Optimal. Leaf size=93 \[ \frac{x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac{1}{n}-p} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]
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Rubi [A] time = 0.0726765, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac{1}{n}-p} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^p*(c + d*x^n)^(-1 - n^(-1) - p),x]
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Rubi in Sympy [A] time = 8.19064, size = 83, normalized size = 0.89 \[ \frac{x \left (\frac{a \left (c + d x^{n}\right )}{c \left (a + b x^{n}\right )}\right )^{p + 1 + \frac{1}{n}} \left (a + b x^{n}\right )^{p + 1} \left (c + d x^{n}\right )^{- p - 1 - \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, p + 1 + \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (a d - b c\right )}{c \left (a + b x^{n}\right )}} \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**p*(c+d*x**n)**(-1-1/n-p),x)
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Mathematica [A] time = 0.558127, size = 93, normalized size = 1. \[ \frac{x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac{n p+1}{n}} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{c} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^p*(c + d*x^n)^(-1 - n^(-1) - p),x]
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Maple [F] time = 0.249, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{-1-{n}^{-1}-p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^p*(c+d*x^n)^(-1-1/n-p),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{-p - \frac{1}{n} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*(d*x^n + c)^(-p - 1/n - 1),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{n} + a\right )}^{p}}{{\left (d x^{n} + c\right )}^{\frac{n p + n + 1}{n}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*(d*x^n + c)^(-p - 1/n - 1),x, algorithm="fricas")
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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**n)**p*(c+d*x**n)**(-1-1/n-p),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{-p - \frac{1}{n} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*(d*x^n + c)^(-p - 1/n - 1),x, algorithm="giac")
[Out]