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

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 = 0.361048, antiderivative size = 167, normalized size of antiderivative = 1.2, number of steps used = 4, number of rules used = 4, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073 \[ \frac{a^2 d \left (b x^{n/2}-a\right )^{\frac{1}{n}-1} \left (a+b x^{n/2}\right )^{\frac{1}{n}-1} \left (1-\frac{b^2 x^n}{a^2}\right )^{-\frac{1-n}{n}} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};-\frac{1-n}{n};\frac{b^2 x^n}{a^2}\right )}{b^2 x}-\frac{\left (\frac{c}{a^2}+\frac{d}{b^2}\right ) \left (b x^{n/2}-a\right )^{\frac{1}{n}-1} \left (a+b x^{n/2}\right )^{\frac{1}{n}-1} \left (a^2-b^2 x^n\right )}{x} \]

Antiderivative was successfully verified.

[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))

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Rubi in Sympy [A]  time = 37.136, size = 143, normalized size = 1.03 \[ \frac{a^{2} d \left (1 - \frac{b^{2} x^{n}}{a^{2}}\right )^{\frac{n - 1}{n}} \left (- a + b x^{\frac{n}{2}}\right )^{- \frac{n - 1}{n}} \left (a + b x^{\frac{n}{2}}\right )^{- \frac{n - 1}{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{n}, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{\frac{b^{2} x^{n}}{a^{2}}} \right )}}{b^{2} x} + \frac{\left (- a + b x^{\frac{n}{2}}\right )^{- \frac{n - 1}{n}} \left (a + b x^{\frac{n}{2}}\right )^{- \frac{n - 1}{n}} \left (- a^{2} + b^{2} x^{n}\right )^{\frac{1}{n}} \left (- a^{2} + b^{2} x^{n}\right )^{\frac{n - 1}{n}} \left (\frac{d}{b^{2}} + \frac{c}{a^{2}}\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

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

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Mathematica [A]  time = 0.140965, 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)/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*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))

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Maple [F]  time = 0.276, size = 0, normalized size = 0. \[ \int{\frac{c+d{x}^{n}}{{x}^{2}} \left ( -a+b{x}^{{\frac{n}{2}}} \right ) ^{{\frac{1-n}{n}}} \left ( a+b{x}^{{\frac{n}{2}}} \right ) ^{{\frac{1-n}{n}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}{\left (b x^{\frac{1}{2} \, n} + a\right )}^{-\frac{n - 1}{n}}{\left (b x^{\frac{1}{2} \, n} - a\right )}^{-\frac{n - 1}{n}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{n} + c}{{\left (b x^{\frac{1}{2} \, n} + a\right )}^{\frac{n - 1}{n}}{\left (b x^{\frac{1}{2} \, n} - a\right )}^{\frac{n - 1}{n}} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{n} + c}{{\left (b x^{\frac{1}{2} \, n} + a\right )}^{\frac{n - 1}{n}}{\left (b x^{\frac{1}{2} \, n} - a\right )}^{\frac{n - 1}{n}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)