3.2667 \(\int x^m (a+b x^n)^3 \, dx\)

Optimal. Leaf size=75 \[ \frac{3 a^2 b x^{m+n+1}}{m+n+1}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+2 n+1}}{m+2 n+1}+\frac{b^3 x^{m+3 n+1}}{m+3 n+1} \]

[Out]

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

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Rubi [A]  time = 0.0340715, antiderivative size = 75, 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} \[ \frac{3 a^2 b x^{m+n+1}}{m+n+1}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+2 n+1}}{m+2 n+1}+\frac{b^3 x^{m+3 n+1}}{m+3 n+1} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^n)^3,x]

[Out]

(a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(1 + m + n))/(1 + m + n) + (3*a*b^2*x^(1 + m + 2*n))/(1 + m + 2*n) + (b^3
*x^(1 + m + 3*n))/(1 + m + 3*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 )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{m+n}+3 a b^2 x^{m+2 n}+b^3 x^{m+3 n}\right ) \, dx\\ &=\frac{a^3 x^{1+m}}{1+m}+\frac{3 a^2 b x^{1+m+n}}{1+m+n}+\frac{3 a b^2 x^{1+m+2 n}}{1+m+2 n}+\frac{b^3 x^{1+m+3 n}}{1+m+3 n}\\ \end{align*}

Mathematica [A]  time = 0.0549593, size = 67, normalized size = 0.89 \[ x^{m+1} \left (\frac{3 a^2 b x^n}{m+n+1}+\frac{a^3}{m+1}+\frac{3 a b^2 x^{2 n}}{m+2 n+1}+\frac{b^3 x^{3 n}}{m+3 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^n)^3,x]

[Out]

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

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Maple [A]  time = 0.015, size = 92, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}x{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+{\frac{{b}^{3}x{{\rm e}^{m\ln \left ( x \right ) }} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+m+3\,n}}+3\,{\frac{{a}^{2}bx{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( x \right ) }}}{m+n+1}}+3\,{\frac{xa{b}^{2}{{\rm e}^{m\ln \left ( x \right ) }} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+m+2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a+b*x^n)^3,x)

[Out]

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

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.07365, size = 779, normalized size = 10.39 \begin{align*} \frac{{\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3} + 2 \,{\left (b^{3} m + b^{3}\right )} n^{2} + 3 \,{\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x x^{m} x^{3 \, n} + 3 \,{\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} + 3 \,{\left (a b^{2} m + a b^{2}\right )} n^{2} + 4 \,{\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x x^{m} x^{2 \, n} + 3 \,{\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 6 \,{\left (a^{2} b m + a^{2} b\right )} n^{2} + 5 \,{\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x x^{m} x^{n} +{\left (a^{3} m^{3} + 6 \, a^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 11 \,{\left (a^{3} m + a^{3}\right )} n^{2} + 6 \,{\left (a^{3} m^{2} + 2 \, a^{3} m + a^{3}\right )} n\right )} x x^{m}}{m^{4} + 6 \,{\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \,{\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \,{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

