3.733 \(\int x^2 (a+b x)^n \, dx\)

Optimal. Leaf size=60 \[ \frac{a^2 (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a (a+b x)^{n+2}}{b^3 (n+2)}+\frac{(a+b x)^{n+3}}{b^3 (n+3)} \]

[Out]

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

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Rubi [A]  time = 0.0504444, antiderivative size = 60, 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 \[ \frac{a^2 (a+b x)^{n+1}}{b^3 (n+1)}-\frac{2 a (a+b x)^{n+2}}{b^3 (n+2)}+\frac{(a+b x)^{n+3}}{b^3 (n+3)} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.2442, size = 51, normalized size = 0.85 \[ \frac{a^{2} \left (a + b x\right )^{n + 1}}{b^{3} \left (n + 1\right )} - \frac{2 a \left (a + b x\right )^{n + 2}}{b^{3} \left (n + 2\right )} + \frac{\left (a + b x\right )^{n + 3}}{b^{3} \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(a + b*x)**(n + 1)/(b**3*(n + 1)) - 2*a*(a + b*x)**(n + 2)/(b**3*(n + 2)) +
(a + b*x)**(n + 3)/(b**3*(n + 3))

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Mathematica [A]  time = 0.0363888, size = 57, normalized size = 0.95 \[ \frac{(a+b x)^{n+1} \left (2 a^2-2 a b (n+1) x+b^2 \left (n^2+3 n+2\right ) x^2\right )}{b^3 (n+1) (n+2) (n+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)^n,x]

[Out]

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

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Maple [A]  time = 0.009, size = 73, normalized size = 1.2 \[{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}{n}^{2}{x}^{2}+3\,{b}^{2}n{x}^{2}-2\,abnx+2\,{b}^{2}{x}^{2}-2\,abx+2\,{a}^{2} \right ) }{{b}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^n,x)

[Out]

(b*x+a)^(1+n)*(b^2*n^2*x^2+3*b^2*n*x^2-2*a*b*n*x+2*b^2*x^2-2*a*b*x+2*a^2)/b^3/(n
^3+6*n^2+11*n+6)

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Maxima [A]  time = 1.35773, size = 92, normalized size = 1.53 \[ \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} +{\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^
n/((n^3 + 6*n^2 + 11*n + 6)*b^3)

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Fricas [A]  time = 0.225279, size = 130, normalized size = 2.17 \[ -\frac{{\left (2 \, a^{2} b n x -{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} - 2 \, a^{3} -{\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )}{\left (b x + a\right )}^{n}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a^2*b*n*x - (b^3*n^2 + 3*b^3*n + 2*b^3)*x^3 - 2*a^3 - (a*b^2*n^2 + a*b^2*n)*
x^2)*(b*x + a)^n/(b^3*n^3 + 6*b^3*n^2 + 11*b^3*n + 6*b^3)

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Sympy [A]  time = 3.22549, size = 597, normalized size = 9.95 \[ \begin{cases} \frac{a^{n} x^{3}}{3} & \text{for}\: b = 0 \\\frac{2 a^{2} \log{\left (\frac{a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{3 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{4 a b x \log{\left (\frac{a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{4 a b x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{2 b^{2} x^{2} \log{\left (\frac{a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} & \text{for}\: n = -3 \\- \frac{2 a^{2} \log{\left (\frac{a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac{2 a^{2}}{a b^{3} + b^{4} x} - \frac{2 a b x \log{\left (\frac{a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac{b^{2} x^{2}}{a b^{3} + b^{4} x} & \text{for}\: n = -2 \\\frac{a^{2} \log{\left (\frac{a}{b} + x \right )}}{b^{3}} - \frac{a x}{b^{2}} + \frac{x^{2}}{2 b} & \text{for}\: n = -1 \\\frac{2 a^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} - \frac{2 a^{2} b n x \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac{a b^{2} n^{2} x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac{a b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac{b^{3} n^{2} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac{3 b^{3} n x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} + \frac{2 b^{3} x^{3} \left (a + b x\right )^{n}}{b^{3} n^{3} + 6 b^{3} n^{2} + 11 b^{3} n + 6 b^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)**n,x)

[Out]

Piecewise((a**n*x**3/3, Eq(b, 0)), (2*a**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*
x + 2*b**5*x**2) + 3*a**2/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*x*log
(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2) + 4*a*b*x/(2*a**2*b**3 + 4*a*
b**4*x + 2*b**5*x**2) + 2*b**2*x**2*log(a/b + x)/(2*a**2*b**3 + 4*a*b**4*x + 2*b
**5*x**2), Eq(n, -3)), (-2*a**2*log(a/b + x)/(a*b**3 + b**4*x) - 2*a**2/(a*b**3
+ b**4*x) - 2*a*b*x*log(a/b + x)/(a*b**3 + b**4*x) + b**2*x**2/(a*b**3 + b**4*x)
, Eq(n, -2)), (a**2*log(a/b + x)/b**3 - a*x/b**2 + x**2/(2*b), Eq(n, -1)), (2*a*
*3*(a + b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) - 2*a**2*b*n*x*(a
+ b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*n**2*x**2*(a
+ b*x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + a*b**2*n*x**2*(a + b*
x)**n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + b**3*n**2*x**3*(a + b*x)*
*n/(b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 3*b**3*n*x**3*(a + b*x)**n/(
b**3*n**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3) + 2*b**3*x**3*(a + b*x)**n/(b**3*n
**3 + 6*b**3*n**2 + 11*b**3*n + 6*b**3), True))

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GIAC/XCAS [A]  time = 0.208666, size = 208, normalized size = 3.47 \[ \frac{b^{3} n^{2} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{2} n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 3 \, b^{3} n x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{2} n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, b^{3} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 2 \, a^{2} b n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, a^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )}}{b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^3*n^2*x^3*e^(n*ln(b*x + a)) + a*b^2*n^2*x^2*e^(n*ln(b*x + a)) + 3*b^3*n*x^3*e
^(n*ln(b*x + a)) + a*b^2*n*x^2*e^(n*ln(b*x + a)) + 2*b^3*x^3*e^(n*ln(b*x + a)) -
2*a^2*b*n*x*e^(n*ln(b*x + a)) + 2*a^3*e^(n*ln(b*x + a)))/(b^3*n^3 + 6*b^3*n^2 +
11*b^3*n + 6*b^3)