Optimal. Leaf size=56 \[ \frac{c^2 x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^3} \]
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Rubi [A] time = 0.0518859, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{c^2 x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^n)^(2 - n^(-1))/(a + b*x^n)^3,x]
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Rubi in Sympy [A] time = 6.46959, size = 44, normalized size = 0.79 \[ \frac{c^{2} x \left (c + d x^{n}\right )^{- \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, 3 \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (- a d + b c\right )}{a \left (c + d x^{n}\right )}} \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*x**n)**(2-1/n)/(a+b*x**n)**3,x)
[Out]
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Mathematica [F] time = 179.999, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In] Integrate[(c + d*x^n)^(2 - n^(-1))/(a + b*x^n)^3,x]
[Out]
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Maple [F] time = 0.115, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{3}} \left ( c+d{x}^{n} \right ) ^{2-{n}^{-1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*x^n)^(2-1/n)/(a+b*x^n)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}^{-\frac{1}{n} + 2}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n + 2)/(b*x^n + a)^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{n} + c\right )}^{\frac{2 \, n - 1}{n}}}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n + 2)/(b*x^n + a)^3,x, algorithm="fricas")
[Out]
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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((c+d*x**n)**(2-1/n)/(a+b*x**n)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}^{-\frac{1}{n} + 2}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^(-1/n + 2)/(b*x^n + a)^3,x, algorithm="giac")
[Out]