\(\int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx\) [629]

Optimal result

Integrand size = 20, antiderivative size = 196 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=-\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{1+b x}}{\sqrt [4]{b} \sqrt [4]{1-a x}}\right )}{\sqrt [4]{a} b^{3/4}}+\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{1+b x}}{\sqrt [4]{b} \sqrt [4]{1-a x}}\right )}{\sqrt [4]{a} b^{3/4}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1+b x}}{\sqrt [4]{1-a x} \left (\sqrt {b}+\frac {\sqrt {a} \sqrt {1+b x}}{\sqrt {1-a x}}\right )}\right )}{\sqrt [4]{a} b^{3/4}} \] Output:

2^(1/2)*arctan(-1+2^(1/2)*a^(1/4)*(b*x+1)^(1/4)/b^(1/4)/(-a*x+1)^(1/4))/a^ 
(1/4)/b^(3/4)+2^(1/2)*arctan(1+2^(1/2)*a^(1/4)*(b*x+1)^(1/4)/b^(1/4)/(-a*x 
+1)^(1/4))/a^(1/4)/b^(3/4)+2^(1/2)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(b*x+1) 
^(1/4)/(-a*x+1)^(1/4)/(b^(1/2)+a^(1/2)*(b*x+1)^(1/2)/(-a*x+1)^(1/2)))/a^(1 
/4)/b^(3/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x} \sqrt [4]{1+b x}}{\sqrt {b} \sqrt {1-a x}-\sqrt {a} \sqrt {1+b x}}\right )+\text {arctanh}\left (\frac {\sqrt {b} \sqrt {1-a x}+\sqrt {a} \sqrt {1+b x}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{1-a x} \sqrt [4]{1+b x}}\right )\right )}{\sqrt [4]{a} b^{3/4}} \] Input:

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

Output:

(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))
 

Rubi [A] (warning: unable to verify)

Time = 0.43 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.78, 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

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

Input:

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

Output:

-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]))
 

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]

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt [4]{1-a x} (1+b x)^{3/4}} \, dx=-\left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a b x - b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) + \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a b x - b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} - {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) - i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, a b x - i \, b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) + i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, a b x + i \, b\right )} \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} + {\left (-a x + 1\right )}^{\frac {3}{4}} {\left (b x + 1\right )}^{\frac {1}{4}}}{a x - 1}\right ) \] Input:

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

Output:

-(-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))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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