((b^3*m^3 + 3*b^3*m^2 + 3*b^3*m + b^3 + 2*(b^3*m + b^3)*n^2 + 3*(b^3*m^2 + 2*b^3*m + b^3)*n)*x*x^m*x^(3*n) + 3
*(a*b^2*m^3 + 3*a*b^2*m^2 + 3*a*b^2*m + a*b^2 + 3*(a*b^2*m + a*b^2)*n^2 + 4*(a*b^2*m^2 + 2*a*b^2*m + a*b^2)*n)
*x*x^m*x^(2*n) + 3*(a^2*b*m^3 + 3*a^2*b*m^2 + 3*a^2*b*m + a^2*b + 6*(a^2*b*m + a^2*b)*n^2 + 5*(a^2*b*m^2 + 2*a
^2*b*m + a^2*b)*n)*x*x^m*x^n + (a^3*m^3 + 6*a^3*n^3 + 3*a^3*m^2 + 3*a^3*m + a^3 + 11*(a^3*m + a^3)*n^2 + 6*(a^
3*m^2 + 2*a^3*m + a^3)*n)*x*x^m)/(m^4 + 6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^
2 + 3*m + 1)*n + 4*m + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a+b*x**n)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.14588, size = 840, normalized size = 11.2 \begin{align*} \frac{b^{3} m^{3} x x^{m} x^{3 \, n} + 3 \, b^{3} m^{2} n x x^{m} x^{3 \, n} + 2 \, b^{3} m n^{2} x x^{m} x^{3 \, n} + 3 \, a b^{2} m^{3} x x^{m} x^{2 \, n} + 12 \, a b^{2} m^{2} n x x^{m} x^{2 \, n} + 9 \, a b^{2} m n^{2} x x^{m} x^{2 \, n} + 3 \, a^{2} b m^{3} x x^{m} x^{n} + 15 \, a^{2} b m^{2} n x x^{m} x^{n} + 18 \, a^{2} b m n^{2} x x^{m} x^{n} + a^{3} m^{3} x x^{m} + 6 \, a^{3} m^{2} n x x^{m} + 11 \, a^{3} m n^{2} x x^{m} + 6 \, a^{3} n^{3} x x^{m} + 3 \, b^{3} m^{2} x x^{m} x^{3 \, n} + 6 \, b^{3} m n x x^{m} x^{3 \, n} + 2 \, b^{3} n^{2} x x^{m} x^{3 \, n} + 9 \, a b^{2} m^{2} x x^{m} x^{2 \, n} + 24 \, a b^{2} m n x x^{m} x^{2 \, n} + 9 \, a b^{2} n^{2} x x^{m} x^{2 \, n} + 9 \, a^{2} b m^{2} x x^{m} x^{n} + 30 \, a^{2} b m n x x^{m} x^{n} + 18 \, a^{2} b n^{2} x x^{m} x^{n} + 3 \, a^{3} m^{2} x x^{m} + 12 \, a^{3} m n x x^{m} + 11 \, a^{3} n^{2} x x^{m} + 3 \, b^{3} m x x^{m} x^{3 \, n} + 3 \, b^{3} n x x^{m} x^{3 \, n} + 9 \, a b^{2} m x x^{m} x^{2 \, n} + 12 \, a b^{2} n x x^{m} x^{2 \, n} + 9 \, a^{2} b m x x^{m} x^{n} + 15 \, a^{2} b n x x^{m} x^{n} + 3 \, a^{3} m x x^{m} + 6 \, a^{3} n x x^{m} + b^{3} x x^{m} x^{3 \, n} + 3 \, a b^{2} x x^{m} x^{2 \, n} + 3 \, a^{2} b x x^{m} x^{n} + a^{3} x x^{m}}{m^{4} + 6 \, m^{3} n + 11 \, m^{2} n^{2} + 6 \, m n^{3} + 4 \, m^{3} + 18 \, m^{2} n + 22 \, m n^{2} + 6 \, n^{3} + 6 \, m^{2} + 18 \, m n + 11 \, n^{2} + 4 \, m + 6 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^n)^3,x, algorithm="giac")

[Out]

(b^3*m^3*x*x^m*x^(3*n) + 3*b^3*m^2*n*x*x^m*x^(3*n) + 2*b^3*m*n^2*x*x^m*x^(3*n) + 3*a*b^2*m^3*x*x^m*x^(2*n) + 1
2*a*b^2*m^2*n*x*x^m*x^(2*n) + 9*a*b^2*m*n^2*x*x^m*x^(2*n) + 3*a^2*b*m^3*x*x^m*x^n + 15*a^2*b*m^2*n*x*x^m*x^n +
18*a^2*b*m*n^2*x*x^m*x^n + a^3*m^3*x*x^m + 6*a^3*m^2*n*x*x^m + 11*a^3*m*n^2*x*x^m + 6*a^3*n^3*x*x^m + 3*b^3*m
^2*x*x^m*x^(3*n) + 6*b^3*m*n*x*x^m*x^(3*n) + 2*b^3*n^2*x*x^m*x^(3*n) + 9*a*b^2*m^2*x*x^m*x^(2*n) + 24*a*b^2*m*
n*x*x^m*x^(2*n) + 9*a*b^2*n^2*x*x^m*x^(2*n) + 9*a^2*b*m^2*x*x^m*x^n + 30*a^2*b*m*n*x*x^m*x^n + 18*a^2*b*n^2*x*
x^m*x^n + 3*a^3*m^2*x*x^m + 12*a^3*m*n*x*x^m + 11*a^3*n^2*x*x^m + 3*b^3*m*x*x^m*x^(3*n) + 3*b^3*n*x*x^m*x^(3*n
) + 9*a*b^2*m*x*x^m*x^(2*n) + 12*a*b^2*n*x*x^m*x^(2*n) + 9*a^2*b*m*x*x^m*x^n + 15*a^2*b*n*x*x^m*x^n + 3*a^3*m*
x*x^m + 6*a^3*n*x*x^m + b^3*x*x^m*x^(3*n) + 3*a*b^2*x*x^m*x^(2*n) + 3*a^2*b*x*x^m*x^n + a^3*x*x^m)/(m^4 + 6*m^
3*n + 11*m^2*n^2 + 6*m*n^3 + 4*m^3 + 18*m^2*n + 22*m*n^2 + 6*n^3 + 6*m^2 + 18*m*n + 11*n^2 + 4*m + 6*n + 1)