3.2662 \(\int x^m \left (a+b x^n\right ) \, dx\)

Optimal. Leaf size=27 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+n+1}}{m+n+1} \]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(1 + m + n))/(1 + m + n)

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Rubi [A]  time = 0.0249212, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+n+1}}{m+n+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^n),x]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(1 + m + n))/(1 + m + n)

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Rubi in Sympy [A]  time = 4.4022, size = 22, normalized size = 0.81 \[ \frac{a x^{m + 1}}{m + 1} + \frac{b x^{m + n + 1}}{m + n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(a+b*x**n),x)

[Out]

a*x**(m + 1)/(m + 1) + b*x**(m + n + 1)/(m + n + 1)

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Mathematica [A]  time = 0.0274097, size = 27, normalized size = 1. \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+n+1}}{m+n+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^n),x]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(1 + m + n))/(1 + m + n)

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Maple [A]  time = 0.021, size = 34, normalized size = 1.3 \[{\frac{ax{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+{\frac{bx{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( x \right ) }}}{1+m+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^n),x)

[Out]

a/(1+m)*x*exp(m*ln(x))+b/(1+m+n)*x*exp(m*ln(x))*exp(n*ln(x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23744, size = 58, normalized size = 2.15 \[ \frac{{\left (b m + b\right )} x x^{m} x^{n} +{\left (a m + a n + a\right )} x x^{m}}{m^{2} +{\left (m + 1\right )} n + 2 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*x^m,x, algorithm="fricas")

[Out]

((b*m + b)*x*x^m*x^n + (a*m + a*n + a)*x*x^m)/(m^2 + (m + 1)*n + 2*m + 1)

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Sympy [A]  time = 12.3647, size = 165, normalized size = 6.11 \[ \begin{cases} \left (a + b\right ) \log{\left (x \right )} & \text{for}\: m = -1 \wedge n = 0 \\a \log{\left (x \right )} + \frac{b x^{n}}{n} & \text{for}\: m = -1 \\\frac{a x x^{m}}{m + 1} + \frac{b m \log{\left (x \right )}}{m + 1} + \frac{b \log{\left (x \right )}}{m + 1} & \text{for}\: n = - m - 1 \\\frac{a m x x^{m}}{m^{2} + m n + 2 m + n + 1} + \frac{a n x x^{m}}{m^{2} + m n + 2 m + n + 1} + \frac{a x x^{m}}{m^{2} + m n + 2 m + n + 1} + \frac{b m x x^{m} x^{n}}{m^{2} + m n + 2 m + n + 1} + \frac{b x x^{m} x^{n}}{m^{2} + m n + 2 m + n + 1} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(a+b*x**n),x)

[Out]

Piecewise(((a + b)*log(x), Eq(m, -1) & Eq(n, 0)), (a*log(x) + b*x**n/n, Eq(m, -1
)), (a*x*x**m/(m + 1) + b*m*log(x)/(m + 1) + b*log(x)/(m + 1), Eq(n, -m - 1)), (
a*m*x*x**m/(m**2 + m*n + 2*m + n + 1) + a*n*x*x**m/(m**2 + m*n + 2*m + n + 1) +
a*x*x**m/(m**2 + m*n + 2*m + n + 1) + b*m*x*x**m*x**n/(m**2 + m*n + 2*m + n + 1)
+ b*x*x**m*x**n/(m**2 + m*n + 2*m + n + 1), True))

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GIAC/XCAS [A]  time = 0.219354, size = 93, normalized size = 3.44 \[ \frac{b m x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + a m x e^{\left (m{\rm ln}\left (x\right )\right )} + a n x e^{\left (m{\rm ln}\left (x\right )\right )} + b x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + a x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{2} + m n + 2 \, m + n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*x^m,x, algorithm="giac")

[Out]

(b*m*x*e^(m*ln(x) + n*ln(x)) + a*m*x*e^(m*ln(x)) + a*n*x*e^(m*ln(x)) + b*x*e^(m*
ln(x) + n*ln(x)) + a*x*e^(m*ln(x)))/(m^2 + m*n + 2*m + n + 1)