Integrand size = 22, antiderivative size = 108 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {\sqrt {1-2 x}}{84 (2+3 x)^4}-\frac {19 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {95 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {95 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {95 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \]
-95/86436*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1/84*(1-2*x)^(1/2)/ (2+3*x)^4-19/252*(1-2*x)^(1/2)/(2+3*x)^3-95/3528*(1-2*x)^(1/2)/(2+3*x)^2-9 5/8232*(1-2*x)^(1/2)/(2+3*x)
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (2790+7942 x+7125 x^2+2565 x^3\right )}{2 (2+3 x)^4}-95 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{86436} \]
((-21*Sqrt[1 - 2*x]*(2790 + 7942*x + 7125*x^2 + 2565*x^3))/(2*(2 + 3*x)^4) - 95*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/86436
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {87, 52, 52, 52, 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.
| \(\displaystyle \int \frac {5 x+3}{\sqrt {1-2 x} (3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {19}{12} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {19}{12} \left (\frac {5}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {19}{12} \left (\frac {5}{21} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {19}{12} \left (\frac {5}{21} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {19}{12} \left (\frac {5}{21} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {19}{12} \left (\frac {5}{21} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x}}{21 (3 x+2)^3}\right )+\frac {\sqrt {1-2 x}}{84 (3 x+2)^4}\) |
Sqrt[1 - 2*x]/(84*(2 + 3*x)^4) + (19*(-1/21*Sqrt[1 - 2*x]/(2 + 3*x)^3 + (5 *(-1/14*Sqrt[1 - 2*x]/(2 + 3*x)^2 + (3*(-1/7*Sqrt[1 - 2*x]/(2 + 3*x) - (2* ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21])))/14))/21))/12
3.21.11.3.1 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] :> 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 = 1.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {5130 x^{4}+11685 x^{3}+8759 x^{2}-2362 x -2790}{8232 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {95 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(56\) |
| pseudoelliptic | \(\frac {-190 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (2565 x^{3}+7125 x^{2}+7942 x +2790\right )}{172872 \left (2+3 x \right )^{4}}\) | \(60\) |
| derivativedivides | \(-\frac {1296 \left (-\frac {95 \left (1-2 x \right )^{\frac {7}{2}}}{197568}+\frac {1045 \left (1-2 x \right )^{\frac {5}{2}}}{254016}-\frac {1387 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {1447 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{4}}-\frac {95 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(66\) |
| default | \(-\frac {1296 \left (-\frac {95 \left (1-2 x \right )^{\frac {7}{2}}}{197568}+\frac {1045 \left (1-2 x \right )^{\frac {5}{2}}}{254016}-\frac {1387 \left (1-2 x \right )^{\frac {3}{2}}}{108864}+\frac {1447 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{4}}-\frac {95 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(66\) |
| trager | \(-\frac {\left (2565 x^{3}+7125 x^{2}+7942 x +2790\right ) \sqrt {1-2 x}}{8232 \left (2+3 x \right )^{4}}+\frac {95 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{172872}\) | \(77\) |
1/8232*(5130*x^4+11685*x^3+8759*x^2-2362*x-2790)/(2+3*x)^4/(1-2*x)^(1/2)-9 5/86436*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.92 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {95 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2565 \, x^{3} + 7125 \, x^{2} + 7942 \, x + 2790\right )} \sqrt {-2 \, x + 1}}{172872 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/172872*(95*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(2565*x^3 + 7125*x^2 + 7942*x + 2790)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (94) = 188\).
Time = 159.15 (sec) , antiderivative size = 469, normalized size of antiderivative = 4.34 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx=- \frac {80 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} - \frac {32 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {35 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{256} - \frac {35 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{256} + \frac {35}{256 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {15}{256 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {5}{192 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} + \frac {1}{128 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{4}} + \frac {35}{256 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {15}{256 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} + \frac {5}{192 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}} - \frac {1}{128 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{4}}\right )}{50421} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right )}{3} \]
-80*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32 + 5*log(s qrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**3))/7203, (sqrt (1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/3 - 32*Piecewise ((sqrt(21)*(35*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/256 - 35*log(sqrt(21)*sqr t(1 - 2*x)/7 + 1)/256 + 35/(256*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 15/(256* (sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 5/(192*(sqrt(21)*sqrt(1 - 2*x)/7 + 1) **3) + 1/(128*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**4) + 35/(256*(sqrt(21)*sqrt( 1 - 2*x)/7 - 1)) - 15/(256*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) + 5/(192*(sq rt(21)*sqrt(1 - 2*x)/7 - 1)**3) - 1/(128*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**4 ))/50421, (sqrt(1 - 2*x) > -sqrt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3)))/3
Time = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {95}{172872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2565 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 21945 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 67963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 70903 \, \sqrt {-2 \, x + 1}}{4116 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
95/172872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(- 2*x + 1))) + 1/4116*(2565*(-2*x + 1)^(7/2) - 21945*(-2*x + 1)^(5/2) + 6796 3*(-2*x + 1)^(3/2) - 70903*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1) ^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {95}{172872} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2565 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 21945 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 67963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 70903 \, \sqrt {-2 \, x + 1}}{65856 \, {\left (3 \, x + 2\right )}^{4}} \]
95/172872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/65856*(2565*(2*x - 1)^3*sqrt(-2*x + 1) + 21945*(2* x - 1)^2*sqrt(-2*x + 1) - 67963*(-2*x + 1)^(3/2) + 70903*sqrt(-2*x + 1))/( 3*x + 2)^4
Time = 1.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \frac {3+5 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {95\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{86436}-\frac {\frac {1447\,\sqrt {1-2\,x}}{6804}-\frac {1387\,{\left (1-2\,x\right )}^{3/2}}{6804}+\frac {1045\,{\left (1-2\,x\right )}^{5/2}}{15876}-\frac {95\,{\left (1-2\,x\right )}^{7/2}}{12348}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]