Integrand size = 24, antiderivative size = 117 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}+\frac {88310}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Output:
1/7*(1-2*x)^(1/2)/(2+3*x)^3+55/49*(1-2*x)^(1/2)/(2+3*x)^2+3840*(1-2*x)^(1/ 2)/(686+1029*x)+88310/2401*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-25 0/11*55^(1/2)*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx=\frac {3 \sqrt {1-2 x} \left (5393+15745 x+11520 x^2\right )}{343 (2+3 x)^3}+\frac {88310}{343} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \] Input:
Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)),x]
Output:
(3*Sqrt[1 - 2*x]*(5393 + 15745*x + 11520*x^2))/(343*(2 + 3*x)^3) + (88310* Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - 250*Sqrt[5/11]*ArcTanh[S qrt[5/11]*Sqrt[1 - 2*x]]
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {114, 27, 168, 27, 168, 174, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^4 (5 x+3)} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{21} \int \frac {15 (4-5 x)}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \int \frac {4-5 x}{\sqrt {1-2 x} (3 x+2)^3 (5 x+3)}dx+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{14} \int \frac {2 (146-165 x)}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {11 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \int \frac {146-165 x}{\sqrt {1-2 x} (3 x+2)^2 (5 x+3)}dx+\frac {11 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (\frac {1}{7} \int \frac {6271-3840 x}{\sqrt {1-2 x} (3 x+2) (5 x+3)}dx+\frac {768 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {11 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (\frac {1}{7} \left (42875 \int \frac {1}{\sqrt {1-2 x} (5 x+3)}dx-26493 \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {768 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {11 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (\frac {1}{7} \left (26493 \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-42875 \int \frac {1}{\frac {11}{2}-\frac {5}{2} (1-2 x)}d\sqrt {1-2 x}\right )+\frac {768 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {11 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{7} \left (\frac {1}{7} \left (\frac {1}{7} \left (17662 \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-17150 \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )+\frac {768 \sqrt {1-2 x}}{7 (3 x+2)}\right )+\frac {11 \sqrt {1-2 x}}{7 (3 x+2)^2}\right )+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}\) |
Input:
Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)),x]
Output:
Sqrt[1 - 2*x]/(7*(2 + 3*x)^3) + (5*((11*Sqrt[1 - 2*x])/(7*(2 + 3*x)^2) + ( (768*Sqrt[1 - 2*x])/(7*(2 + 3*x)) + (17662*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqr t[1 - 2*x]] - 17150*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/7)/7))/7
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] :> 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*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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 = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(-\frac {3 \left (23040 x^{3}+19970 x^{2}-4959 x -5393\right )}{343 \left (2+3 x \right )^{3} \sqrt {1-2 x}}+\frac {88310 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2401}-\frac {250 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{11}\) | \(69\) |
| derivativedivides | \(-\frac {250 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{11}-\frac {162 \left (\frac {1280 \left (1-2 x \right )^{\frac {5}{2}}}{1029}-\frac {7790 \left (1-2 x \right )^{\frac {3}{2}}}{1323}+\frac {1318 \sqrt {1-2 x}}{189}\right )}{\left (-4-6 x \right )^{3}}+\frac {88310 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2401}\) | \(75\) |
| default | \(-\frac {250 \sqrt {55}\, \operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right )}{11}-\frac {162 \left (\frac {1280 \left (1-2 x \right )^{\frac {5}{2}}}{1029}-\frac {7790 \left (1-2 x \right )^{\frac {3}{2}}}{1323}+\frac {1318 \sqrt {1-2 x}}{189}\right )}{\left (-4-6 x \right )^{3}}+\frac {88310 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{2401}\) | \(75\) |
| pseudoelliptic | \(\frac {971410 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}-600250 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{3} \sqrt {55}+231 \sqrt {1-2 x}\, \left (11520 x^{2}+15745 x +5393\right )}{26411 \left (2+3 x \right )^{3}}\) | \(80\) |
| trager | \(\frac {3 \left (11520 x^{2}+15745 x +5393\right ) \sqrt {1-2 x}}{343 \left (2+3 x \right )^{3}}+\frac {125 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{11}-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1637717781\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1637717781\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1637717781\right )+185451 \sqrt {1-2 x}}{2+3 x}\right )}{2401}\) | \(116\) |
Input:
int(1/(1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x,method=_RETURNVERBOSE)
Output:
-3/343*(23040*x^3+19970*x^2-4959*x-5393)/(2+3*x)^3/(1-2*x)^(1/2)+88310/240 1*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))-250/11*55^(1/2)*arctanh(1/1 1*55^(1/2)*(1-2*x)^(1/2))
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx=\frac {42875 \, \sqrt {\frac {5}{11}} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {5 \, x + 11 \, \sqrt {\frac {5}{11}} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 44155 \, \sqrt {\frac {3}{7}} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x - 7 \, \sqrt {\frac {3}{7}} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 3 \, {\left (11520 \, x^{2} + 15745 \, x + 5393\right )} \sqrt {-2 \, x + 1}}{343 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x, algorithm="fricas")
Output:
1/343*(42875*sqrt(5/11)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + 11*sqrt(5/ 11)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 44155*sqrt(3/7)*(27*x^3 + 54*x^2 + 36 *x + 8)*log((3*x - 7*sqrt(3/7)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 3*(11520*x ^2 + 15745*x + 5393)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
Result contains complex when optimal does not.
