Integrand size = 23, antiderivative size = 78 \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx=-\frac {2 \sqrt [6]{2} b (a-b x)^{3/2} \sqrt [6]{a+b x} \operatorname {AppellF1}\left (\frac {3}{2},2,-\frac {1}{6},\frac {5}{2},1-\frac {b x}{a},\frac {a-b x}{2 a}\right )}{3 a^2 \sqrt [6]{\frac {a+b x}{a}}} \] Output:
-2/3*2^(1/6)*b*(-b*x+a)^(3/2)*(b*x+a)^(1/6)*AppellF1(3/2,-1/6,2,5/2,1/2*(- b*x+a)/a,1-b*x/a)/a^2/((b*x+a)/a)^(1/6)
Leaf count is larger than twice the leaf count of optimal. \(268\) vs. \(2(78)=156\).
Time = 10.56 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.44 \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx=\frac {-\frac {2 a^2}{x}+2 b^2 x-b^2 \sqrt {1-\frac {a}{b x}} \left (1+\frac {a}{b x}\right )^{5/6} x \left (\operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},\frac {5}{6},\frac {4}{3},\frac {a}{b x},-\frac {a}{b x}\right )-\operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},-\frac {1}{6},\frac {4}{3},\frac {a}{b x},-\frac {a}{b x}\right )\right )+\frac {b (a-b x) (a+b x) \left (\sqrt {1-\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{6},\frac {1}{3},\frac {2 a}{a-b x}\right )-4 \sqrt {2} \sqrt [6]{\frac {a+b x}{-a+b x}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6},\frac {7}{6},\frac {a+b x}{2 a}\right )\right )}{a \sqrt [6]{\frac {a+b x}{-a+b x}} \sqrt {1-\frac {b x}{a}}}}{2 \sqrt {a-b x} (a+b x)^{5/6}} \] Input:
Integrate[(Sqrt[a - b*x]*(a + b*x)^(1/6))/x^2,x]
Output:
((-2*a^2)/x + 2*b^2*x - b^2*Sqrt[1 - a/(b*x)]*(1 + a/(b*x))^(5/6)*x*(Appel lF1[1/3, -1/2, 5/6, 4/3, a/(b*x), -(a/(b*x))] - AppellF1[1/3, 1/2, -1/6, 4 /3, a/(b*x), -(a/(b*x))]) + (b*(a - b*x)*(a + b*x)*(Sqrt[1 - (b*x)/a]*Hype rgeometric2F1[-2/3, -1/6, 1/3, (2*a)/(a - b*x)] - 4*Sqrt[2]*((a + b*x)/(-a + b*x))^(1/6)*Hypergeometric2F1[-1/2, 1/6, 7/6, (a + b*x)/(2*a)]))/(a*((a + b*x)/(-a + b*x))^(1/6)*Sqrt[1 - (b*x)/a]))/(2*Sqrt[a - b*x]*(a + b*x)^( 5/6))
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {149, 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.
\(\displaystyle \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx\) |
\(\Big \downarrow \) 149 |
\(\displaystyle \frac {6 \int \frac {\sqrt {a-b x} (a+b x)}{x^2}d\sqrt [6]{a+b x}}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 6 b \int \frac {\sqrt {a-b x} (a+b x)}{b^2 x^2}d\sqrt [6]{a+b x}\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {6 \sqrt {2} b \sqrt {a-b x} \int \frac {(a+b x) \sqrt {2-\frac {a+b x}{a}}}{\sqrt {2} b^2 x^2}d\sqrt [6]{a+b x}}{\sqrt {2-\frac {a+b x}{a}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {6 b \sqrt {a-b x} \int \frac {(a+b x) \sqrt {2-\frac {a+b x}{a}}}{b^2 x^2}d\sqrt [6]{a+b x}}{\sqrt {2-\frac {a+b x}{a}}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {6 \sqrt {2} b \sqrt {a-b x} (a+b x)^{7/6} \operatorname {AppellF1}\left (\frac {7}{6},2,-\frac {1}{2},\frac {13}{6},\frac {a+b x}{a},\frac {a+b x}{2 a}\right )}{7 a^2 \sqrt {2-\frac {a+b x}{a}}}\) |
Input:
Int[(Sqrt[a - b*x]*(a + b*x)^(1/6))/x^2,x]
Output:
(6*Sqrt[2]*b*Sqrt[a - b*x]*(a + b*x)^(7/6)*AppellF1[7/6, 2, -1/2, 13/6, (a + b*x)/a, (a + b*x)/(2*a)])/(7*a^2*Sqrt[2 - (a + b*x)/a])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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])
\[\int \frac {\sqrt {-b x +a}\, \left (b x +a \right )^{\frac {1}{6}}}{x^{2}}d x\]
Input:
int((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x^2,x)
Output:
int((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x^2,x)
Timed out. \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx=\text {Timed out} \] Input:
integrate((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx=\int \frac {\sqrt {a - b x} \sqrt [6]{a + b x}}{x^{2}}\, dx \] Input:
integrate((-b*x+a)**(1/2)*(b*x+a)**(1/6)/x**2,x)
Output:
Integral(sqrt(a - b*x)*(a + b*x)**(1/6)/x**2, x)
\[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}} \sqrt {-b x + a}}{x^{2}} \,d x } \] Input:
integrate((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x^2,x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/6)*sqrt(-b*x + a)/x^2, x)
\[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{6}} \sqrt {-b x + a}}{x^{2}} \,d x } \] Input:
integrate((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x^2,x, algorithm="giac")
Output:
integrate((b*x + a)^(1/6)*sqrt(-b*x + a)/x^2, x)
Timed out. \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/6}\,\sqrt {a-b\,x}}{x^2} \,d x \] Input:
int(((a + b*x)^(1/6)*(a - b*x)^(1/2))/x^2,x)
Output:
int(((a + b*x)^(1/6)*(a - b*x)^(1/2))/x^2, x)
Time = 0.55 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a-b x} \sqrt [6]{a+b x}}{x^2} \, dx=\frac {\sqrt {b x +a}\, \left (-\sqrt {-b x +a}\, a -\sqrt {-b x +a}\, b x -\sqrt {a}\, \mathrm {log}\left (\sqrt {-b x +a}-\sqrt {a}\right ) b x +\sqrt {a}\, \mathrm {log}\left (\sqrt {-b x +a}+\sqrt {a}\right ) b x \right )}{\left (b x +a \right )^{\frac {1}{3}} a x} \] Input:
int((-b*x+a)^(1/2)*(b*x+a)^(1/6)/x^2,x)
Output:
(sqrt(a + b*x)*( - sqrt(a - b*x)*a - sqrt(a - b*x)*b*x - sqrt(a)*log(sqrt( a - b*x) - sqrt(a))*b*x + sqrt(a)*log(sqrt(a - b*x) + sqrt(a))*b*x))/((a + b*x)**(1/3)*a*x)