Integrand size = 22, antiderivative size = 96 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {5}{49 (1-2 x)^{3/2}}+\frac {45}{343 \sqrt {1-2 x}}+\frac {1}{42 (1-2 x)^{3/2} (2+3 x)^2}-\frac {3}{14 (1-2 x)^{3/2} (2+3 x)}-\frac {45}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
5/49/(1-2*x)^(3/2)+1/42/(1-2*x)^(3/2)/(2+3*x)^2-3/14/(1-2*x)^(3/2)/(2+3*x) -45/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+45/343/(1-2*x)^(1/2)
Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {-\frac {7 \left (-1087-2277 x+2160 x^2+4860 x^3\right )}{2 (1-2 x)^{3/2} (2+3 x)^2}-135 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7203} \]
((-7*(-1087 - 2277*x + 2160*x^2 + 4860*x^3))/(2*(1 - 2*x)^(3/2)*(2 + 3*x)^ 2) - 135*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7203
Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {87, 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.
| \(\displaystyle \int \frac {5 x+3}{(1-2 x)^{5/2} (3 x+2)^3} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {3}{2} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^2}dx+\frac {1}{42 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {3}{2} \left (\frac {5}{7} \int \frac {1}{(1-2 x)^{5/2} (3 x+2)}dx-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )+\frac {1}{42 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {3}{2} \left (\frac {5}{7} \left (\frac {3}{7} \int \frac {1}{(1-2 x)^{3/2} (3 x+2)}dx+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )+\frac {1}{42 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {3}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {3}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {2}{7 \sqrt {1-2 x}}\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )+\frac {1}{42 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {3}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )+\frac {1}{42 (1-2 x)^{3/2} (3 x+2)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{2} \left (\frac {5}{7} \left (\frac {3}{7} \left (\frac {2}{7 \sqrt {1-2 x}}-\frac {2}{7} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right )+\frac {2}{21 (1-2 x)^{3/2}}\right )-\frac {1}{7 (1-2 x)^{3/2} (3 x+2)}\right )+\frac {1}{42 (1-2 x)^{3/2} (3 x+2)^2}\) |
1/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (3*(-1/7*1/((1 - 2*x)^(3/2)*(2 + 3*x) ) + (5*(2/(21*(1 - 2*x)^(3/2)) + (3*(2/(7*Sqrt[1 - 2*x]) - (2*Sqrt[3/7]*Ar cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7))/7))/7))/2
3.22.43.3.1 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 = 1.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.60
| method | result | size |
| risch | \(\frac {4860 x^{3}+2160 x^{2}-2277 x -1087}{2058 \sqrt {1-2 x}\, \left (2+3 x \right )^{2} \left (-1+2 x \right )}-\frac {45 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) | \(58\) |
| derivativedivides | \(\frac {\frac {531 \left (1-2 x \right )^{\frac {3}{2}}}{2401}-\frac {171 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {45 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {44}{1029 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {256}{2401 \sqrt {1-2 x}}\) | \(66\) |
| default | \(\frac {\frac {531 \left (1-2 x \right )^{\frac {3}{2}}}{2401}-\frac {171 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {45 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}+\frac {44}{1029 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {256}{2401 \sqrt {1-2 x}}\) | \(66\) |
| pseudoelliptic | \(\frac {\frac {45 \sqrt {1-2 x}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (-1+2 x \right ) \left (2+3 x \right )^{2} \sqrt {21}}{2401}-\frac {810 x^{3}}{343}-\frac {360 x^{2}}{343}+\frac {759 x}{686}+\frac {1087}{2058}}{\left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{2}}\) | \(69\) |
| trager | \(-\frac {\left (4860 x^{3}+2160 x^{2}-2277 x -1087\right ) \sqrt {1-2 x}}{2058 \left (6 x^{2}+x -2\right )^{2}}-\frac {45 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{4802}\) | \(80\) |
1/2058*(4860*x^3+2160*x^2-2277*x-1087)/(1-2*x)^(1/2)/(2+3*x)^2/(-1+2*x)-45 /2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.09 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {135 \, \sqrt {7} \sqrt {3} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (4860 \, x^{3} + 2160 \, x^{2} - 2277 \, x - 1087\right )} \sqrt {-2 \, x + 1}}{14406 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
1/14406*(135*sqrt(7)*sqrt(3)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log((sqr t(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 7*(4860*x^3 + 2160*x^2 - 2277*x - 1087)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)
Time = 88.01 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.69 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {128 \sqrt {21} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {21}}{3} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {21}}{3} \right )}\right )}{16807} - \frac {372 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{343} - \frac {24 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{49} + \frac {256}{2401 \sqrt {1 - 2 x}} + \frac {44}{1029 \left (1 - 2 x\right )^{\frac {3}{2}}} \]
128*sqrt(21)*(log(sqrt(1 - 2*x) - sqrt(21)/3) - log(sqrt(1 - 2*x) + sqrt(2 1)/3))/16807 - 372*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)))/343 - 24*Piecewise((sqrt(21)*(3*lo g(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/1 6 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/ 7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt( 1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/49 + 256/(2401*sqrt(1 - 2*x)) + 44/(1029*(1 - 2*x)**(3/2))
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {45}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1215 \, {\left (2 \, x - 1\right )}^{3} + 4725 \, {\left (2 \, x - 1\right )}^{2} + 7056 \, x - 5684}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
45/4802*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2* x + 1))) - 1/1029*(1215*(2*x - 1)^3 + 4725*(2*x - 1)^2 + 7056*x - 5684)/(9 *(-2*x + 1)^(7/2) - 42*(-2*x + 1)^(5/2) + 49*(-2*x + 1)^(3/2))
Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=\frac {45}{4802} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (384 \, x - 269\right )}}{7203 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {9 \, {\left (59 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 133 \, \sqrt {-2 \, x + 1}\right )}}{9604 \, {\left (3 \, x + 2\right )}^{2}} \]
45/4802*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3 *sqrt(-2*x + 1))) + 4/7203*(384*x - 269)/((2*x - 1)*sqrt(-2*x + 1)) + 9/96 04*(59*(-2*x + 1)^(3/2) - 133*sqrt(-2*x + 1))/(3*x + 2)^2
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.75 \[ \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^3} \, dx=-\frac {45\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {16\,x}{21}+\frac {25\,{\left (2\,x-1\right )}^2}{49}+\frac {45\,{\left (2\,x-1\right )}^3}{343}-\frac {116}{189}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \]