Integrand size = 24, antiderivative size = 127 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {11237 \sqrt {1-2 x}}{111132 (2+3 x)}-\frac {53 \sqrt {1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac {\sqrt {1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}+\frac {11237 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{55566 \sqrt {21}} \]
11237/1166886*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+11237/111132*(1 -2*x)^(1/2)/(2+3*x)-53/630*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^4-1/15*(3+5*x)^ 3*(1-2*x)^(1/2)/(2+3*x)^5-1/79380*(37224+59665*x)*(1-2*x)^(1/2)/(2+3*x)^3
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-1984928-8471518 x-10100352 x^2+240615 x^3+4550985 x^4\right )}{2 (2+3 x)^5}+56185 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5834430} \]
((21*Sqrt[1 - 2*x]*(-1984928 - 8471518*x - 10100352*x^2 + 240615*x^3 + 455 0985*x^4))/(2*(2 + 3*x)^5) + 56185*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x ]])/5834430
Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {108, 166, 27, 162, 52, 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.
| \(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{15} \int \frac {(12-35 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^5}dx-\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{84} \int \frac {2 (173-1655 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{42} \int \frac {(173-1655 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{42} \left (-\frac {56185}{126} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (59665 x+37224)}{126 (3 x+2)^3}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{42} \left (-\frac {56185}{126} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (59665 x+37224)}{126 (3 x+2)^3}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{42} \left (-\frac {56185}{126} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (59665 x+37224)}{126 (3 x+2)^3}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{15} \left (\frac {1}{42} \left (-\frac {56185}{126} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (59665 x+37224)}{126 (3 x+2)^3}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{42 (3 x+2)^4}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{15 (3 x+2)^5}\) |
-1/15*(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^5 + ((-53*Sqrt[1 - 2*x]*(3 + 5 *x)^2)/(42*(2 + 3*x)^4) + (-1/126*(Sqrt[1 - 2*x]*(37224 + 59665*x))/(2 + 3 *x)^3 - (56185*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/126)/42)/15
3.19.25.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
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 = 2.99 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(-\frac {9101970 x^{5}-4069755 x^{4}-20441319 x^{3}-6842684 x^{2}+4501662 x +1984928}{555660 \left (2+3 x \right )^{5} \sqrt {1-2 x}}+\frac {11237 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1166886}\) | \(61\) |
| pseudoelliptic | \(\frac {112370 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}+21 \sqrt {1-2 x}\, \left (4550985 x^{4}+240615 x^{3}-10100352 x^{2}-8471518 x -1984928\right )}{11668860 \left (2+3 x \right )^{5}}\) | \(65\) |
| derivativedivides | \(\frac {-\frac {11237 \left (1-2 x \right )^{\frac {9}{2}}}{686}+\frac {4237 \left (1-2 x \right )^{\frac {7}{2}}}{63}+\frac {39632 \left (1-2 x \right )^{\frac {5}{2}}}{945}-\frac {263117 \left (1-2 x \right )^{\frac {3}{2}}}{567}+\frac {78659 \sqrt {1-2 x}}{162}}{\left (-4-6 x \right )^{5}}+\frac {11237 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1166886}\) | \(75\) |
| default | \(\frac {-\frac {11237 \left (1-2 x \right )^{\frac {9}{2}}}{686}+\frac {4237 \left (1-2 x \right )^{\frac {7}{2}}}{63}+\frac {39632 \left (1-2 x \right )^{\frac {5}{2}}}{945}-\frac {263117 \left (1-2 x \right )^{\frac {3}{2}}}{567}+\frac {78659 \sqrt {1-2 x}}{162}}{\left (-4-6 x \right )^{5}}+\frac {11237 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1166886}\) | \(75\) |
| trager | \(\frac {\left (4550985 x^{4}+240615 x^{3}-10100352 x^{2}-8471518 x -1984928\right ) \sqrt {1-2 x}}{555660 \left (2+3 x \right )^{5}}+\frac {11237 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2333772}\) | \(82\) |
-1/555660*(9101970*x^5-4069755*x^4-20441319*x^3-6842684*x^2+4501662*x+1984 928)/(2+3*x)^5/(1-2*x)^(1/2)+11237/1166886*arctanh(1/7*21^(1/2)*(1-2*x)^(1 /2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {56185 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (4550985 \, x^{4} + 240615 \, x^{3} - 10100352 \, x^{2} - 8471518 \, x - 1984928\right )} \sqrt {-2 \, x + 1}}{11668860 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/11668860*(56185*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(4550985*x^ 4 + 240615*x^3 - 10100352*x^2 - 8471518*x - 1984928)*sqrt(-2*x + 1))/(243* x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=-\frac {11237}{2333772} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4550985 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 18685170 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 11651808 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 128927330 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 134900185 \, \sqrt {-2 \, x + 1}}{277830 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
-11237/2333772*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s qrt(-2*x + 1))) + 1/277830*(4550985*(-2*x + 1)^(9/2) - 18685170*(-2*x + 1) ^(7/2) - 11651808*(-2*x + 1)^(5/2) + 128927330*(-2*x + 1)^(3/2) - 13490018 5*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=-\frac {11237}{2333772} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4550985 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 18685170 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - 11651808 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + 128927330 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 134900185 \, \sqrt {-2 \, x + 1}}{8890560 \, {\left (3 \, x + 2\right )}^{5}} \]
-11237/2333772*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt( 21) + 3*sqrt(-2*x + 1))) + 1/8890560*(4550985*(2*x - 1)^4*sqrt(-2*x + 1) + 18685170*(2*x - 1)^3*sqrt(-2*x + 1) - 11651808*(2*x - 1)^2*sqrt(-2*x + 1) + 128927330*(-2*x + 1)^(3/2) - 134900185*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 1.56 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx=\frac {11237\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1166886}-\frac {\frac {78659\,\sqrt {1-2\,x}}{39366}-\frac {263117\,{\left (1-2\,x\right )}^{3/2}}{137781}+\frac {39632\,{\left (1-2\,x\right )}^{5/2}}{229635}+\frac {4237\,{\left (1-2\,x\right )}^{7/2}}{15309}-\frac {11237\,{\left (1-2\,x\right )}^{9/2}}{166698}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
(11237*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1166886 - ((78659*(1 - 2*x)^(1/2))/39366 - (263117*(1 - 2*x)^(3/2))/137781 + (39632*(1 - 2*x)^( 5/2))/229635 + (4237*(1 - 2*x)^(7/2))/15309 - (11237*(1 - 2*x)^(9/2))/1666 98)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)