Integrand size = 22, antiderivative size = 76 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {76}{1815 (1-2 x)^{3/2}}+\frac {76}{1331 \sqrt {1-2 x}}-\frac {1}{55 (1-2 x)^{3/2} (3+5 x)}-\frac {76 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
76/1815/(1-2*x)^(3/2)-1/55/(1-2*x)^(3/2)/(3+5*x)-76/14641*arctanh(1/11*55^ (1/2)*(1-2*x)^(1/2))*55^(1/2)+76/1331/(1-2*x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {2 \left (-\frac {11 \left (-1113-608 x+2280 x^2\right )}{2 (1-2 x)^{3/2} (3+5 x)}-114 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )}{43923} \]
(2*((-11*(-1113 - 608*x + 2280*x^2))/(2*(1 - 2*x)^(3/2)*(3 + 5*x)) - 114*S qrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/43923
Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {87, 61, 61, 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.
\(\displaystyle \int \frac {3 x+2}{(1-2 x)^{5/2} (5 x+3)^2} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {38}{55} \int \frac {1}{(1-2 x)^{5/2} (5 x+3)}dx-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {38}{55} \left (\frac {5}{11} \int \frac {1}{(1-2 x)^{3/2} (5 x+3)}dx+\frac {2}{33 (1-2 x)^{3/2}}\right )-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {38}{55} \left (\frac {5}{11} \left (\frac {5}{11} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{11 \sqrt {1-2 x}}\right )+\frac {2}{33 (1-2 x)^{3/2}}\right )-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {38}{55} \left (\frac {5}{11} \left (\frac {2}{11 \sqrt {1-2 x}}-\frac {5}{11} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{33 (1-2 x)^{3/2}}\right )-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {38}{55} \left (\frac {5}{11} \left (\frac {2}{11 \sqrt {1-2 x}}-\frac {2}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {2}{33 (1-2 x)^{3/2}}\right )-\frac {1}{55 (1-2 x)^{3/2} (5 x+3)}\) |
-1/55*1/((1 - 2*x)^(3/2)*(3 + 5*x)) + (38*(2/(33*(1 - 2*x)^(3/2)) + (5*(2/ (11*Sqrt[1 - 2*x]) - (2*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11)) /11))/55
3.22.85.3.1 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 + 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((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 = 1.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\frac {2280 x^{2}-608 x -1113}{3993 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {76 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}\) | \(53\) |
derivativedivides | \(\frac {2 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}-\frac {76 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {14}{363 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {74}{1331 \sqrt {1-2 x}}\) | \(54\) |
default | \(\frac {2 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}-\frac {76 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {14}{363 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {74}{1331 \sqrt {1-2 x}}\) | \(54\) |
pseudoelliptic | \(\frac {228 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (10 x^{2}+x -3\right ) \sqrt {55}-25080 x^{2}+6688 x +12243}{\left (1-2 x \right )^{\frac {3}{2}} \left (131769+219615 x \right )}\) | \(60\) |
trager | \(-\frac {\left (2280 x^{2}-608 x -1113\right ) \sqrt {1-2 x}}{3993 \left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {38 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{14641}\) | \(79\) |
1/3993*(2280*x^2-608*x-1113)/(3+5*x)/(1-2*x)^(1/2)/(-1+2*x)-76/14641*arcta nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.18 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {114 \, \sqrt {11} \sqrt {5} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (2280 \, x^{2} - 608 \, x - 1113\right )} \sqrt {-2 \, x + 1}}{43923 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
1/43923*(114*sqrt(11)*sqrt(5)*(20*x^3 - 8*x^2 - 7*x + 3)*log((sqrt(11)*sqr t(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 11*(2280*x^2 - 608*x - 1113)*s qrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)
Time = 31.99 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.43 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {37 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{14641} - \frac {20 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{121} + \frac {74}{1331 \sqrt {1 - 2 x}} + \frac {14}{363 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
37*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55 )/5))/14641 - 20*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/ 4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqr t(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/121 + 74/(1331*sqrt(1 - 2*x)) + 14/(363*(1 - 2*x)**(3/2))
Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {38}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (570 \, {\left (2 \, x - 1\right )}^{2} + 1672 \, x - 1683\right )}}{3993 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
38/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2 *x + 1))) + 2/3993*(570*(2*x - 1)^2 + 1672*x - 1683)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=\frac {38}{14641} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (111 \, x - 94\right )}}{3993 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {5 \, \sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} \]
38/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4/3993*(111*x - 94)/((2*x - 1)*sqrt(-2*x + 1)) - 5/1 331*sqrt(-2*x + 1)/(5*x + 3)
Time = 1.63 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {2+3 x}{(1-2 x)^{5/2} (3+5 x)^2} \, dx=-\frac {\frac {304\,x}{1815}+\frac {76\,{\left (2\,x-1\right )}^2}{1331}-\frac {102}{605}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {76\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641} \]