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

Optimal. Leaf size=25 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+2}}{m+2} \]

[Out]

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

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Rubi [A]  time = 0.0196175, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+2}}{m+2} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.54535, size = 19, normalized size = 0.76 \[ \frac{a x^{m + 1}}{m + 1} + \frac{b x^{m + 2}}{m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.017617, size = 23, normalized size = 0.92 \[ x^m \left (\frac{a x}{m+1}+\frac{b x^2}{m+2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x),x]

[Out]

x^m*((a*x)/(1 + m) + (b*x^2)/(2 + m))

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Maple [A]  time = 0., size = 31, normalized size = 1.2 \[{\frac{{x}^{1+m} \left ( bmx+am+bx+2\,a \right ) }{ \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a),x)

[Out]

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

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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 + a)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224939, size = 45, normalized size = 1.8 \[ \frac{{\left ({\left (b m + b\right )} x^{2} +{\left (a m + 2 \, a\right )} x\right )} x^{m}}{m^{2} + 3 \, m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b*m + b)*x^2 + (a*m + 2*a)*x)*x^m/(m^2 + 3*m + 2)

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Sympy [A]  time = 0.737113, size = 87, normalized size = 3.48 \[ \begin{cases} - \frac{a}{x} + b \log{\left (x \right )} & \text{for}\: m = -2 \\a \log{\left (x \right )} + b x & \text{for}\: m = -1 \\\frac{a m x x^{m}}{m^{2} + 3 m + 2} + \frac{2 a x x^{m}}{m^{2} + 3 m + 2} + \frac{b m x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac{b x^{2} x^{m}}{m^{2} + 3 m + 2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x+a),x)

[Out]

Piecewise((-a/x + b*log(x), Eq(m, -2)), (a*log(x) + b*x, Eq(m, -1)), (a*m*x*x**m
/(m**2 + 3*m + 2) + 2*a*x*x**m/(m**2 + 3*m + 2) + b*m*x**2*x**m/(m**2 + 3*m + 2)
+ b*x**2*x**m/(m**2 + 3*m + 2), True))

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GIAC/XCAS [A]  time = 0.206867, size = 69, normalized size = 2.76 \[ \frac{b m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + a m x e^{\left (m{\rm ln}\left (x\right )\right )} + b x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{2} + 3 \, m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*m*x^2*e^(m*ln(x)) + a*m*x*e^(m*ln(x)) + b*x^2*e^(m*ln(x)) + 2*a*x*e^(m*ln(x))
)/(m^2 + 3*m + 2)