Optimal. Leaf size=54 \[ \frac{x^{m+1} \left (a+b x^n\right )^{p+1} \, _2F_1\left (1,\frac{m+1}{n}+p+1;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]
[Out]
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Rubi [A] time = 0.0595514, antiderivative size = 62, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Int[x^m*(a + b*x^n)^p,x]
[Out]
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Rubi in Sympy [A] time = 8.20038, size = 46, normalized size = 0.85 \[ \frac{x^{m + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(a+b*x**n)**p,x)
[Out]
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Mathematica [A] time = 0.0650122, size = 63, normalized size = 1.17 \[ \frac{x^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Integrate[x^m*(a + b*x^n)^p,x]
[Out]
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Maple [F] time = 0.158, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+b{x}^{n} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(a+b*x^n)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*x^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{n} + a\right )}^{p} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*x^m,x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(a+b*x**n)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{p} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^p*x^m,x, algorithm="giac")
[Out]