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

Optimal. Leaf size=139 \[ \frac{\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}}}{x}-\frac{d \left (b x^{n/2}-a\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}} \left (1-\frac{b^2 x^n}{a^2}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};-\frac{1-n}{n};\frac{b^2 x^n}{a^2}\right )}{b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.114513, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.085, Rules used = {519, 452, 365, 364} \[ \frac{\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}}}{x}-\frac{d \left (b x^{n/2}-a\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}} \left (1-\frac{b^2 x^n}{a^2}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};-\frac{1-n}{n};\frac{b^2 x^n}{a^2}\right )}{b^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 519

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)^FracP
art[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])

Rule 452

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 365

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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \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 &=\left (\left (-a+b x^{n/2}\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}} \left (-a^2+b^2 x^n\right )^{-1/n}\right ) \int \frac{\left (-a^2+b^2 x^n\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{\left (d \left (-a+b x^{n/2}\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}} \left (-a^2+b^2 x^n\right )^{-1/n}\right ) \int \frac{\left (-a^2+b^2 x^n\right )^{\frac{1}{n}}}{x^2} \, dx}{b^2}\\ &=\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{\left (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}\right ) \int \frac{\left (1-\frac{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 (-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} \, _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]  time = 0.15465, size = 124, normalized size = 0.89 \[ \frac{\left (b x^{n/2}-a\right )^{\frac{1}{n}} \left (a+b x^{n/2}\right )^{\frac{1}{n}} \left (1-\frac{b^2 x^n}{a^2}\right )^{-1/n} \left (c (n-1) \left (1-\frac{b^2 x^n}{a^2}\right )^{\frac{1}{n}}-d x^n \, _2F_1\left (\frac{n-1}{n},\frac{n-1}{n};2-\frac{1}{n};\frac{b^2 x^n}{a^2}\right )\right )}{a^2 (n-1) x} \]

Antiderivative was successfully verified.

[In]

Integrate[((-a + b*x^(n/2))^(-1 + n^(-1))*(a + b*x^(n/2))^(-1 + n^(-1))*(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))

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Maple [F]  time = 0.795, size = 0, normalized size = 0. \begin{align*} \int{\frac{c+d{x}^{n}}{{x}^{2}} \left ( -a+b{x}^{{\frac{n}{2}}} \right ) ^{-1+{n}^{-1}} \left ( a+b{x}^{{\frac{n}{2}}} \right ) ^{-1+{n}^{-1}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-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)

[Out]

int((-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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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