Optimal. Leaf size=47 \[ \frac{x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{b x}{a}\right )}{m+1} \]
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Rubi [A] time = 0.0333883, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{b x}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Int[x^m*(a + b*x)^n,x]
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Rubi in Sympy [A] time = 5.91468, size = 36, normalized size = 0.77 \[ \frac{x^{m + 1} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(b*x+a)**n,x)
[Out]
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Mathematica [A] time = 0.0423101, size = 47, normalized size = 1. \[ \frac{x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{b x}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Integrate[x^m*(a + b*x)^n,x]
[Out]
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Maple [F] time = 0.146, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( bx+a \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(b*x+a)^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{n} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.0083, size = 34, normalized size = 0.72 \[ \frac{a^{n} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(b*x+a)**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^m,x, algorithm="giac")
[Out]