3.702 \(\int x^m (a+b x)^2 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+2}}{m+2}+\frac{b^2 x^{m+3}}{m+3} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(2 + m))/(2 + m) + (b^2*x^(3 + m))/(3 + m)

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Rubi [A]  time = 0.0342321, antiderivative size = 43, 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 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+2}}{m+2}+\frac{b^2 x^{m+3}}{m+3} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x)^2,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(2 + m))/(2 + m) + (b^2*x^(3 + m))/(3 + m)

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Rubi in Sympy [A]  time = 6.72856, size = 36, normalized size = 0.84 \[ \frac{a^{2} x^{m + 1}}{m + 1} + \frac{2 a b x^{m + 2}}{m + 2} + \frac{b^{2} x^{m + 3}}{m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x+a)**2,x)

[Out]

a**2*x**(m + 1)/(m + 1) + 2*a*b*x**(m + 2)/(m + 2) + b**2*x**(m + 3)/(m + 3)

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Mathematica [A]  time = 0.0255346, size = 39, normalized size = 0.91 \[ x^m \left (\frac{a^2 x}{m+1}+\frac{2 a b x^2}{m+2}+\frac{b^2 x^3}{m+3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x)^2,x]

[Out]

x^m*((a^2*x)/(1 + m) + (2*a*b*x^2)/(2 + m) + (b^2*x^3)/(3 + m))

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Maple [A]  time = 0., size = 87, normalized size = 2. \[{\frac{{x}^{1+m} \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+3\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+8\,abmx+2\,{b}^{2}{x}^{2}+5\,{a}^{2}m+6\,abx+6\,{a}^{2} \right ) }{ \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a)^2,x)

[Out]

x^(1+m)*(b^2*m^2*x^2+2*a*b*m^2*x+3*b^2*m*x^2+a^2*m^2+8*a*b*m*x+2*b^2*x^2+5*a^2*m
+6*a*b*x+6*a^2)/(3+m)/(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)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226931, size = 115, normalized size = 2.67 \[ \frac{{\left ({\left (b^{2} m^{2} + 3 \, b^{2} m + 2 \, b^{2}\right )} x^{3} + 2 \,{\left (a b m^{2} + 4 \, a b m + 3 \, a b\right )} x^{2} +{\left (a^{2} m^{2} + 5 \, a^{2} m + 6 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*x^m,x, algorithm="fricas")

[Out]

((b^2*m^2 + 3*b^2*m + 2*b^2)*x^3 + 2*(a*b*m^2 + 4*a*b*m + 3*a*b)*x^2 + (a^2*m^2
+ 5*a^2*m + 6*a^2)*x)*x^m/(m^3 + 6*m^2 + 11*m + 6)

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Sympy [A]  time = 1.43253, size = 299, normalized size = 6.95 \[ \begin{cases} - \frac{a^{2}}{2 x^{2}} - \frac{2 a b}{x} + b^{2} \log{\left (x \right )} & \text{for}\: m = -3 \\- \frac{a^{2}}{x} + 2 a b \log{\left (x \right )} + b^{2} x & \text{for}\: m = -2 \\a^{2} \log{\left (x \right )} + 2 a b x + \frac{b^{2} x^{2}}{2} & \text{for}\: m = -1 \\\frac{a^{2} m^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{5 a^{2} m x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 a^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 a b m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{8 a b m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 a b x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{b^{2} m^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 b^{2} m x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 b^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x+a)**2,x)

[Out]

Piecewise((-a**2/(2*x**2) - 2*a*b/x + b**2*log(x), Eq(m, -3)), (-a**2/x + 2*a*b*
log(x) + b**2*x, Eq(m, -2)), (a**2*log(x) + 2*a*b*x + b**2*x**2/2, Eq(m, -1)), (
a**2*m**2*x*x**m/(m**3 + 6*m**2 + 11*m + 6) + 5*a**2*m*x*x**m/(m**3 + 6*m**2 + 1
1*m + 6) + 6*a**2*x*x**m/(m**3 + 6*m**2 + 11*m + 6) + 2*a*b*m**2*x**2*x**m/(m**3
+ 6*m**2 + 11*m + 6) + 8*a*b*m*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + 6*a*b*x**
2*x**m/(m**3 + 6*m**2 + 11*m + 6) + b**2*m**2*x**3*x**m/(m**3 + 6*m**2 + 11*m +
6) + 3*b**2*m*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6) + 2*b**2*x**3*x**m/(m**3 + 6*
m**2 + 11*m + 6), True))

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GIAC/XCAS [A]  time = 0.209904, size = 182, normalized size = 4.23 \[ \frac{b^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a b m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, b^{2} m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, a b m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, b^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, a b x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*x^m,x, algorithm="giac")

[Out]

(b^2*m^2*x^3*e^(m*ln(x)) + 2*a*b*m^2*x^2*e^(m*ln(x)) + 3*b^2*m*x^3*e^(m*ln(x)) +
a^2*m^2*x*e^(m*ln(x)) + 8*a*b*m*x^2*e^(m*ln(x)) + 2*b^2*x^3*e^(m*ln(x)) + 5*a^2
*m*x*e^(m*ln(x)) + 6*a*b*x^2*e^(m*ln(x)) + 6*a^2*x*e^(m*ln(x)))/(m^3 + 6*m^2 + 1
1*m + 6)