Integrand size = 47, antiderivative size = 139 \[ \int \frac {\left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx=\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}}}{x}-\frac {d \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-\frac {1}{n},-\frac {1-n}{n},\frac {b^2 x^n}{a^2}\right )}{b^2 x} \]
(c/a^2+d/b^2)*(-a+b*x^(1/2*n))^(1/n)*(a+b*x^(1/2*n))^(1/n)/x-d*(-a+b*x^(1/ 2*n))^(1/n)*(a+b*x^(1/2*n))^(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))
Time = 0.41 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx=\frac {\left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \left (c (-1+n) \left (1-\frac {b^2 x^n}{a^2}\right )^{\frac {1}{n}}-d x^n \operatorname {Hypergeometric2F1}\left (\frac {-1+n}{n},\frac {-1+n}{n},2-\frac {1}{n},\frac {b^2 x^n}{a^2}\right )\right )}{a^2 (-1+n) x} \]
((-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))
Time = 0.39 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, 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.
| \(\displaystyle \int \frac {\left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (c+d x^n\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2038 |
| \(\displaystyle \left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (b^2 x^n-a^2\right )^{-1/n} \int \frac {\left (b^2 x^n-a^2\right )^{\frac {1}{n}-1} \left (d x^n+c\right )}{x^2}dx\) |
| \(\Big \downarrow \) 954 |
| \(\displaystyle \left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (b^2 x^n-a^2\right )^{-1/n} \left (\frac {d \int \frac {\left (b^2 x^n-a^2\right )^{\frac {1}{n}}}{x^2}dx}{b^2}+\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b^2 x^n-a^2\right )^{\frac {1}{n}}}{x}\right )\) |
| \(\Big \downarrow \) 882 |
| \(\displaystyle \left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (b^2 x^n-a^2\right )^{-1/n} \left (\frac {d \left (-\frac {x^n}{a^2-b^2 x^n}\right )^{\frac {1}{n}} \left (b^2 x^n-a^2\right )^{\frac {1}{n}} \int \frac {\left (-\frac {x^n}{a^2-b^2 x^n}\right )^{-1-\frac {1}{n}}}{\frac {b^2 x^n}{a^2-b^2 x^n}+1}d\left (-\frac {x^n}{a^2-b^2 x^n}\right )}{b^2 n x}+\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b^2 x^n-a^2\right )^{\frac {1}{n}}}{x}\right )\) |
| \(\Big \downarrow \) 74 |
| \(\displaystyle \left (b x^{n/2}-a\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (b^2 x^n-a^2\right )^{-1/n} \left (\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b^2 x^n-a^2\right )^{\frac {1}{n}}}{x}-\frac {d \left (b^2 x^n-a^2\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {b^2 x^n}{a^2-b^2 x^n}\right )}{b^2 x}\right )\) |
((-a + b*x^(n/2))^n^(-1)*(a + b*x^(n/2))^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)*Hypergeometric2F1[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 )^n^(-1)
3.1.45.3.1 Defintions of rubi rules used
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])))
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]]
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]
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])
\[\int \frac {\left (-a +b \,x^{\frac {n}{2}}\right )^{-1+\frac {1}{n}} \left (a +b \,x^{\frac {n}{2}}\right )^{-1+\frac {1}{n}} \left (c +d \,x^{n}\right )}{x^{2}}d x\]
\[ \int \frac {\left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{\frac {1}{2} \, n} + a\right )}^{\frac {1}{n} - 1} {\left (b x^{\frac {1}{2} \, n} - a\right )}^{\frac {1}{n} - 1}}{x^{2}} \,d x } \]
integrate((-a+b*x^(1/2*n))^(-1+1/n)*(a+b*x^(1/2*n))^(-1+1/n)*(c+d*x^n)/x^2 ,x, algorithm="fricas")
Timed out. \[ \int \frac {\left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{\frac {1}{2} \, n} + a\right )}^{\frac {1}{n} - 1} {\left (b x^{\frac {1}{2} \, n} - a\right )}^{\frac {1}{n} - 1}}{x^{2}} \,d x } \]
integrate((-a+b*x^(1/2*n))^(-1+1/n)*(a+b*x^(1/2*n))^(-1+1/n)*(c+d*x^n)/x^2 ,x, algorithm="maxima")
\[ \int \frac {\left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{\frac {1}{2} \, n} + a\right )}^{\frac {1}{n} - 1} {\left (b x^{\frac {1}{2} \, n} - a\right )}^{\frac {1}{n} - 1}}{x^{2}} \,d x } \]
integrate((-a+b*x^(1/2*n))^(-1+1/n)*(a+b*x^(1/2*n))^(-1+1/n)*(c+d*x^n)/x^2 ,x, algorithm="giac")
Timed out. \[ \int \frac {\left (-a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (a+b x^{n/2}\right )^{-1+\frac {1}{n}} \left (c+d x^n\right )}{x^2} \, dx=\int \frac {{\left (a+b\,x^{n/2}\right )}^{\frac {1}{n}-1}\,{\left (b\,x^{n/2}-a\right )}^{\frac {1}{n}-1}\,\left (c+d\,x^n\right )}{x^2} \,d x \]