Integrand size = 24, antiderivative size = 80 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (541+857 x)}{2058 (2+3 x)^2}+\frac {2245 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
2245/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+11/7*(3+5*x)^2/(2+ 3*x)^2/(1-2*x)^(1/2)+5/2058*(541+857*x)*(1-2*x)^(1/2)/(2+3*x)^2
Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\frac {195657-168175 (1-2 x)+36140 (1-2 x)^2}{1029 (-7+3 (1-2 x))^2 \sqrt {1-2 x}}+\frac {2245 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
(195657 - 168175*(1 - 2*x) + 36140*(1 - 2*x)^2)/(1029*(-7 + 3*(1 - 2*x))^2 *Sqrt[1 - 2*x]) + (2245*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])
Time = 0.18 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {109, 27, 162, 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.
\(\displaystyle \int \frac {(5 x+3)^3}{(1-2 x)^{3/2} (3 x+2)^3} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {1}{7} \int \frac {5 (x+5) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \int \frac {(x+5) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^3}dx\) |
\(\Big \downarrow \) 162 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \left (\frac {449}{294} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x} (857 x+541)}{294 (3 x+2)^2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \left (-\frac {449}{294} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x} (857 x+541)}{294 (3 x+2)^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^2}-\frac {5}{7} \left (-\frac {449 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}-\frac {\sqrt {1-2 x} (857 x+541)}{294 (3 x+2)^2}\right )\) |
(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*(-1/294*(Sqrt[1 - 2*x] *(541 + 857*x))/(2 + 3*x)^2 - (449*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147* Sqrt[21])))/7
3.22.1.3.1 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[(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_) )*((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_)^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 = 1.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.58
method | result | size |
risch | \(\frac {72280 x^{2}+95895 x +31811}{2058 \left (2+3 x \right )^{2} \sqrt {1-2 x}}+\frac {2245 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}\) | \(46\) |
pseudoelliptic | \(\frac {\frac {2245 \sqrt {21}\, \left (\frac {2}{3}+x \right )^{2} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2401}+\frac {36140 x^{2}}{1029}+\frac {31965 x}{686}+\frac {31811}{2058}}{\left (2+3 x \right )^{2} \sqrt {1-2 x}}\) | \(56\) |
derivativedivides | \(-\frac {18 \left (-\frac {203 \left (1-2 x \right )^{\frac {3}{2}}}{54}+\frac {469 \sqrt {1-2 x}}{54}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {2245 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}+\frac {1331}{343 \sqrt {1-2 x}}\) | \(57\) |
default | \(-\frac {18 \left (-\frac {203 \left (1-2 x \right )^{\frac {3}{2}}}{54}+\frac {469 \sqrt {1-2 x}}{54}\right )}{343 \left (-4-6 x \right )^{2}}+\frac {2245 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{21609}+\frac {1331}{343 \sqrt {1-2 x}}\) | \(57\) |
trager | \(-\frac {\left (72280 x^{2}+95895 x +31811\right ) \sqrt {1-2 x}}{2058 \left (2+3 x \right )^{2} \left (-1+2 x \right )}+\frac {2245 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{43218}\) | \(79\) |
1/2058*(72280*x^2+95895*x+31811)/(2+3*x)^2/(1-2*x)^(1/2)+2245/21609*arctan h(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\frac {2245 \, \sqrt {21} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (72280 \, x^{2} + 95895 \, x + 31811\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]
1/43218*(2245*sqrt(21)*(18*x^3 + 15*x^2 - 4*x - 4)*log((3*x - sqrt(21)*sqr t(-2*x + 1) - 5)/(3*x + 2)) - 21*(72280*x^2 + 95895*x + 31811)*sqrt(-2*x + 1))/(18*x^3 + 15*x^2 - 4*x - 4)
Time = 140.71 (sec) , antiderivative size = 342, normalized size of antiderivative = 4.28 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=- \frac {3469 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{64827} - \frac {412 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{441} - \frac {8 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{63} + \frac {1331}{343 \sqrt {1 - 2 x}} \]
-3469*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt (21)/3))/64827 - 412*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/ 7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqr t(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/441 - 8*Piecewise((sqrt(21)*(3*l og(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/ 16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x) /7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt (1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/63 + 1331/(343*sqrt(1 - 2*x))
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=-\frac {2245}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (18070 \, {\left (2 \, x - 1\right )}^{2} + 168175 \, x + 13741\right )}}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 49 \, \sqrt {-2 \, x + 1}\right )}} \]
-2245/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt (-2*x + 1))) + 2/1029*(18070*(2*x - 1)^2 + 168175*x + 13741)/(9*(-2*x + 1) ^(5/2) - 42*(-2*x + 1)^(3/2) + 49*sqrt(-2*x + 1))
Time = 0.34 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=-\frac {2245}{43218} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1331}{343 \, \sqrt {-2 \, x + 1}} + \frac {29 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 67 \, \sqrt {-2 \, x + 1}}{588 \, {\left (3 \, x + 2\right )}^{2}} \]
-2245/43218*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1331/343/sqrt(-2*x + 1) + 1/588*(29*(-2*x + 1)^(3/ 2) - 67*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^3} \, dx=\frac {2245\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}+\frac {\frac {48050\,x}{1323}+\frac {36140\,{\left (2\,x-1\right )}^2}{9261}+\frac {3926}{1323}}{\frac {49\,\sqrt {1-2\,x}}{9}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{3}+{\left (1-2\,x\right )}^{5/2}} \]