\(\int \frac {(-a+b x^{n/2})^{\frac {1-n}{n}} (a+b x^{n/2})^{\frac {1-n}{n}} (c+d x^n)}{x^2} \, dx\) [91]

Optimal result

Integrand size = 55, antiderivative size = 167 \[ \int \frac {\left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx=-\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a^2-b^2 x^n\right )}{x}+\frac {a^2 d \left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-\frac {1-n}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-\frac {1}{n},-\frac {1-n}{n},\frac {b^2 x^n}{a^2}\right )}{b^2 x} \] Output:

-(c/a^2+d/b^2)*(-a+b*x^(1/2*n))^(-1+1/n)*(a+b*x^(1/2*n))^(-1+1/n)*(a^2-b^2 
*x^n)/x+a^2*d*(-a+b*x^(1/2*n))^(-1+1/n)*(a+b*x^(1/2*n))^(-1+1/n)*hypergeom 
([-1/n, -1/n],[-(1-n)/n],b^2*x^n/a^2)/b^2/x/((1-b^2*x^n/a^2)^((1-n)/n))
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx=\frac {\left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \left (c (-1+n) \left (1-\frac {b^2 x^n}{a^2}\right )^{\frac {1}{n}}-d x^n \operatorname {Hypergeometric2F1}\left (\frac {-1+n}{n},\frac {-1+n}{n},2-\frac {1}{n},\frac {b^2 x^n}{a^2}\right )\right )}{a^2 (-1+n) x} \] Input:

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

Output:

((-a + b*x^(n/2))^n^(-1)*(a + b*x^(n/2))^n^(-1)*(c*(-1 + n)*(1 - (b^2*x^n) 
/a^2)^n^(-1) - d*x^n*Hypergeometric2F1[(-1 + n)/n, (-1 + n)/n, 2 - n^(-1), 
 (b^2*x^n)/a^2]))/(a^2*(-1 + n)*x*(1 - (b^2*x^n)/a^2)^n^(-1))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {2038, 954, 882, 74}

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.

\(\displaystyle \int \frac {\left (b x^{n/2}-a\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx\)

\(\Big \downarrow \) 2038

\(\displaystyle \left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (b^2 x^n-a^2\right )^{-\frac {1-n}{n}} \int \frac {\left (b^2 x^n-a^2\right )^{\frac {1}{n}-1} \left (d x^n+c\right )}{x^2}dx\)

\(\Big \downarrow \) 954

\(\displaystyle \left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (b^2 x^n-a^2\right )^{-\frac {1-n}{n}} \left (\frac {d \int \frac {\left (b^2 x^n-a^2\right )^{\frac {1}{n}}}{x^2}dx}{b^2}+\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b^2 x^n-a^2\right )^{\frac {1}{n}}}{x}\right )\)

\(\Big \downarrow \) 882

\(\displaystyle \left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (b^2 x^n-a^2\right )^{-\frac {1-n}{n}} \left (\frac {d \left (-\frac {x^n}{a^2-b^2 x^n}\right )^{\frac {1}{n}} \left (b^2 x^n-a^2\right )^{\frac {1}{n}} \int \frac {\left (-\frac {x^n}{a^2-b^2 x^n}\right )^{-1-\frac {1}{n}}}{\frac {b^2 x^n}{a^2-b^2 x^n}+1}d\left (-\frac {x^n}{a^2-b^2 x^n}\right )}{b^2 n x}+\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b^2 x^n-a^2\right )^{\frac {1}{n}}}{x}\right )\)

\(\Big \downarrow \) 74

\(\displaystyle \left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (b^2 x^n-a^2\right )^{-\frac {1-n}{n}} \left (\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b^2 x^n-a^2\right )^{\frac {1}{n}}}{x}-\frac {d \left (b^2 x^n-a^2\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {b^2 x^n}{a^2-b^2 x^n}\right )}{b^2 x}\right )\)

Input:

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

Output:

((-a + b*x^(n/2))^(-1 + n^(-1))*(a + b*x^(n/2))^(-1 + n^(-1))*(((c/a^2 + d 
/b^2)*(-a^2 + b^2*x^n)^n^(-1))/x - (d*(-a^2 + b^2*x^n)^n^(-1)*Hypergeometr 
ic2F1[1, -n^(-1), -((1 - n)/n), -((b^2*x^n)/(a^2 - b^2*x^n))])/(b^2*x)))/( 
-a^2 + b^2*x^n)^((1 - n)/n)
 

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 882

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^Simplify[ 
(m + 1)/n + p]*x^m*(a + b*x^n)^p*((x^n/(a + b*x^n))^p/(n*x^Simplify[m + n*p 
]))   Subst[Int[x^((m + 1)/n - 1)/(1 - b*x)^(Simplify[(m + 1)/n + p] + 1), 
x], x, x^n/(a + b*x^n)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simpli 
fy[(m + 1)/n + p]]
 

rule 954

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(b*c - a*d)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b* 
e*(m + 1))), x] + Simp[d/b   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; Fre 
eQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 
 1, 0] && NeQ[m, -1]
 

rule 2038

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p 
_)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_), x_Symbol] :> Simp[(a1 + b1*x^(n/2)) 
^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPart[p] 
)   Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, 
 b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(Eq 
Q[n, 2] && IGtQ[q, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x^n + c)/((b*x^(1/2*n) + a)^((n - 1)/n)*(b*x^(1/2*n) - a)^((n 
- 1)/n)*x^2), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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