\(\int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx\) [630]

Optimal result

Integrand size = 20, antiderivative size = 136 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=-\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{a}+\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )}{a}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1+a x}}{\sqrt [4]{1-a x} \left (1+\frac {\sqrt {1+a x}}{\sqrt {1-a x}}\right )}\right )}{a} \] Output:

-2^(1/2)*arctan(1-2^(1/2)*(a*x+1)^(1/4)/(-a*x+1)^(1/4))/a+2^(1/2)*arctan(1 
+2^(1/2)*(a*x+1)^(1/4)/(-a*x+1)^(1/4))/a+2^(1/2)*arctanh(2^(1/2)*(a*x+1)^( 
1/4)/(-a*x+1)^(1/4)/(1+(a*x+1)^(1/2)/(-a*x+1)^(1/2)))/a
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a^2 x^2}}{\sqrt {1-a x}-\sqrt {1+a x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-a^2 x^2}}{\sqrt {1-a x}+\sqrt {1+a x}}\right )\right )}{a} \] Input:

Integrate[1/((1 - a*x)^(1/4)*(1 + a*x)^(3/4)),x]
 

Output:

(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
 

Rubi [A] (warning: unable to verify)

Time = 0.30 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.33, 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

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}\)

Input:

Int[1/((1 - a*x)^(1/4)*(1 + a*x)^(3/4)),x]
 

Output:

(-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
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

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]
 

rule 217

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])
 

rule 826

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]]))
 

rule 854

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

Maple [F]

\[\int \frac {1}{\left (-a x +1\right )^{\frac {1}{4}} \left (a x +1\right )^{\frac {3}{4}}}d x\]

Input:

int(1/(-a*x+1)^(1/4)/(a*x+1)^(3/4),x)
 

Output:

int(1/(-a*x+1)^(1/4)/(a*x+1)^(3/4),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=-\frac {2 \, \sqrt {2} \arctan \left (\frac {a x + \sqrt {2} {\left (a x + 1\right )}^{\frac {1}{4}} {\left (-a x + 1\right )}^{\frac {3}{4}} - 1}{a x - 1}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {a x - \sqrt {2} {\left (a x + 1\right )}^{\frac {1}{4}} {\left (-a x + 1\right )}^{\frac {3}{4}} - 1}{a x - 1}\right ) + \sqrt {2} \log \left (\frac {a x + \sqrt {2} {\left (a x + 1\right )}^{\frac {1}{4}} {\left (-a x + 1\right )}^{\frac {3}{4}} - \sqrt {a x + 1} \sqrt {-a x + 1} - 1}{a x - 1}\right ) - \sqrt {2} \log \left (\frac {a x - \sqrt {2} {\left (a x + 1\right )}^{\frac {1}{4}} {\left (-a x + 1\right )}^{\frac {3}{4}} - \sqrt {a x + 1} \sqrt {-a x + 1} - 1}{a x - 1}\right )}{2 \, a} \] Input:

integrate(1/(-a*x+1)^(1/4)/(a*x+1)^(3/4),x, algorithm="fricas")
 

Output:

-1/2*(2*sqrt(2)*arctan((a*x + sqrt(2)*(a*x + 1)^(1/4)*(-a*x + 1)^(3/4) - 1 
)/(a*x - 1)) + 2*sqrt(2)*arctan(-(a*x - sqrt(2)*(a*x + 1)^(1/4)*(-a*x + 1) 
^(3/4) - 1)/(a*x - 1)) + sqrt(2)*log((a*x + sqrt(2)*(a*x + 1)^(1/4)*(-a*x 
+ 1)^(3/4) - sqrt(a*x + 1)*sqrt(-a*x + 1) - 1)/(a*x - 1)) - sqrt(2)*log((a 
*x - sqrt(2)*(a*x + 1)^(1/4)*(-a*x + 1)^(3/4) - sqrt(a*x + 1)*sqrt(-a*x + 
1) - 1)/(a*x - 1)))/a
 

Sympy [F]

\[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int \frac {1}{\sqrt [4]{- a x + 1} \left (a x + 1\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(1/(-a*x+1)**(1/4)/(a*x+1)**(3/4),x)
 

Output:

Integral(1/((-a*x + 1)**(1/4)*(a*x + 1)**(3/4)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int { \frac {1}{{\left (a x + 1\right )}^{\frac {3}{4}} {\left (-a x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(1/(-a*x+1)^(1/4)/(a*x+1)^(3/4),x, algorithm="maxima")
 

Output:

integrate(1/((a*x + 1)^(3/4)*(-a*x + 1)^(1/4)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int { \frac {1}{{\left (a x + 1\right )}^{\frac {3}{4}} {\left (-a x + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(1/(-a*x+1)^(1/4)/(a*x+1)^(3/4),x, algorithm="giac")
 

Output:

integrate(1/((a*x + 1)^(3/4)*(-a*x + 1)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int \frac {1}{{\left (1-a\,x\right )}^{1/4}\,{\left (a\,x+1\right )}^{3/4}} \,d x \] Input:

int(1/((1 - a*x)^(1/4)*(a*x + 1)^(3/4)),x)
                                                                                    
                                                                                    
 

Output:

int(1/((1 - a*x)^(1/4)*(a*x + 1)^(3/4)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [4]{1-a x} (1+a x)^{3/4}} \, dx=\int \frac {1}{\left (a x +1\right )^{\frac {3}{4}} \left (-a x +1\right )^{\frac {1}{4}}}d x \] Input:

int(1/(-a*x+1)^(1/4)/(a*x+1)^(3/4),x)
 

Output:

int(1/((a*x + 1)**(3/4)*( - a*x + 1)**(1/4)),x)