Integrand size = 22, antiderivative size = 81 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {213}{6655 \sqrt {1-2 x}}-\frac {1}{110 \sqrt {1-2 x} (3+5 x)^2}-\frac {71}{1210 \sqrt {1-2 x} (3+5 x)}-\frac {213 \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331 \sqrt {55}} \]
-213/73205*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+213/6655/(1-2*x)^ (1/2)-1/110/(3+5*x)^2/(1-2*x)^(1/2)-71/1210/(3+5*x)/(1-2*x)^(1/2)
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {\frac {55 \left (274+1775 x+2130 x^2\right )}{\sqrt {1-2 x} (3+5 x)^2}-426 \sqrt {55} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{146410} \]
((55*(274 + 1775*x + 2130*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^2) - 426*Sqrt[55] *ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/146410
Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {87, 52, 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.
\(\displaystyle \int \frac {3 x+2}{(1-2 x)^{3/2} (5 x+3)^3} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {71}{110} \int \frac {1}{(1-2 x)^{3/2} (5 x+3)^2}dx-\frac {1}{110 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {71}{110} \left (\frac {3}{11} \int \frac {1}{(1-2 x)^{3/2} (5 x+3)}dx-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {1}{110 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {71}{110} \left (\frac {3}{11} \left (\frac {5}{11} \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx+\frac {2}{11 \sqrt {1-2 x}}\right )-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {1}{110 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {71}{110} \left (\frac {3}{11} \left (\frac {2}{11 \sqrt {1-2 x}}-\frac {5}{11} \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {1}{110 \sqrt {1-2 x} (5 x+3)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {71}{110} \left (\frac {3}{11} \left (\frac {2}{11 \sqrt {1-2 x}}-\frac {2}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )-\frac {1}{11 \sqrt {1-2 x} (5 x+3)}\right )-\frac {1}{110 \sqrt {1-2 x} (5 x+3)^2}\) |
-1/110*1/(Sqrt[1 - 2*x]*(3 + 5*x)^2) + (71*(-1/11*1/(Sqrt[1 - 2*x]*(3 + 5* x)) + (3*(2/(11*Sqrt[1 - 2*x]) - (2*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11))/11))/110
3.22.30.3.1 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_))*((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 = 1.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {2130 x^{2}+1775 x +274}{2662 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {213 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}\) | \(46\) |
pseudoelliptic | \(-\frac {1065 \left (\sqrt {55}\, \left (x +\frac {3}{5}\right )^{2} \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )-11 x^{2}-\frac {55 x}{6}-\frac {1507}{1065}\right )}{14641 \sqrt {1-2 x}\, \left (3+5 x \right )^{2}}\) | \(56\) |
derivativedivides | \(\frac {\frac {365 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {75 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {213 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}+\frac {28}{1331 \sqrt {1-2 x}}\) | \(57\) |
default | \(\frac {\frac {365 \left (1-2 x \right )^{\frac {3}{2}}}{1331}-\frac {75 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {213 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{73205}+\frac {28}{1331 \sqrt {1-2 x}}\) | \(57\) |
trager | \(-\frac {\left (2130 x^{2}+1775 x +274\right ) \sqrt {1-2 x}}{2662 \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {213 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{146410}\) | \(79\) |
1/2662*(2130*x^2+1775*x+274)/(3+5*x)^2/(1-2*x)^(1/2)-213/73205*arctanh(1/1 1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.04 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {213 \, \sqrt {55} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (2130 \, x^{2} + 1775 \, x + 274\right )} \sqrt {-2 \, x + 1}}{146410 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
1/146410*(213*sqrt(55)*(50*x^3 + 35*x^2 - 12*x - 9)*log((5*x + sqrt(55)*sq rt(-2*x + 1) - 8)/(5*x + 3)) - 55*(2130*x^2 + 1775*x + 274)*sqrt(-2*x + 1) )/(50*x^3 + 35*x^2 - 12*x - 9)
Time = 73.08 (sec) , antiderivative size = 342, normalized size of antiderivative = 4.22 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {14 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{14641} - \frac {140 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{121} + \frac {8 \left (\begin {cases} \frac {\sqrt {55} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )^{2}}\right )}{6655} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{11} + \frac {28}{1331 \sqrt {1 - 2 x}} \]
14*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55 )/5))/14641 - 140*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)/1 1 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sq rt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/121 + 8*Piecewise((sqrt(55)*(3* log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)) - 1/(16*(sqrt(55 )*sqrt(1 - 2*x)/11 - 1)**2))/6655, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/11 + 28/(1331*sqrt(1 - 2*x))
Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {213}{146410} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1065 \, {\left (2 \, x - 1\right )}^{2} + 7810 \, x - 517}{1331 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \]
213/146410*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt( -2*x + 1))) + 1/1331*(1065*(2*x - 1)^2 + 7810*x - 517)/(25*(-2*x + 1)^(5/2 ) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {213}{146410} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {28}{1331 \, \sqrt {-2 \, x + 1}} + \frac {5 \, {\left (73 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 165 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \]
213/146410*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 28/1331/sqrt(-2*x + 1) + 5/5324*(73*(-2*x + 1)^(3/ 2) - 165*sqrt(-2*x + 1))/(5*x + 3)^2
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.77 \[ \int \frac {2+3 x}{(1-2 x)^{3/2} (3+5 x)^3} \, dx=\frac {\frac {142\,x}{605}+\frac {213\,{\left (2\,x-1\right )}^2}{6655}-\frac {47}{3025}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\left (1-2\,x\right )}^{3/2}}{5}+{\left (1-2\,x\right )}^{5/2}}-\frac {213\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{73205} \]