Integrand size = 24, antiderivative size = 120 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac {5293 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}} \]
-5293/453789*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-5293/18522*(1-2* x)^(1/2)/(2+3*x)^2-5293/43218*(1-2*x)^(1/2)/(2+3*x)+1/105*(3+5*x)^2*(1-2*x )^(1/2)/(2+3*x)^5+1/6615*(1255+1971*x)*(1-2*x)^(1/2)/(2+3*x)^4
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {-\frac {21 \sqrt {1-2 x} \left (816938+4450198 x+8806422 x^2+7383735 x^3+2143665 x^4\right )}{2 (2+3 x)^5}-26465 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268945} \]
((-21*Sqrt[1 - 2*x]*(816938 + 4450198*x + 8806422*x^2 + 7383735*x^3 + 2143 665*x^4))/(2*(2 + 3*x)^5) - 26465*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x] ])/2268945
Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {109, 27, 162, 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.
| \(\displaystyle \int \frac {(5 x+3)^3}{\sqrt {1-2 x} (3 x+2)^6} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}-\frac {1}{105} \int -\frac {2 (5 x+3) (425 x+244)}{\sqrt {1-2 x} (3 x+2)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{105} \int \frac {(5 x+3) (425 x+244)}{\sqrt {1-2 x} (3 x+2)^5}dx+\frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {2}{105} \left (\frac {26465}{126} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^3}dx+\frac {\sqrt {1-2 x} (1971 x+1255)}{126 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2}{105} \left (\frac {26465}{126} \left (\frac {3}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (1971 x+1255)}{126 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {2}{105} \left (\frac {26465}{126} \left (\frac {3}{14} \left (\frac {1}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (1971 x+1255)}{126 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2}{105} \left (\frac {26465}{126} \left (\frac {3}{14} \left (-\frac {1}{7} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (1971 x+1255)}{126 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2}{105} \left (\frac {26465}{126} \left (\frac {3}{14} \left (-\frac {2 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{7 \sqrt {21}}-\frac {\sqrt {1-2 x}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x}}{14 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x} (1971 x+1255)}{126 (3 x+2)^4}\right )+\frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}\) |
(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(105*(2 + 3*x)^5) + (2*((Sqrt[1 - 2*x]*(1255 + 1971*x))/(126*(2 + 3*x)^4) + (26465*(-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*S qrt[21])))/14))/126))/105
3.21.34.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
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 = 61, normalized size of antiderivative = 0.51
| method | result | size |
| risch | \(\frac {4287330 x^{5}+12623805 x^{4}+10229109 x^{3}+93974 x^{2}-2816322 x -816938}{216090 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {5293 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\) | \(61\) |
| pseudoelliptic | \(\frac {-52930 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{5} \sqrt {21}-21 \sqrt {1-2 x}\, \left (2143665 x^{4}+7383735 x^{3}+8806422 x^{2}+4450198 x +816938\right )}{4537890 \left (2+3 x \right )^{5}}\) | \(65\) |
| derivativedivides | \(\frac {\frac {47637 \left (1-2 x \right )^{\frac {9}{2}}}{2401}-\frac {10586 \left (1-2 x \right )^{\frac {7}{2}}}{49}+\frac {628504 \left (1-2 x \right )^{\frac {5}{2}}}{735}-\frac {648694 \left (1-2 x \right )^{\frac {3}{2}}}{441}+\frac {58781 \sqrt {1-2 x}}{63}}{\left (-4-6 x \right )^{5}}-\frac {5293 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\) | \(75\) |
| default | \(\frac {\frac {47637 \left (1-2 x \right )^{\frac {9}{2}}}{2401}-\frac {10586 \left (1-2 x \right )^{\frac {7}{2}}}{49}+\frac {628504 \left (1-2 x \right )^{\frac {5}{2}}}{735}-\frac {648694 \left (1-2 x \right )^{\frac {3}{2}}}{441}+\frac {58781 \sqrt {1-2 x}}{63}}{\left (-4-6 x \right )^{5}}-\frac {5293 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\) | \(75\) |
| trager | \(-\frac {\left (2143665 x^{4}+7383735 x^{3}+8806422 x^{2}+4450198 x +816938\right ) \sqrt {1-2 x}}{216090 \left (2+3 x \right )^{5}}-\frac {5293 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{907578}\) | \(82\) |
1/216090*(4287330*x^5+12623805*x^4+10229109*x^3+93974*x^2-2816322*x-816938 )/(2+3*x)^5/(1-2*x)^(1/2)-5293/453789*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))* 21^(1/2)
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {26465 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2143665 \, x^{4} + 7383735 \, x^{3} + 8806422 \, x^{2} + 4450198 \, x + 816938\right )} \sqrt {-2 \, x + 1}}{4537890 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
1/4537890*(26465*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(2143665*x^4 + 7383735*x^3 + 8806422*x^2 + 4450198*x + 816938)*sqrt(-2*x + 1))/(243*x^ 5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
Timed out. \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.07 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {5293}{907578} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2143665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 23342130 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 92390088 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 158930030 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 100809415 \, \sqrt {-2 \, x + 1}}{108045 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
5293/907578*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt (-2*x + 1))) - 1/108045*(2143665*(-2*x + 1)^(9/2) - 23342130*(-2*x + 1)^(7 /2) + 92390088*(-2*x + 1)^(5/2) - 158930030*(-2*x + 1)^(3/2) + 100809415*s qrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 3 0870*(2*x - 1)^2 + 72030*x - 19208)
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx=\frac {5293}{907578} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {2143665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 23342130 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 92390088 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 158930030 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 100809415 \, \sqrt {-2 \, x + 1}}{3457440 \, {\left (3 \, x + 2\right )}^{5}} \]
5293/907578*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/3457440*(2143665*(2*x - 1)^4*sqrt(-2*x + 1) + 23 342130*(2*x - 1)^3*sqrt(-2*x + 1) + 92390088*(2*x - 1)^2*sqrt(-2*x + 1) - 158930030*(-2*x + 1)^(3/2) + 100809415*sqrt(-2*x + 1))/(3*x + 2)^5
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90 \[ \int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx=-\frac {5293\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{453789}-\frac {\frac {58781\,\sqrt {1-2\,x}}{15309}-\frac {648694\,{\left (1-2\,x\right )}^{3/2}}{107163}+\frac {628504\,{\left (1-2\,x\right )}^{5/2}}{178605}-\frac {10586\,{\left (1-2\,x\right )}^{7/2}}{11907}+\frac {5293\,{\left (1-2\,x\right )}^{9/2}}{64827}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \]
- (5293*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/453789 - ((58781*(1 - 2*x)^(1/2))/15309 - (648694*(1 - 2*x)^(3/2))/107163 + (628504*(1 - 2*x)^ (5/2))/178605 - (10586*(1 - 2*x)^(7/2))/11907 + (5293*(1 - 2*x)^(9/2))/648 27)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/243)