Integrand size = 24, antiderivative size = 108 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {\sqrt {1-2 x}}{252 (2+3 x)^4}+\frac {13 \sqrt {1-2 x}}{252 (2+3 x)^3}-\frac {635 \sqrt {1-2 x}}{3528 (2+3 x)^2}-\frac {635 \sqrt {1-2 x}}{8232 (2+3 x)}-\frac {635 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{4116 \sqrt {21}} \]
-635/86436*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1/252*(1-2*x)^(1/2 )/(2+3*x)^4+13/252*(1-2*x)^(1/2)/(2+3*x)^3-635/3528*(1-2*x)^(1/2)/(2+3*x)^ 2-635/8232*(1-2*x)^(1/2)/(2+3*x)
Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.60 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (10190+39366 x+47625 x^2+17145 x^3\right )}{2 (2+3 x)^4}-635 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{86436} \]
((-21*Sqrt[1 - 2*x]*(10190 + 39366*x + 47625*x^2 + 17145*x^3))/(2*(2 + 3*x )^4) - 635*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/86436
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {100, 27, 87, 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.
\(\displaystyle \int \frac {(5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^5} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {1}{252} \int \frac {7 (300 x+161)}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{36} \int \frac {300 x+161}{\sqrt {1-2 x} (3 x+2)^4}dx-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{36} \left (\frac {635}{7} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {13 \sqrt {1-2 x}}{7 (3 x+2)^3}\right )-\frac {\sqrt {1-2 x}}{252 (3 x+2)^4}\) |
-1/252*Sqrt[1 - 2*x]/(2 + 3*x)^4 + ((13*Sqrt[1 - 2*x])/(7*(2 + 3*x)^3) + ( 635*(-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))/7)/36
3.21.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 = 1.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {34290 x^{4}+78105 x^{3}+31107 x^{2}-18986 x -10190}{8232 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {635 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(56\) |
pseudoelliptic | \(\frac {-1270 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}-21 \sqrt {1-2 x}\, \left (17145 x^{3}+47625 x^{2}+39366 x +10190\right )}{172872 \left (2+3 x \right )^{4}}\) | \(60\) |
derivativedivides | \(\frac {\frac {5715 \left (1-2 x \right )^{\frac {7}{2}}}{1372}-\frac {6985 \left (1-2 x \right )^{\frac {5}{2}}}{196}+\frac {2717 \left (1-2 x \right )^{\frac {3}{2}}}{28}-\frac {7171 \sqrt {1-2 x}}{84}}{\left (-4-6 x \right )^{4}}-\frac {635 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(66\) |
default | \(\frac {\frac {5715 \left (1-2 x \right )^{\frac {7}{2}}}{1372}-\frac {6985 \left (1-2 x \right )^{\frac {5}{2}}}{196}+\frac {2717 \left (1-2 x \right )^{\frac {3}{2}}}{28}-\frac {7171 \sqrt {1-2 x}}{84}}{\left (-4-6 x \right )^{4}}-\frac {635 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{86436}\) | \(66\) |
trager | \(-\frac {\left (17145 x^{3}+47625 x^{2}+39366 x +10190\right ) \sqrt {1-2 x}}{8232 \left (2+3 x \right )^{4}}+\frac {635 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{172872}\) | \(77\) |
1/8232*(34290*x^4+78105*x^3+31107*x^2-18986*x-10190)/(2+3*x)^4/(1-2*x)^(1/ 2)-635/86436*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 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {635 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (17145 \, x^{3} + 47625 \, x^{2} + 39366 \, x + 10190\right )} \sqrt {-2 \, x + 1}}{172872 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/172872*(635*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*(17145*x^3 + 47625*x^2 + 393 66*x + 10190)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {635}{172872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {17145 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 146685 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 399399 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 351379 \, \sqrt {-2 \, x + 1}}{4116 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
635/172872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt( -2*x + 1))) + 1/4116*(17145*(-2*x + 1)^(7/2) - 146685*(-2*x + 1)^(5/2) + 3 99399*(-2*x + 1)^(3/2) - 351379*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {635}{172872} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {17145 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 146685 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 399399 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 351379 \, \sqrt {-2 \, x + 1}}{65856 \, {\left (3 \, x + 2\right )}^{4}} \]
635/172872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/65856*(17145*(2*x - 1)^3*sqrt(-2*x + 1) + 146685* (2*x - 1)^2*sqrt(-2*x + 1) - 399399*(-2*x + 1)^(3/2) + 351379*sqrt(-2*x + 1))/(3*x + 2)^4
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {635\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{86436}-\frac {\frac {7171\,\sqrt {1-2\,x}}{6804}-\frac {2717\,{\left (1-2\,x\right )}^{3/2}}{2268}+\frac {6985\,{\left (1-2\,x\right )}^{5/2}}{15876}-\frac {635\,{\left (1-2\,x\right )}^{7/2}}{12348}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \]