\(\int \frac {(-a+b x^{n/2})^{\frac {1-n}{n}} (a+b x^{n/2})^{\frac {1-n}{n}} (c+d x^n)}{x^2} \, dx\) [46]

Optimal result

Integrand size = 55, antiderivative size = 139 \[ \int \frac {\left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx=\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}}}{x}-\frac {d \left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-\frac {1}{n},-\frac {1-n}{n},\frac {b^2 x^n}{a^2}\right )}{b^2 x} \]

[Out]

(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)*hy
pergeom([-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))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {533, 463, 372, 371} \[ \int \frac {\left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx=\frac {a^2 d \left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (1-\frac {b^2 x^n}{a^2}\right )^{-\frac {1-n}{n}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{n},-\frac {1}{n},-\frac {1-n}{n},\frac {b^2 x^n}{a^2}\right )}{b^2 x}-\frac {\left (\frac {c}{a^2}+\frac {d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac {1}{n}-1} \left (a+b x^{n/2}\right )^{\frac {1}{n}-1} \left (a^2-b^2 x^n\right )}{x} \]

[In]

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

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 463

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] + Dist[d/b, Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /;
FreeQ[{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 533

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPa
rt[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] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rubi steps \begin{align*} \text {integral}& = \left (\left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (-a^2+b^2 x^n\right )^{-\frac {1-n}{n}}\right ) \int \frac {\left (-a^2+b^2 x^n\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 {\left (d \left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (-a^2+b^2 x^n\right )^{-\frac {1-n}{n}}\right ) \int \frac {\left (-a^2+b^2 x^n\right )^{1+\frac {1-n}{n}}}{x^2} \, dx}{b^2} \\ & = -\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 {\left (a^2 d \left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-\frac {1-n}{n}}\right ) \int \frac {\left (1-\frac {b^2 x^n}{a^2}\right )^{1+\frac {1-n}{n}}}{x^2} \, dx}{b^2} \\ & = -\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}} \, _2F_1\left (-\frac {1}{n},-\frac {1}{n};-\frac {1-n}{n};\frac {b^2 x^n}{a^2}\right )}{b^2 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (a+b x^{n/2}\right )^{\frac {1-n}{n}} \left (c+d x^n\right )}{x^2} \, dx=\frac {\left (-a+b x^{n/2}\right )^{\frac {1}{n}} \left (a+b x^{n/2}\right )^{\frac {1}{n}} \left (1-\frac {b^2 x^n}{a^2}\right )^{-1/n} \left (c (-1+n) \left (1-\frac {b^2 x^n}{a^2}\right )^{\frac {1}{n}}-d x^n \operatorname {Hypergeometric2F1}\left (\frac {-1+n}{n},\frac {-1+n}{n},2-\frac {1}{n},\frac {b^2 x^n}{a^2}\right )\right )}{a^2 (-1+n) x} \]

[In]

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

[Out]

((-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*Hypergeometric2
F1[(-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))

Maple [F]

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

[In]

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)

[Out]

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]

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

[In]

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")

[Out]

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

[In]

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)

[Out]

Timed out

Maxima [F]

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

[In]

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")

[Out]

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]

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

[In]

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")

[Out]

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

[In]

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

[Out]

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