\(\int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx\) [292]

Optimal result

Integrand size = 23, antiderivative size = 77 \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx=-\frac {2 \sqrt [6]{2} (a-b x)^{3/2} \sqrt [6]{a+b x} \operatorname {AppellF1}\left (\frac {3}{2},1,-\frac {1}{6},\frac {5}{2},1-\frac {b x}{a},\frac {a-b x}{2 a}\right )}{3 a \sqrt [6]{\frac {a+b x}{a}}} \] Output:

-2/3*2^(1/6)*(-b*x+a)^(3/2)*(b*x+a)^(1/6)*AppellF1(3/2,-1/6,1,5/2,1/2*(-b* 
x+a)/a,1-b*x/a)/a/((b*x+a)/a)^(1/6)
 

Mathematica [A] (verified)

Time = 10.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.79 \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx=\frac {-3 a^2 \sqrt {1-\frac {a}{b x}} \left (1+\frac {a}{b x}\right )^{5/6} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {5}{6},\frac {7}{3},\frac {a}{b x},-\frac {a}{b x}\right )+6 (a+b x) \left (a-b x+a \sqrt {2-\frac {2 b x}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {a+b x}{2 a}\right )\right )}{4 \sqrt {a-b x} (a+b x)^{5/6}} \] Input:

Integrate[(Sqrt[a - b*x]*(a + b*x)^(1/6))/x,x]
 

Output:

(-3*a^2*Sqrt[1 - a/(b*x)]*(1 + a/(b*x))^(5/6)*AppellF1[4/3, 1/2, 5/6, 7/3, 
 a/(b*x), -(a/(b*x))] + 6*(a + b*x)*(a - b*x + a*Sqrt[2 - (2*b*x)/a]*Hyper 
geometric2F1[1/6, 1/2, 7/6, (a + b*x)/(2*a)]))/(4*Sqrt[a - b*x]*(a + b*x)^ 
(5/6))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {149, 25, 27, 1013, 27, 1012}

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.

\(\displaystyle \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx\)

\(\Big \downarrow \) 149

\(\displaystyle \frac {6 \int \frac {\sqrt {a-b x} (a+b x)}{x}d\sqrt [6]{a+b x}}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {6 \int -\frac {\sqrt {a-b x} (a+b x)}{x}d\sqrt [6]{a+b x}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -6 \int -\frac {\sqrt {a-b x} (a+b x)}{b x}d\sqrt [6]{a+b x}\)

\(\Big \downarrow \) 1013

\(\displaystyle -\frac {6 \sqrt {2} \sqrt {a-b x} \int -\frac {(a+b x) \sqrt {2-\frac {a+b x}{a}}}{\sqrt {2} b x}d\sqrt [6]{a+b x}}{\sqrt {2-\frac {a+b x}{a}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {6 \sqrt {a-b x} \int -\frac {(a+b x) \sqrt {2-\frac {a+b x}{a}}}{b x}d\sqrt [6]{a+b x}}{\sqrt {2-\frac {a+b x}{a}}}\)

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {6 \sqrt {2} \sqrt {a-b x} (a+b x)^{7/6} \operatorname {AppellF1}\left (\frac {7}{6},1,-\frac {1}{2},\frac {13}{6},\frac {a+b x}{a},\frac {a+b x}{2 a}\right )}{7 a \sqrt {2-\frac {a+b x}{a}}}\)

Input:

Int[(Sqrt[a - b*x]*(a + b*x)^(1/6))/x,x]
 

Output:

(-6*Sqrt[2]*Sqrt[a - b*x]*(a + b*x)^(7/6)*AppellF1[7/6, 1, -1/2, 13/6, (a 
+ b*x)/a, (a + b*x)/(2*a)])/(7*a*Sqrt[2 - (a + b*x)/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 149

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b   Subst[Int[x^(k*(m + 1 
) - 1)*(c - a*(d/b) + d*(x^k/b))^n*(e - a*(f/b) + f*(x^k/b))^p, x], x, (a + 
 b*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && FractionQ[m] && 
IntegerQ[2*n] && IntegerQ[p]
 

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

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

Maple [F]

\[\int \frac {\sqrt {-b x +a}\, \left (b x +a \right )^{\frac {1}{6}}}{x}d x\]

Input:

int((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x,x)
 

Output:

int((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x,x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx=\text {Timed out} \] Input:

integrate((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx=\int \frac {\sqrt {a - b x} \sqrt [6]{a + b x}}{x}\, dx \] Input:

integrate((-b*x+a)**(1/2)*(b*x+a)**(1/6)/x,x)
 

Output:

Integral(sqrt(a - b*x)*(a + b*x)**(1/6)/x, x)
 

Maxima [F]

\[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}} \sqrt {-b x + a}}{x} \,d x } \] Input:

integrate((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x,x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(1/6)*sqrt(-b*x + a)/x, x)
 

Giac [F]

\[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}} \sqrt {-b x + a}}{x} \,d x } \] Input:

integrate((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x,x, algorithm="giac")
 

Output:

integrate((b*x + a)^(1/6)*sqrt(-b*x + a)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/6}\,\sqrt {a-b\,x}}{x} \,d x \] Input:

int(((a + b*x)^(1/6)*(a - b*x)^(1/2))/x,x)
 

Output:

int(((a + b*x)^(1/6)*(a - b*x)^(1/2))/x, x)
 

Reduce [F]

\[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x} \, dx=\int \frac {\sqrt {-b x +a}\, \left (b x +a \right )^{\frac {1}{6}}}{x}d x \] Input:

int((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x,x)
 

Output:

int((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x,x)