3.730 \(\int x^m (a+b x)^n \, dx\)

Optimal. Leaf size=47 \[ \frac{x^{m+1} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{b x}{a}\right )}{m+1} \]

[Out]

(x^(1 + m)*(a + b*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((b*x)/a)])/((1 + m)
*(1 + (b*x)/a)^n)

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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]

[Out]

(x^(1 + m)*(a + b*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((b*x)/a)])/((1 + m)
*(1 + (b*x)/a)^n)

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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]

x**(m + 1)*(1 + b*x/a)**(-n)*(a + b*x)**n*hyper((-n, m + 1), (m + 2,), -b*x/a)/(
m + 1)

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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]

(x^(1 + m)*(a + b*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((b*x)/a)])/((1 + m)
*(1 + (b*x)/a)^n)

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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]

int(x^m*(b*x+a)^n,x)

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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]

integrate((b*x + a)^n*x^m, x)

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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]

integral((b*x + a)^n*x^m, x)

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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]

a**n*x*x**m*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), b*x*exp_polar(I*pi)/a)/gam
ma(m + 2)

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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]

integrate((b*x + a)^n*x^m, x)