Time = 12.14 (sec) , antiderivative size = 8978, normalized size of antiderivative = 76.74 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-2*x)**(1/2)/(2+3*x)**4/(3+5*x),x)
Output:
-36294822144000*sqrt(55)*I*(x - 1/2)**(35/2)*atan(sqrt(110)*sqrt(x - 1/2)/ 11)/(1596972174336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31/2) + 304311919887360*(x - 1/2)**(29/2) + 621 303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)**(25/2) + 84566 3101446240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 2466517 37921820*(x - 1/2)**(19/2) + 63946746868620*(x - 1/2)**(17/2) + 7460453801 339*(x - 1/2)**(15/2)) + 1753277239296*sqrt(21)*I*(x - 1/2)**(35/2)*atan(s qrt(42)/(6*sqrt(x - 1/2)))/(1596972174336*(x - 1/2)**(35/2) + 186313420339 20*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31/2) + 304311919887360* (x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)**(25/2) + 845663101446240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(19/2) + 63946746868620*(x - 1/ 2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) + 60490725267456*sqrt(21)*I* (x - 1/2)**(35/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(1596972174336*(x - 1/2)* *(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31 /2) + 304311919887360*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2 ) + 869824904344704*(x - 1/2)**(25/2) + 845663101446240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(19/2) + 63946746868620*(x - 1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) - 3024 5362633728*sqrt(21)*I*pi*(x - 1/2)**(35/2)/(1596972174336*(x - 1/2)**(3...
Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx=\frac {125}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {44155}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12 \, {\left (5760 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 27265 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 32291 \, \sqrt {-2 \, x + 1}\right )}}{343 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x, algorithm="maxima")
Output:
125/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 44155/2401*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 12/343*(5760*(-2*x + 1)^(5/2) - 27265*(-2*x + 1)^( 3/2) + 32291*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 9 8)
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx=\frac {125}{11} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {44155}{2401} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {3 \, {\left (5760 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 27265 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 32291 \, \sqrt {-2 \, x + 1}\right )}}{686 \, {\left (3 \, x + 2\right )}^{3}} \] Input:
integrate(1/(1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x, algorithm="giac")
Output:
125/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5 *sqrt(-2*x + 1))) - 44155/2401*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(- 2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 3/686*(5760*(2*x - 1)^2*sqrt(-2 *x + 1) - 27265*(-2*x + 1)^(3/2) + 32291*sqrt(-2*x + 1))/(3*x + 2)^3
Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx=\frac {88310\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {2636\,\sqrt {1-2\,x}}{63}-\frac {15580\,{\left (1-2\,x\right )}^{3/2}}{441}+\frac {2560\,{\left (1-2\,x\right )}^{5/2}}{343}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \] Input:
int(1/((1 - 2*x)^(1/2)*(3*x + 2)^4*(5*x + 3)),x)
Output:
(88310*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (250*55^(1/2)* atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/11 + ((2636*(1 - 2*x)^(1/2))/63 - (1 5580*(1 - 2*x)^(3/2))/441 + (2560*(1 - 2*x)^(5/2))/343)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 98/27)
Time = 0.16 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.97 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx=\frac {2661120 \sqrt {-2 x +1}\, x^{2}+3637095 \sqrt {-2 x +1}\, x +1245783 \sqrt {-2 x +1}+8103375 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{3}+16206750 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x^{2}+10804500 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right ) x +2401000 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}-\sqrt {55}\right )-8103375 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{3}-16206750 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x^{2}-10804500 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right ) x -2401000 \sqrt {55}\, \mathrm {log}\left (5 \sqrt {-2 x +1}+\sqrt {55}\right )-13114035 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}-26228070 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}-17485380 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x -3885640 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )+13114035 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}+26228070 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}+17485380 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x +3885640 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{713097 x^{3}+1426194 x^{2}+950796 x +211288} \] Input:
int(1/(1-2*x)^(1/2)/(2+3*x)^4/(3+5*x),x)
Output:
(2661120*sqrt( - 2*x + 1)*x**2 + 3637095*sqrt( - 2*x + 1)*x + 1245783*sqrt ( - 2*x + 1) + 8103375*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**3 + 16206750*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x**2 + 10804500*sqrt( 55)*log(5*sqrt( - 2*x + 1) - sqrt(55))*x + 2401000*sqrt(55)*log(5*sqrt( - 2*x + 1) - sqrt(55)) - 8103375*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55)) *x**3 - 16206750*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x**2 - 108045 00*sqrt(55)*log(5*sqrt( - 2*x + 1) + sqrt(55))*x - 2401000*sqrt(55)*log(5* sqrt( - 2*x + 1) + sqrt(55)) - 13114035*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**3 - 26228070*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x**2 - 17485380*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x - 3885640*sqrt(2 1)*log(3*sqrt( - 2*x + 1) - sqrt(21)) + 13114035*sqrt(21)*log(3*sqrt( - 2* x + 1) + sqrt(21))*x**3 + 26228070*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt( 21))*x**2 + 17485380*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21))*x + 38856 40*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(26411*(27*x**3 + 54*x**2 + 36*x + 8))