\(\int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx\) [498]

Optimal result

Integrand size = 22, antiderivative size = 131 \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}-\frac {(b c-a d) \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}} \] Output:

-(b*x+a)^(1/4)*(d*x+c)^(3/4)/c/x-1/2*(-a*d+b*c)*arctan(c^(1/4)*(b*x+a)^(1/ 
4)/a^(1/4)/(d*x+c)^(1/4))/a^(3/4)/c^(5/4)-1/2*(-a*d+b*c)*arctanh(c^(1/4)*( 
b*x+a)^(1/4)/a^(1/4)/(d*x+c)^(1/4))/a^(3/4)/c^(5/4)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 3.58 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\frac {-2 a^{3/4} \sqrt [4]{c} \sqrt [4]{a+b x} (c+d x)^{3/4}+(b c-a d) x \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{c+d x}}{\sqrt [4]{c} \sqrt [4]{a+b x}}\right )+(-b c x+a d x) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4} x} \] Input:

Integrate[(a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)),x]
 

Output:

(-2*a^(3/4)*c^(1/4)*(a + b*x)^(1/4)*(c + d*x)^(3/4) + (b*c - a*d)*x*ArcTan 
[(a^(1/4)*(c + d*x)^(1/4))/(c^(1/4)*(a + b*x)^(1/4))] + (-(b*c*x) + a*d*x) 
*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a^(3/4)* 
c^(5/4)*x)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 104, 756, 218, 221}

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.

\(\displaystyle \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx\)

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

\(\displaystyle \frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {(b c-a d) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a}+\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}\right )}{c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(b c-a d) \left (-\frac {\int \frac {1}{\sqrt {a}-\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c+d x}}}d\frac {\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(b c-a d) \left (-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} \sqrt [4]{c}}\right )}{c}-\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}\)

Input:

Int[(a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)),x]
 

Output:

-(((a + b*x)^(1/4)*(c + d*x)^(3/4))/(c*x)) + ((b*c - a*d)*(-1/2*ArcTan[(c^ 
(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))]/(a^(3/4)*c^(1/4)) - ArcT 
anh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))]/(2*a^(3/4)*c^(1/4 
))))/c
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{2} \left (x d +c \right )^{\frac {1}{4}}}d x\]

Input:

int((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x)
 

Output:

int((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 693, normalized size of antiderivative = 5.29 \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x, algorithm="fricas")
 

Output:

-1/4*(c*x*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 
a^4*d^4)/(a^3*c^5))^(1/4)*log(-((b*c - a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4 
) + (a*c*d*x + a*c^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^ 
3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4))/(d*x + c)) - c*x*((b^4*c^4 - 4*a*b^ 
3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4)*lo 
g(-((b*c - a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a*c*d*x + a*c^2)*((b^4* 
c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^ 
5))^(1/4))/(d*x + c)) + I*c*x*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^ 
2 - 4*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4)*log(-((b*c - a*d)*(b*x + a)^ 
(1/4)*(d*x + c)^(3/4) + (I*a*c*d*x + I*a*c^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 
6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4))/(d*x + c)) 
- I*c*x*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^ 
4*d^4)/(a^3*c^5))^(1/4)*log(-((b*c - a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) 
+ (-I*a*c*d*x - I*a*c^2)*((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4 
*a^3*b*c*d^3 + a^4*d^4)/(a^3*c^5))^(1/4))/(d*x + c)) + 4*(b*x + a)^(1/4)*( 
d*x + c)^(3/4))/(c*x)
 

Sympy [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int \frac {\sqrt [4]{a + b x}}{x^{2} \sqrt [4]{c + d x}}\, dx \] Input:

integrate((b*x+a)**(1/4)/x**2/(d*x+c)**(1/4),x)
 

Output:

Integral((a + b*x)**(1/4)/(x**2*(c + d*x)**(1/4)), x)
 

Maxima [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{2}} \,d x } \] Input:

integrate((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^2), x)
 

Giac [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{{\left (d x + c\right )}^{\frac {1}{4}} x^{2}} \,d x } \] Input:

integrate((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((b*x + a)^(1/4)/((d*x + c)^(1/4)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/4}}{x^2\,{\left (c+d\,x\right )}^{1/4}} \,d x \] Input:

int((a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)),x)
 

Output:

int((a + b*x)^(1/4)/(x^2*(c + d*x)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx=\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{\left (d x +c \right )^{\frac {1}{4}} x^{2}}d x \] Input:

int((b*x+a)^(1/4)/x^2/(d*x+c)^(1/4),x)
 

Output:

int((a + b*x)**(1/4)/((c + d*x)**(1/4)*x**2),x)