Integrand size = 22, antiderivative size = 76 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {20}{147 (1-2 x)^{3/2}}+\frac {60}{343 \sqrt {1-2 x}}+\frac {1}{21 (1-2 x)^{3/2} (2+3 x)}-\frac {60}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
20/147/(1-2*x)^(3/2)+1/21/(1-2*x)^(3/2)/(2+3*x)-60/2401*arctanh(1/7*21^(1/ 2)*(1-2*x)^(1/2))*21^(1/2)+60/343/(1-2*x)^(1/2)
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {2 \left (-539-420 (1-2 x)+270 (1-2 x)^2\right )}{1029 (-7+3 (1-2 x)) (1-2 x)^{3/2}}-\frac {60}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
(2*(-539 - 420*(1 - 2*x) + 270*(1 - 2*x)^2))/(1029*(-7 + 3*(1 - 2*x))*(1 - 2*x)^(3/2)) - (60*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343
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.
\(\displaystyle \int \frac {5 x+3}{(1-2 x)^{5/2} (3 x+2)^2} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {10}{7} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)}dx+\frac {1}{21 (1-2 x)^{3/2} (3 x+2)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {10}{7} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx+\frac {2}{21 (1-2 x)^{3/2}}\right )+\frac {1}{21 (1-2 x)^{3/2} (3 x+2)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {10}{7} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )+\frac {1}{21 (1-2 x)^{3/2} (3 x+2)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {10}{7} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )+\frac {1}{21 (1-2 x)^{3/2} (3 x+2)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {10}{7} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )+\frac {1}{21 (1-2 x)^{3/2} (3 x+2)}\) |
1/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) + (10*(2/(21*(1 - 2*x)^(3/2)) + (3*(2/(7* Sqrt[1 - 2*x]) - (2*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7))/7))/7
3.22.42.3.1 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 = 3.46 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\frac {1080 x^{2}-240 x -689}{1029 \left (-1+2 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )}-\frac {60 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(53\) |
derivativedivides | \(-\frac {2 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}-\frac {60 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {22}{147 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {62}{343 \sqrt {1-2 x}}\) | \(54\) |
default | \(-\frac {2 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}-\frac {60 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {22}{147 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {62}{343 \sqrt {1-2 x}}\) | \(54\) |
pseudoelliptic | \(\frac {180 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (6 x^{2}+x -2\right ) \sqrt {21}-7560 x^{2}+1680 x +4823}{\left (1-2 x \right )^{\frac {3}{2}} \left (14406+21609 x \right )}\) | \(60\) |
trager | \(-\frac {\left (1080 x^{2}-240 x -689\right ) \sqrt {1-2 x}}{1029 \left (-1+2 x \right )^{2} \left (2+3 x \right )}+\frac {30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2401}\) | \(79\) |
1/1029*(1080*x^2-240*x-689)/(-1+2*x)/(1-2*x)^(1/2)/(2+3*x)-60/2401*arctanh (1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.18 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {90 \, \sqrt {7} \sqrt {3} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (1080 \, x^{2} - 240 \, x - 689\right )} \sqrt {-2 \, x + 1}}{7203 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
1/7203*(90*sqrt(7)*sqrt(3)*(12*x^3 - 4*x^2 - 5*x + 2)*log((sqrt(7)*sqrt(3) *sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 7*(1080*x^2 - 240*x - 689)*sqrt(-2 *x + 1))/(12*x^3 - 4*x^2 - 5*x + 2)
Time = 32.22 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.43 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {31 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{2401} + \frac {12 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{49} + \frac {62}{343 \sqrt {1 - 2 x}} + \frac {22}{147 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
31*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(21 )/3))/2401 + 12*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1 )) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21) /3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/49 + 62/(343*sqrt(1 - 2*x)) + 22/(147 *(1 - 2*x)**(3/2))
Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {30}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (270 \, {\left (2 \, x - 1\right )}^{2} + 840 \, x - 959\right )}}{1029 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
30/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* x + 1))) + 2/1029*(270*(2*x - 1)^2 + 840*x - 959)/(3*(-2*x + 1)^(5/2) - 7* (-2*x + 1)^(3/2))
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=\frac {30}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (93 \, x - 85\right )}}{1029 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {3 \, \sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} \]
30/2401*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 4/1029*(93*x - 85)/((2*x - 1)*sqrt(-2*x + 1)) + 3/343* sqrt(-2*x + 1)/(3*x + 2)
Time = 1.62 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^2} \, dx=-\frac {\frac {80\,x}{147}+\frac {60\,{\left (2\,x-1\right )}^2}{343}-\frac {274}{441}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}-\frac {60\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401} \]