Integrand size = 20, antiderivative size = 279 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{1+b x}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {1+b x}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (\sqrt {a}+\frac {\sqrt {b} \sqrt {1-a x}}{\sqrt {1+b x}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{1+b x}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/4}} \]
-1/2*ln(-a^(1/4)*b^(1/4)*(-a*x+1)^(1/4)*2^(1/2)/(b*x+1)^(1/4)+a^(1/2)+b^(1 /2)*(-a*x+1)^(1/2)/(b*x+1)^(1/2))/a^(1/4)/b^(3/4)*2^(1/2)+1/2*ln(a^(1/4)*b ^(1/4)*(-a*x+1)^(1/4)*2^(1/2)/(b*x+1)^(1/4)+a^(1/2)+b^(1/2)*(-a*x+1)^(1/2) /(b*x+1)^(1/2))/a^(1/4)/b^(3/4)*2^(1/2)+arctan(1-b^(1/4)*(-a*x+1)^(1/4)*2^ (1/2)/a^(1/4)/(b*x+1)^(1/4))*2^(1/2)/a^(1/4)/b^(3/4)-arctan(1+b^(1/4)*(-a* x+1)^(1/4)*2^(1/2)/a^(1/4)/(b*x+1)^(1/4))*2^(1/2)/a^(1/4)/b^(3/4)
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x} \sqrt [4]{1+b x}}{\sqrt {b} \sqrt {1-a x}-\sqrt {a} \sqrt {1+b x}}\right )+\text {arctanh}\left (\frac {\sqrt {b} \sqrt {1-a x}+\sqrt {a} \sqrt {1+b x}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x} \sqrt [4]{1+b x}}\right )\right )}{\sqrt [4]{a} b^{3/4}} \]
(Sqrt[2]*(ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*(1 - a*x)^(1/4)*(1 + b*x)^(1/4)) /(Sqrt[b]*Sqrt[1 - a*x] - Sqrt[a]*Sqrt[1 + b*x])] + ArcTanh[(Sqrt[b]*Sqrt[ 1 - a*x] + Sqrt[a]*Sqrt[1 + b*x])/(Sqrt[2]*a^(1/4)*b^(1/4)*(1 - a*x)^(1/4) *(1 + b*x)^(1/4))]))/(a^(1/4)*b^(3/4))
Time = 0.49 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.25, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {73, 854, 27, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{\sqrt [4]{1-a x} (b x+1)^{3/4}} \, dx\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {4 \int \frac {\sqrt {1-a x}}{\left (\frac {a+b}{a}-\frac {b (1-a x)}{a}\right )^{3/4}}d\sqrt [4]{1-a x}}{a}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {4 \int \frac {a \sqrt {1-a x}}{a+b (1-a x)}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \int \frac {\sqrt {1-a x}}{a+b (1-a x)}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -4 \left (\frac {\int \frac {\sqrt {a}+\sqrt {b} \sqrt {1-a x}}{a+b (1-a x)}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} \sqrt {1-a x}}{a+b (1-a x)}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -4 \left (\frac {\frac {\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{1-a x} \sqrt [4]{a}}{\sqrt [4]{b} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {1-a x}}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {b}}+\frac {\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{1-a x} \sqrt [4]{a}}{\sqrt [4]{b} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {1-a x}}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} \sqrt {1-a x}}{a+b (1-a x)}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -4 \left (\frac {\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} \sqrt {1-a x}}{a+b (1-a x)}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -4 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} \sqrt {1-a x}}{a+b (1-a x)}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -4 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{\sqrt [4]{b} \left (\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{1-a x} \sqrt [4]{a}}{\sqrt [4]{b} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {1-a x}\right )}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\right )}{\sqrt [4]{b} \left (\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{1-a x} \sqrt [4]{a}}{\sqrt [4]{b} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {1-a x}\right )}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{\sqrt [4]{b} \left (\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{1-a x} \sqrt [4]{a}}{\sqrt [4]{b} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {1-a x}\right )}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\right )}{\sqrt [4]{b} \left (\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{1-a x} \sqrt [4]{a}}{\sqrt [4]{b} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {1-a x}\right )}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-\frac {2 \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{1-a x} \sqrt [4]{a}}{\sqrt [4]{b} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {1-a x}}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt [4]{a}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{1-a x} \sqrt [4]{a}}{\sqrt [4]{b} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {1-a x}}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -4 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{a} \sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {b} \sqrt {1-a x}+\sqrt {a}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x}}{\sqrt [4]{\frac {a+b}{a}-\frac {b (1-a x)}{a}}}+\sqrt {b} \sqrt {1-a x}+\sqrt {a}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
-4*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*(1 - a*x)^(1/4))/(a^(1/4)*((a + b)/a - (b*(1 - a*x))/a)^(1/4))]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]* b^(1/4)*(1 - a*x)^(1/4))/(a^(1/4)*((a + b)/a - (b*(1 - a*x))/a)^(1/4))]/(S qrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b]) - (-1/2*Log[Sqrt[a] + Sqrt[b]*Sqrt[1 - a*x] - (Sqrt[2]*a^(1/4)*b^(1/4)*(1 - a*x)^(1/4))/((a + b)/a - (b*(1 - a* x))/a)^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[b]*Sqrt[1 - a *x] + (Sqrt[2]*a^(1/4)*b^(1/4)*(1 - a*x)^(1/4))/((a + b)/a - (b*(1 - a*x)) /a)^(1/4)]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b]))
3.18.29.3.1 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] :> 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_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
\[\int \frac {1}{\left (-a x +1\right )^{\frac {1}{4}} \left (b x +1\right )^{\frac {3}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=-\left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a b x - b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) + \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a b x - b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} - {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) - i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, a b x - i \, b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) + i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, a b x + i \, b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) \]
-(-1/(a*b^3))^(1/4)*log(((a*b*x - b)*(-1/(a*b^3))^(1/4) + (-a*x + 1)^(3/4) *(b*x + 1)^(1/4))/(a*x - 1)) + (-1/(a*b^3))^(1/4)*log(-((a*b*x - b)*(-1/(a *b^3))^(1/4) - (-a*x + 1)^(3/4)*(b*x + 1)^(1/4))/(a*x - 1)) - I*(-1/(a*b^3 ))^(1/4)*log(((I*a*b*x - I*b)*(-1/(a*b^3))^(1/4) + (-a*x + 1)^(3/4)*(b*x + 1)^(1/4))/(a*x - 1)) + I*(-1/(a*b^3))^(1/4)*log(((-I*a*b*x + I*b)*(-1/(a* b^3))^(1/4) + (-a*x + 1)^(3/4)*(b*x + 1)^(1/4))/(a*x - 1))
\[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\int \frac {1}{\sqrt [4]{- a x + 1} \left (b x + 1\right )^{\frac {3}{4}}}\, dx \]
\[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\int { \frac {1}{{\left (-a x + 1\right )}^{\frac {1}{4}} {\left (b x + 1\right )}^{\frac {3}{4}}} \,d x } \]
\[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\int { \frac {1}{{\left (-a x + 1\right )}^{\frac {1}{4}} {\left (b x + 1\right )}^{\frac {3}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\int \frac {1}{{\left (1-a\,x\right )}^{1/4}\,{\left (b\,x+1\right )}^{3/4}} \,d x \]