Integrand size = 17, antiderivative size = 96 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {35}{3993 (1-2 x)^{3/2}}+\frac {175}{14641 \sqrt {1-2 x}}-\frac {1}{22 (1-2 x)^{3/2} (3+5 x)^2}-\frac {7}{242 (1-2 x)^{3/2} (3+5 x)}-\frac {175 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]
35/3993/(1-2*x)^(3/2)-1/22/(1-2*x)^(3/2)/(3+5*x)^2-7/242/(1-2*x)^(3/2)/(3+ 5*x)-175/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+175/14641/(1 -2*x)^(1/2)
Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {-\frac {11 \left (-4764-22995 x+17500 x^2+52500 x^3\right )}{2 (1-2 x)^{3/2} (3+5 x)^2}-525 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{483153} \]
((-11*(-4764 - 22995*x + 17500*x^2 + 52500*x^3))/(2*(1 - 2*x)^(3/2)*(3 + 5 *x)^2) - 525*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/483153
Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {52, 52, 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.
\(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {7}{22} \int \frac {1}{(1-2 x)^{5/2} (5 x+3)^2}dx-\frac {1}{22 (1-2 x)^{3/2} (5 x+3)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {7}{22} \left (\frac {5}{11} \int \frac {1}{(1-2 x)^{5/2} (5 x+3)}dx-\frac {1}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1}{22 (1-2 x)^{3/2} (5 x+3)^2}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {7}{22} \left (\frac {5}{11} \left (\frac {5}{11} \int \frac {1}{(1-2 x)^{3/2} (5 x+3)}dx+\frac {2}{33 (1-2 x)^{3/2}}\right )-\frac {1}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1}{22 (1-2 x)^{3/2} (5 x+3)^2}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {7}{22} \left (\frac {5}{11} \left (\frac {5}{11} \left (\frac {5}{11} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{11 \sqrt {1-2 x}}\right )+\frac {2}{33 (1-2 x)^{3/2}}\right )-\frac {1}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1}{22 (1-2 x)^{3/2} (5 x+3)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {7}{22} \left (\frac {5}{11} \left (\frac {5}{11} \left (\frac {2}{11 \sqrt {1-2 x}}-\frac {5}{11} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{33 (1-2 x)^{3/2}}\right )-\frac {1}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1}{22 (1-2 x)^{3/2} (5 x+3)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {7}{22} \left (\frac {5}{11} \left (\frac {5}{11} \left (\frac {2}{11 \sqrt {1-2 x}}-\frac {2}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {2}{33 (1-2 x)^{3/2}}\right )-\frac {1}{11 (1-2 x)^{3/2} (5 x+3)}\right )-\frac {1}{22 (1-2 x)^{3/2} (5 x+3)^2}\) |
-1/22*1/((1 - 2*x)^(3/2)*(3 + 5*x)^2) + (7*(-1/11*1/((1 - 2*x)^(3/2)*(3 + 5*x)) + (5*(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))/11))/22
3.22.96.3.1 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] && ILtQ[m, -1] && FractionQ[n] && LtQ[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] && 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_)^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 = 1.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.60
method | result | size |
risch | \(\frac {52500 x^{3}+17500 x^{2}-22995 x -4764}{87846 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {175 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}\) | \(58\) |
derivativedivides | \(\frac {\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {325 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {175 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {8}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {120}{14641 \sqrt {1-2 x}}\) | \(66\) |
default | \(\frac {\frac {125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {325 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {175 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {8}{3993 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {120}{14641 \sqrt {1-2 x}}\) | \(66\) |
pseudoelliptic | \(\frac {\frac {175 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (-1+2 x \right ) \left (3+5 x \right )^{2} \sqrt {55}}{161051}-\frac {8750 x^{3}}{14641}-\frac {8750 x^{2}}{43923}+\frac {7665 x}{29282}+\frac {794}{14641}}{\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{2}}\) | \(69\) |
trager | \(-\frac {\left (52500 x^{3}+17500 x^{2}-22995 x -4764\right ) \sqrt {1-2 x}}{87846 \left (10 x^{2}+x -3\right )^{2}}+\frac {175 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{322102}\) | \(80\) |
1/87846*(52500*x^3+17500*x^2-22995*x-4764)/(3+5*x)^2/(1-2*x)^(1/2)/(-1+2*x )-175/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {525 \, \sqrt {11} \sqrt {5} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (52500 \, x^{3} + 17500 \, x^{2} - 22995 \, x - 4764\right )} \sqrt {-2 \, x + 1}}{966306 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
1/966306*(525*sqrt(11)*sqrt(5)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(( sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 11*(52500*x^3 + 17 500*x^2 - 22995*x - 4764)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
Result contains complex when optimal does not.
Time = 5.65 (sec) , antiderivative size = 983, normalized size of antiderivative = 10.24 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\text {Too large to display} \]
Piecewise((-105000*sqrt(55)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2 )*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/(96630600*sqrt(-1 + 11/(10*(x + 3/5) ))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)** (153/2)) + 52500*sqrt(55)*I*pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(15 5/2)/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660 *sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 115500*sqrt(55)*sqrt(- 1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)*acosh(sqrt(110)/(10*sqrt(x + 3/5 )))/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660* sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 57750*sqrt(55)*I*pi*sqr t(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 577500*sqrt(2)*(x + 3/5)**77/(96630600*sqrt(-1 + 11/(10* (x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 847000*sqrt(2)*(x + 3/5)**76/(96630600*sqrt(-1 + 11/(1 0*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))* (x + 3/5)**(153/2)) + 139755*sqrt(2)*(x + 3/5)**75/(96630600*sqrt(-1 + 11/ (10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)) )*(x + 3/5)**(153/2)) + 43923*sqrt(2)*(x + 3/5)**74/(96630600*sqrt(-1 + 11 /(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5) ))*(x + 3/5)**(153/2)), 1/Abs(x + 3/5) > 10/11), (105000*sqrt(55)*I*sqr...
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {175}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {13125 \, {\left (2 \, x - 1\right )}^{3} + 48125 \, {\left (2 \, x - 1\right )}^{2} + 67760 \, x - 44528}{43923 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
175/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) - 1/43923*(13125*(2*x - 1)^3 + 48125*(2*x - 1)^2 + 67760*x - 4 4528)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=\frac {175}{322102} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (45 \, x - 28\right )}}{43923 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {25 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \]
175/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 16/43923*(45*x - 28)/((2*x - 1)*sqrt(-2*x + 1)) + 25/5324*(5*(-2*x + 1)^(3/2) - 13*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx=-\frac {175\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051}-\frac {\frac {112\,x}{1815}+\frac {175\,{\left (2\,x-1\right )}^2}{3993}+\frac {175\,{\left (2\,x-1\right )}^3}{14641}-\frac {368}{9075}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}} \]