+3 x)^4}+26486 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696} \]
((-21*Sqrt[1 - 2*x]*(74810 + 401850*x + 788415*x^2 + 661639*x^3 + 196000*x ^4))/(2 + 3*x)^4 + 26486*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/24696
Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {108, 27, 166, 27, 166, 27, 163, 73, 219}
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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^3}{(3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{12} \int \frac {3 (2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {(2-15 x) \sqrt {1-2 x} (5 x+3)^2}{(3 x+2)^4}dx-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{4} \left (\frac {4 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}-\frac {1}{9} \int -\frac {9 (5 x+3)^2 (30 x+7)}{\sqrt {1-2 x} (3 x+2)^3}dx\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {(5 x+3)^2 (30 x+7)}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {4 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{42} \int \frac {3 (5 x+3) (765 x+173)}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {4 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}+\frac {13 \sqrt {1-2 x} (5 x+3)^2}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \int \frac {(5 x+3) (765 x+173)}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {4 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}+\frac {13 \sqrt {1-2 x} (5 x+3)^2}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 163 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (-\frac {13243}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x} (26775 x+18187)}{21 (3 x+2)}\right )+\frac {4 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}+\frac {13 \sqrt {1-2 x} (5 x+3)^2}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (\frac {13243}{21} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x} (26775 x+18187)}{21 (3 x+2)}\right )+\frac {4 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}+\frac {13 \sqrt {1-2 x} (5 x+3)^2}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{14} \left (\frac {26486 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}-\frac {\sqrt {1-2 x} (26775 x+18187)}{21 (3 x+2)}\right )+\frac {4 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^3}+\frac {13 \sqrt {1-2 x} (5 x+3)^2}{14 (3 x+2)^2}\right )-\frac {(1-2 x)^{3/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
-1/12*((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(2 + 3*x)^4 + ((13*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(14*(2 + 3*x)^2) + (4*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^3 + (-1 /21*(Sqrt[1 - 2*x]*(18187 + 26775*x))/(2 + 3*x) + (26486*ArcTanh[Sqrt[3/7] *Sqrt[1 - 2*x]])/(21*Sqrt[21]))/14)/4
3.19.91.3.1 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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* (m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f *h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* d*(b*c - a*d)*(m + 1)*(m + n + 3)) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 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*((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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.47
| method | result | size |
| risch | \(\frac {392000 x^{5}+1127278 x^{4}+915191 x^{3}+15285 x^{2}-252230 x -74810}{1176 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {13243 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\) | \(61\) |
| pseudoelliptic | \(\frac {26486 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (196000 x^{4}+661639 x^{3}+788415 x^{2}+401850 x +74810\right )}{24696 \left (2+3 x \right )^{4}}\) | \(65\) |
| derivativedivides | \(-\frac {500 \sqrt {1-2 x}}{243}-\frac {4 \left (-\frac {416917 \left (1-2 x \right )^{\frac {7}{2}}}{2352}+\frac {406463 \left (1-2 x \right )^{\frac {5}{2}}}{336}-\frac {1189171 \left (1-2 x \right )^{\frac {3}{2}}}{432}+\frac {2706781 \sqrt {1-2 x}}{1296}\right )}{3 \left (-4-6 x \right )^{4}}+\frac {13243 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\) | \(75\) |
| default | \(-\frac {500 \sqrt {1-2 x}}{243}-\frac {4 \left (-\frac {416917 \left (1-2 x \right )^{\frac {7}{2}}}{2352}+\frac {406463 \left (1-2 x \right )^{\frac {5}{2}}}{336}-\frac {1189171 \left (1-2 x \right )^{\frac {3}{2}}}{432}+\frac {2706781 \sqrt {1-2 x}}{1296}\right )}{3 \left (-4-6 x \right )^{4}}+\frac {13243 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{12348}\) | \(75\) |
| trager | \(-\frac {\left (196000 x^{4}+661639 x^{3}+788415 x^{2}+401850 x +74810\right ) \sqrt {1-2 x}}{1176 \left (2+3 x \right )^{4}}+\frac {13243 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{24696}\) | \(82\) |
1/1176*(392000*x^5+1127278*x^4+915191*x^3+15285*x^2-252230*x-74810)/(2+3*x )^4/(1-2*x)^(1/2)+13243/12348*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {13243 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (196000 \, x^{4} + 661639 \, x^{3} + 788415 \, x^{2} + 401850 \, x + 74810\right )} \sqrt {-2 \, x + 1}}{24696 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/24696*(13243*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(196000*x^4 + 661639*x^3 + 788415*x^2 + 401850*x + 74810)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {13243}{24696} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {500}{243} \, \sqrt {-2 \, x + 1} + \frac {11256759 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 76821507 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 174808137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 132632269 \, \sqrt {-2 \, x + 1}}{47628 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
-13243/24696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr t(-2*x + 1))) - 500/243*sqrt(-2*x + 1) + 1/47628*(11256759*(-2*x + 1)^(7/2 ) - 76821507*(-2*x + 1)^(5/2) + 174808137*(-2*x + 1)^(3/2) - 132632269*sqr t(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {13243}{24696} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {500}{243} \, \sqrt {-2 \, x + 1} - \frac {11256759 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 76821507 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 174808137 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 132632269 \, \sqrt {-2 \, x + 1}}{762048 \, {\left (3 \, x + 2\right )}^{4}} \]
-13243/24696*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21 ) + 3*sqrt(-2*x + 1))) - 500/243*sqrt(-2*x + 1) - 1/762048*(11256759*(2*x - 1)^3*sqrt(-2*x + 1) + 76821507*(2*x - 1)^2*sqrt(-2*x + 1) - 174808137*(- 2*x + 1)^(3/2) + 132632269*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.38 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {13243\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{12348}-\frac {500\,\sqrt {1-2\,x}}{243}-\frac {\frac {2706781\,\sqrt {1-2\,x}}{78732}-\frac {1189171\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {406463\,{\left (1-2\,x\right )}^{5/2}}{20412}-\frac {416917\,{\left (1-2\,x\right )}^{7/2}}{142884}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]
(13243*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/12348 - (500*(1 - 2*x )^(1/2))/243 - ((2706781*(1 - 2*x)^(1/2))/78732 - (1189171*(1 - 2*x)^(3/2) )/26244 + (406463*(1 - 2*x)^(5/2))/20412 - (416917*(1 - 2*x)^(7/2))/142884 )/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1 715/81)