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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]