Integrand size = 22, antiderivative size = 110 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx=\frac {(1-2 x)^{3/2}}{84 (2+3 x)^4}-\frac {5 \sqrt {1-2 x}}{28 (2+3 x)^3}+\frac {5 \sqrt {1-2 x}}{392 (2+3 x)^2}+\frac {15 \sqrt {1-2 x}}{2744 (2+3 x)}+\frac {5 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1372} \]
1/84*(1-2*x)^(3/2)/(2+3*x)^4+5/9604*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21 ^(1/2)-5/28*(1-2*x)^(1/2)/(2+3*x)^3+5/392*(1-2*x)^(1/2)/(2+3*x)^2+15/2744* (1-2*x)^(1/2)/(2+3*x)
Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx=\frac {\frac {7 \sqrt {1-2 x} \left (-2062-1726 x+3375 x^2+1215 x^3\right )}{2 (2+3 x)^4}+15 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{28812} \]
((7*Sqrt[1 - 2*x]*(-2062 - 1726*x + 3375*x^2 + 1215*x^3))/(2*(2 + 3*x)^4) + 15*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28812
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {87, 51, 52, 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.
\(\displaystyle \int \frac {\sqrt {1-2 x} (5 x+3)}{(3 x+2)^5} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {45}{28} \int \frac {\sqrt {1-2 x}}{(3 x+2)^4}dx+\frac {(1-2 x)^{3/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {45}{28} \left (-\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {45}{28} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {45}{28} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {45}{28} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{84 (3 x+2)^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {45}{28} \left (\frac {1}{9} \left (\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}-\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )\right )-\frac {\sqrt {1-2 x}}{9 (3 x+2)^3}\right )+\frac {(1-2 x)^{3/2}}{84 (3 x+2)^4}\) |
(1 - 2*x)^(3/2)/(84*(2 + 3*x)^4) + (45*(-1/9*Sqrt[1 - 2*x]/(2 + 3*x)^3 + ( Sqrt[1 - 2*x]/(14*(2 + 3*x)^2) - (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2*Arc Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14)/9))/28
3.19.1.3.1 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] :> 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_))*((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 = 0.97 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51
method | result | size |
risch | \(-\frac {2430 x^{4}+5535 x^{3}-6827 x^{2}-2398 x +2062}{8232 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\) | \(56\) |
pseudoelliptic | \(\frac {30 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}+7 \sqrt {1-2 x}\, \left (1215 x^{3}+3375 x^{2}-1726 x -2062\right )}{57624 \left (2+3 x \right )^{4}}\) | \(60\) |
derivativedivides | \(-\frac {1296 \left (\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{21952}-\frac {55 \left (1-2 x \right )^{\frac {5}{2}}}{28224}+\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {5 \sqrt {1-2 x}}{1728}\right )}{\left (-4-6 x \right )^{4}}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\) | \(66\) |
default | \(-\frac {1296 \left (\frac {5 \left (1-2 x \right )^{\frac {7}{2}}}{21952}-\frac {55 \left (1-2 x \right )^{\frac {5}{2}}}{28224}+\frac {209 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {5 \sqrt {1-2 x}}{1728}\right )}{\left (-4-6 x \right )^{4}}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{9604}\) | \(66\) |
trager | \(\frac {\left (1215 x^{3}+3375 x^{2}-1726 x -2062\right ) \sqrt {1-2 x}}{8232 \left (2+3 x \right )^{4}}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{19208}\) | \(77\) |
-1/8232*(2430*x^4+5535*x^3-6827*x^2-2398*x+2062)/(2+3*x)^4/(1-2*x)^(1/2)+5 /9604*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx=\frac {15 \, \sqrt {7} \sqrt {3} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 7 \, {\left (1215 \, x^{3} + 3375 \, x^{2} - 1726 \, x - 2062\right )} \sqrt {-2 \, x + 1}}{57624 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/57624*(15*sqrt(7)*sqrt(3)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(- (sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 7*(1215*x^3 + 3375 *x^2 - 1726*x - 2062)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx=-\frac {5}{19208} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1215 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 10395 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 10241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 15435 \, \sqrt {-2 \, x + 1}}{4116 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
-5/19208*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2 *x + 1))) - 1/4116*(1215*(-2*x + 1)^(7/2) - 10395*(-2*x + 1)^(5/2) + 10241 *(-2*x + 1)^(3/2) + 15435*sqrt(-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 = 100, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx=-\frac {5}{19208} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1215 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 10395 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 10241 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 15435 \, \sqrt {-2 \, x + 1}}{65856 \, {\left (3 \, x + 2\right )}^{4}} \]
-5/19208*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/65856*(1215*(2*x - 1)^3*sqrt(-2*x + 1) + 10395*(2*x - 1)^2*sqrt(-2*x + 1) - 10241*(-2*x + 1)^(3/2) - 15435*sqrt(-2*x + 1))/(3 *x + 2)^4
Time = 0.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {1-2 x} (3+5 x)}{(2+3 x)^5} \, dx=\frac {5\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{9604}-\frac {\frac {5\,\sqrt {1-2\,x}}{108}+\frac {209\,{\left (1-2\,x\right )}^{3/2}}{6804}-\frac {55\,{\left (1-2\,x\right )}^{5/2}}{1764}+\frac {5\,{\left (1-2\,x\right )}^{7/2}}{1372}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]