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3.2668 \(\int x^m (a+b x^n)^2 \, dx\)

Optimal. Leaf size=51 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+n+1}}{m+n+1}+\frac{b^2 x^{m+2 n+1}}{m+2 n+1} \]

[Out]

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

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Rubi [A]  time = 0.0182556, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+n+1}}{m+n+1}+\frac{b^2 x^{m+2 n+1}}{m+2 n+1} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^n)^2,x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
 x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^m \left (a+b x^n\right )^2 \, dx &=\int \left (a^2 x^m+2 a b x^{m+n}+b^2 x^{m+2 n}\right ) \, dx\\ &=\frac{a^2 x^{1+m}}{1+m}+\frac{2 a b x^{1+m+n}}{1+m+n}+\frac{b^2 x^{1+m+2 n}}{1+m+2 n}\\ \end{align*}

Mathematica [A]  time = 0.0328734, size = 46, normalized size = 0.9 \[ x^{m+1} \left (\frac{a^2}{m+1}+\frac{2 a b x^n}{m+n+1}+\frac{b^2 x^{2 n}}{m+2 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^n)^2,x]

[Out]

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

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Maple [A]  time = 0.012, size = 63, normalized size = 1.2 \begin{align*}{\frac{{a}^{2}x{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+{\frac{{b}^{2}x{{\rm e}^{m\ln \left ( x \right ) }} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+m+2\,n}}+2\,{\frac{xab{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( x \right ) }}}{m+n+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a+b*x^n)^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.04344, size = 336, normalized size = 6.59 \begin{align*} \frac{{\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2} +{\left (b^{2} m + b^{2}\right )} n\right )} x x^{m} x^{2 \, n} + 2 \,{\left (a b m^{2} + 2 \, a b m + a b + 2 \,{\left (a b m + a b\right )} n\right )} x x^{m} x^{n} +{\left (a^{2} m^{2} + 2 \, a^{2} n^{2} + 2 \, a^{2} m + a^{2} + 3 \,{\left (a^{2} m + a^{2}\right )} n\right )} x x^{m}}{m^{3} + 2 \,{\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \,{\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 64.266, size = 1280, normalized size = 25.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a+b*x**n)**2,x)

[Out]

Piecewise(((a + b)**2*log(x), Eq(m, -1) & Eq(n, 0)), (a**2*log(x) + 2*a*b*x**n/n + b**2*x**(2*n)/(2*n), Eq(m,
-1)), (-2*a**2*n/(4*n**2*x**(2*n) + 2*n*x**(2*n)) - a**2/(4*n**2*x**(2*n) + 2*n*x**(2*n)) - 8*a*b*n*x**n/(4*n*
*2*x**(2*n) + 2*n*x**(2*n)) - 4*a*b*x**n/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 4*b**2*n**2*x**(2*n)*log(x)/(4*n**
2*x**(2*n) + 2*n*x**(2*n)) + 2*b**2*n*x**(2*n)*log(x)/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 2*b**2*n*x**(2*n)/(4*
n**2*x**(2*n) + 2*n*x**(2*n)), Eq(m, -2*n - 1)), (-a**2*n/(n**2*x**n + n*x**n) - a**2/(n**2*x**n + n*x**n) + 2
*a*b*n**2*x**n*log(x)/(n**2*x**n + n*x**n) + 2*a*b*n*x**n*log(x)/(n**2*x**n + n*x**n) + 2*a*b*n*x**n/(n**2*x**
n + n*x**n) + b**2*n*x**(2*n)/(n**2*x**n + n*x**n) + b**2*x**(2*n)/(n**2*x**n + n*x**n), Eq(m, -n - 1)), (a**2
*m**2*x*x**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 3*a**2*m*n*x*x**m/(m**3
+ 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*a**2*m*x*x**m/(m**3 + 3*m**2*n + 3*m**2 +
 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*a**2*n**2*x*x**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n
+ 3*m + 2*n**2 + 3*n + 1) + 3*a**2*n*x*x**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n
+ 1) + a**2*x*x**m/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*a*b*m**2*x*x**m*
x**n/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 4*a*b*m*n*x*x**m*x**n/(m**3 + 3*
m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 4*a*b*m*x*x**m*x**n/(m**3 + 3*m**2*n + 3*m**2 +
 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 4*a*b*n*x*x**m*x**n/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n
 + 3*m + 2*n**2 + 3*n + 1) + 2*a*b*x*x**m*x**n/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3
*n + 1) + b**2*m**2*x*x**m*x**(2*n)/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + b
**2*m*n*x*x**m*x**(2*n)/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + 2*b**2*m*x*x*
*m*x**(2*n)/(m**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + b**2*n*x*x**m*x**(2*n)/(m
**3 + 3*m**2*n + 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1) + b**2*x*x**m*x**(2*n)/(m**3 + 3*m**2*n +
 3*m**2 + 2*m*n**2 + 6*m*n + 3*m + 2*n**2 + 3*n + 1), True))

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Giac [B]  time = 1.11933, size = 321, normalized size = 6.29 \begin{align*} \frac{b^{2} m^{2} x x^{m} x^{2 \, n} + b^{2} m n x x^{m} x^{2 \, n} + 2 \, a b m^{2} x x^{m} x^{n} + 4 \, a b m n x x^{m} x^{n} + a^{2} m^{2} x x^{m} + 3 \, a^{2} m n x x^{m} + 2 \, a^{2} n^{2} x x^{m} + 2 \, b^{2} m x x^{m} x^{2 \, n} + b^{2} n x x^{m} x^{2 \, n} + 4 \, a b m x x^{m} x^{n} + 4 \, a b n x x^{m} x^{n} + 2 \, a^{2} m x x^{m} + 3 \, a^{2} n x x^{m} + b^{2} x x^{m} x^{2 \, n} + 2 \, a b x x^{m} x^{n} + a^{2} x x^{m}}{m^{3} + 3 \, m^{2} n + 2 \, m n^{2} + 3 \, m^{2} + 6 \, m n + 2 \, n^{2} + 3 \, m + 3 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*m^2*x*x^m*x^(2*n) + b^2*m*n*x*x^m*x^(2*n) + 2*a*b*m^2*x*x^m*x^n + 4*a*b*m*n*x*x^m*x^n + a^2*m^2*x*x^m + 3
*a^2*m*n*x*x^m + 2*a^2*n^2*x*x^m + 2*b^2*m*x*x^m*x^(2*n) + b^2*n*x*x^m*x^(2*n) + 4*a*b*m*x*x^m*x^n + 4*a*b*n*x
*x^m*x^n + 2*a^2*m*x*x^m + 3*a^2*n*x*x^m + b^2*x*x^m*x^(2*n) + 2*a*b*x*x^m*x^n + a^2*x*x^m)/(m^3 + 3*m^2*n + 2
*m*n^2 + 3*m^2 + 6*m*n + 2*n^2 + 3*m + 3*n + 1)

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