Integrand size = 22, antiderivative size = 108 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {(1-2 x)^{7/2}}{84 (2+3 x)^4}-\frac {139 (1-2 x)^{5/2}}{756 (2+3 x)^3}+\frac {695 (1-2 x)^{3/2}}{4536 (2+3 x)^2}-\frac {695 \sqrt {1-2 x}}{4536 (2+3 x)}+\frac {695 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268 \sqrt {21}} \]
1/84*(1-2*x)^(7/2)/(2+3*x)^4-139/756*(1-2*x)^(5/2)/(2+3*x)^3+695/4536*(1-2 *x)^(3/2)/(2+3*x)^2+695/47628*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2) -695/4536*(1-2*x)^(1/2)/(2+3*x)
Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (4394+18394 x+43971 x^2+41715 x^3\right )}{2 (2+3 x)^4}+695 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{47628} \]
((-21*Sqrt[1 - 2*x]*(4394 + 18394*x + 43971*x^2 + 41715*x^3))/(2*(2 + 3*x) ^4) + 695*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/47628
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {87, 51, 51, 51, 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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)}{(3 x+2)^5} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {139}{84} \int \frac {(1-2 x)^{5/2}}{(3 x+2)^4}dx+\frac {(1-2 x)^{7/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {139}{84} \left (-\frac {5}{9} \int \frac {(1-2 x)^{3/2}}{(3 x+2)^3}dx-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{7/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {139}{84} \left (-\frac {5}{9} \left (-\frac {1}{2} \int \frac {\sqrt {1-2 x}}{(3 x+2)^2}dx-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{7/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {139}{84} \left (-\frac {5}{9} \left (\frac {1}{2} \left (\frac {1}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {\sqrt {1-2 x}}{3 (3 x+2)}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{7/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {139}{84} \left (-\frac {5}{9} \left (\frac {1}{2} \left (\frac {\sqrt {1-2 x}}{3 (3 x+2)}-\frac {1}{3} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{7/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {139}{84} \left (-\frac {5}{9} \left (\frac {1}{2} \left (\frac {\sqrt {1-2 x}}{3 (3 x+2)}-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}\right )-\frac {(1-2 x)^{3/2}}{6 (3 x+2)^2}\right )-\frac {(1-2 x)^{5/2}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{7/2}}{84 (3 x+2)^4}\) |
(1 - 2*x)^(7/2)/(84*(2 + 3*x)^4) + (139*(-1/9*(1 - 2*x)^(5/2)/(2 + 3*x)^3 - (5*(-1/6*(1 - 2*x)^(3/2)/(2 + 3*x)^2 + (Sqrt[1 - 2*x]/(3*(2 + 3*x)) - (2 *ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]))/2))/9))/84
3.20.39.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 3.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {83430 x^{4}+46227 x^{3}-7183 x^{2}-9606 x -4394}{4536 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {695 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{47628}\) | \(56\) |
pseudoelliptic | \(\frac {1390 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (41715 x^{3}+43971 x^{2}+18394 x +4394\right )}{95256 \left (2+3 x \right )^{4}}\) | \(60\) |
derivativedivides | \(-\frac {1296 \left (-\frac {515 \left (1-2 x \right )^{\frac {7}{2}}}{36288}+\frac {10147 \left (1-2 x \right )^{\frac {5}{2}}}{139968}-\frac {53515 \left (1-2 x \right )^{\frac {3}{2}}}{419904}+\frac {34055 \sqrt {1-2 x}}{419904}\right )}{\left (-4-6 x \right )^{4}}+\frac {695 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{47628}\) | \(66\) |
default | \(-\frac {1296 \left (-\frac {515 \left (1-2 x \right )^{\frac {7}{2}}}{36288}+\frac {10147 \left (1-2 x \right )^{\frac {5}{2}}}{139968}-\frac {53515 \left (1-2 x \right )^{\frac {3}{2}}}{419904}+\frac {34055 \sqrt {1-2 x}}{419904}\right )}{\left (-4-6 x \right )^{4}}+\frac {695 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{47628}\) | \(66\) |
trager | \(-\frac {\left (41715 x^{3}+43971 x^{2}+18394 x +4394\right ) \sqrt {1-2 x}}{4536 \left (2+3 x \right )^{4}}+\frac {695 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{95256}\) | \(77\) |
1/4536*(83430*x^4+46227*x^3-7183*x^2-9606*x-4394)/(2+3*x)^4/(1-2*x)^(1/2)+ 695/47628*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.22 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {695 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (41715 \, x^{3} + 43971 \, x^{2} + 18394 \, x + 4394\right )} \sqrt {-2 \, x + 1}}{95256 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/95256*(695*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*(41715*x^3 + 43971*x^2 + 1839 4*x + 4394)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx=-\frac {695}{95256} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {41715 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 213087 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 374605 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 238385 \, \sqrt {-2 \, x + 1}}{2268 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
-695/95256*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( -2*x + 1))) + 1/2268*(41715*(-2*x + 1)^(7/2) - 213087*(-2*x + 1)^(5/2) + 3 74605*(-2*x + 1)^(3/2) - 238385*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx=-\frac {695}{95256} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {41715 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 213087 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 374605 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 238385 \, \sqrt {-2 \, x + 1}}{36288 \, {\left (3 \, x + 2\right )}^{4}} \]
-695/95256*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/36288*(41715*(2*x - 1)^3*sqrt(-2*x + 1) + 213087* (2*x - 1)^2*sqrt(-2*x + 1) - 374605*(-2*x + 1)^(3/2) + 238385*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 1.48 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)}{(2+3 x)^5} \, dx=\frac {695\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{47628}-\frac {\frac {34055\,\sqrt {1-2\,x}}{26244}-\frac {53515\,{\left (1-2\,x\right )}^{3/2}}{26244}+\frac {10147\,{\left (1-2\,x\right )}^{5/2}}{8748}-\frac {515\,{\left (1-2\,x\right )}^{7/2}}{2268}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]