\(\int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx\) [497]

Optimal result

Integrand size = 22, antiderivative size = 169 \[ \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{\sqrt [4]{d}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx=\frac {4 \sqrt [4]{a+b x} \left (\sqrt [4]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {d (a+b x)}{-b c+a d}\right )-\operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {c (a+b x)}{a (c+d x)}\right )\right )}{\sqrt [4]{c+d x}} \] Input:

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

Output:

(4*(a + b*x)^(1/4)*(((b*(c + d*x))/(b*c - a*d))^(1/4)*Hypergeometric2F1[1/ 
4, 1/4, 5/4, (d*(a + b*x))/(-(b*c) + a*d)] - Hypergeometric2F1[1/4, 1, 5/4 
, (c*(a + b*x))/(a*(c + d*x))]))/(c + d*x)^(1/4)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.25, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {140, 27, 73, 104, 756, 218, 221, 770, 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.

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 140

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

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 1 
/n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt [4]{a+b x}}{x \sqrt [4]{c+d x}} \, dx=-\left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - i \, \left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, d x + i \, c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + i \, \left (\frac {a}{c}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, d x - i \, c\right )} \left (\frac {a}{c}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) + i \, \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (i \, d x + i \, c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) - i \, \left (\frac {b}{d}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-i \, d x - i \, c\right )} \left (\frac {b}{d}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{d x + c}\right ) \] Input:

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

Output:

-(a/c)^(1/4)*log(((d*x + c)*(a/c)^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4)) 
/(d*x + c)) + (a/c)^(1/4)*log(-((d*x + c)*(a/c)^(1/4) - (b*x + a)^(1/4)*(d 
*x + c)^(3/4))/(d*x + c)) - I*(a/c)^(1/4)*log(((I*d*x + I*c)*(a/c)^(1/4) + 
 (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) + I*(a/c)^(1/4)*log(((-I*d*x 
- I*c)*(a/c)^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) + (b/d)^( 
1/4)*log(((d*x + c)*(b/d)^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + 
c)) - (b/d)^(1/4)*log(-((d*x + c)*(b/d)^(1/4) - (b*x + a)^(1/4)*(d*x + c)^ 
(3/4))/(d*x + c)) + I*(b/d)^(1/4)*log(((I*d*x + I*c)*(b/d)^(1/4) + (b*x + 
a)^(1/4)*(d*x + c)^(3/4))/(d*x + c)) - I*(b/d)^(1/4)*log(((-I*d*x - I*c)*( 
b/d)^(1/4) + (b*x + a)^(1/4)*(d*x + c)^(3/4))/(d*x + c))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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