Integrand size = 20, antiderivative size = 193 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}+\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a} \]
-1/2*ln(1-(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/(a*x+1)^(1/2 ))/a*2^(1/2)+1/2*ln(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/ (a*x+1)^(1/2))/a*2^(1/2)-arctan(-1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))*2 ^(1/2)/a-arctan(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))*2^(1/2)/a
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a^2 x^2}}{\sqrt {1-a x}-\sqrt {1+a x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-a^2 x^2}}{\sqrt {1-a x}+\sqrt {1+a x}}\right )\right )}{a} \]
(Sqrt[2]*(ArcTan[(Sqrt[2]*(1 - a^2*x^2)^(1/4))/(Sqrt[1 - a*x] - Sqrt[1 + a *x])] + ArcTanh[(Sqrt[2]*(1 - a^2*x^2)^(1/4))/(Sqrt[1 - a*x] + Sqrt[1 + a* x])]))/a
Time = 0.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {73, 854, 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.
\(\displaystyle \int \frac {1}{\sqrt [4]{1-a x} (a x+1)^{3/4}} \, dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {4 \int \frac {\sqrt {1-a x}}{(a x+1)^{3/4}}d\sqrt [4]{1-a x}}{a}\) |
\(\Big \downarrow \) 854 |
\(\displaystyle -\frac {4 \int \frac {\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{a}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}\) |
(-4*((-(ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2]) + A rcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/Sqrt[2])/2 + (Log[1 + Sqrt[1 - a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[1 - a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt [2]))/2))/a
3.18.30.3.1 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]
\[\int \frac {1}{\left (-a x +1\right )^{\frac {1}{4}} \left (a x +1\right )^{\frac {3}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=-\left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a^{2} x - a\right )} \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} + {\left (a x + 1\right )}^{\frac {1}{4}} {\left (-a x + 1\right )}^{\frac {3}{4}}}{a x - 1}\right ) + \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a^{2} x - a\right )} \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} - {\left (a x + 1\right )}^{\frac {1}{4}} {\left (-a x + 1\right )}^{\frac {3}{4}}}{a x - 1}\right ) - i \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, a^{2} x - i \, a\right )} \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} + {\left (a x + 1\right )}^{\frac {1}{4}} {\left (-a x + 1\right )}^{\frac {3}{4}}}{a x - 1}\right ) + i \, \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, a^{2} x + i \, a\right )} \left (-\frac {1}{a^{4}}\right )^{\frac {1}{4}} + {\left (a x + 1\right )}^{\frac {1}{4}} {\left (-a x + 1\right )}^{\frac {3}{4}}}{a x - 1}\right ) \]
-(-1/a^4)^(1/4)*log(((a^2*x - a)*(-1/a^4)^(1/4) + (a*x + 1)^(1/4)*(-a*x + 1)^(3/4))/(a*x - 1)) + (-1/a^4)^(1/4)*log(-((a^2*x - a)*(-1/a^4)^(1/4) - ( a*x + 1)^(1/4)*(-a*x + 1)^(3/4))/(a*x - 1)) - I*(-1/a^4)^(1/4)*log(((I*a^2 *x - I*a)*(-1/a^4)^(1/4) + (a*x + 1)^(1/4)*(-a*x + 1)^(3/4))/(a*x - 1)) + I*(-1/a^4)^(1/4)*log(((-I*a^2*x + I*a)*(-1/a^4)^(1/4) + (a*x + 1)^(1/4)*(- a*x + 1)^(3/4))/(a*x - 1))
\[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int \frac {1}{\sqrt [4]{- a x + 1} \left (a x + 1\right )^{\frac {3}{4}}}\, dx \]
\[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int { \frac {1}{{\left (a x + 1\right )}^{\frac {3}{4}} {\left (-a x + 1\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int { \frac {1}{{\left (a x + 1\right )}^{\frac {3}{4}} {\left (-a x + 1\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int \frac {1}{{\left (1-a\,x\right )}^{1/4}\,{\left (a\,x+1\right )}^{3/4}} \,d x \]