3.2740 \(\int x^m (a+b x^{2+2 m})^3 \, dx\)

Optimal. Leaf size=71 \[ \frac{a^2 b x^{3 (m+1)}}{m+1}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{5 (m+1)}}{5 (m+1)}+\frac{b^3 x^{7 (m+1)}}{7 (m+1)} \]

[Out]

(a^3*x^(1 + m))/(1 + m) + (a^2*b*x^(3*(1 + m)))/(1 + m) + (3*a*b^2*x^(5*(1 + m)))/(5*(1 + m)) + (b^3*x^(7*(1 +
m)))/(7*(1 + m))

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Rubi [A]  time = 0.0361272, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {270} \[ \frac{a^2 b x^{3 (m+1)}}{m+1}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{5 (m+1)}}{5 (m+1)}+\frac{b^3 x^{7 (m+1)}}{7 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^(2 + 2*m))^3,x]

[Out]

(a^3*x^(1 + m))/(1 + m) + (a^2*b*x^(3*(1 + m)))/(1 + m) + (3*a*b^2*x^(5*(1 + m)))/(5*(1 + m)) + (b^3*x^(7*(1 +
m)))/(7*(1 + m))

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^{2+2 m}\right )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{2+3 m}+3 a b^2 x^{4+5 m}+b^3 x^{6+7 m}\right ) \, dx\\ &=\frac{a^3 x^{1+m}}{1+m}+\frac{a^2 b x^{3 (1+m)}}{1+m}+\frac{3 a b^2 x^{5 (1+m)}}{5 (1+m)}+\frac{b^3 x^{7 (1+m)}}{7 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0450491, size = 57, normalized size = 0.8 \[ \frac{35 a^2 b x^{3 m+3}+35 a^3 x^{m+1}+21 a b^2 x^{5 m+5}+5 b^3 x^{7 m+7}}{35 m+35} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^(2 + 2*m))^3,x]

[Out]

(35*a^3*x^(1 + m) + 35*a^2*b*x^(3 + 3*m) + 21*a*b^2*x^(5 + 5*m) + 5*b^3*x^(7 + 7*m))/(35 + 35*m)

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Maple [A]  time = 0.016, size = 70, normalized size = 1. \begin{align*}{\frac{{b}^{3}{x}^{7} \left ({x}^{m} \right ) ^{7}}{7+7\,m}}+{\frac{3\,{x}^{5}a{b}^{2} \left ({x}^{m} \right ) ^{5}}{5+5\,m}}+{\frac{{a}^{2}b{x}^{3} \left ({x}^{m} \right ) ^{3}}{1+m}}+{\frac{{a}^{3}x{x}^{m}}{1+m}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a+b*x^(2+2*m))^3,x)

[Out]

1/7*b^3*x^7/(1+m)*(x^m)^7+3/5*x^5*a*b^2/(1+m)*(x^m)^5+a^2*b*x^3/(1+m)*(x^m)^3+a^3/(1+m)*x*x^m

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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^(2+2*m))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3323, size = 128, normalized size = 1.8 \begin{align*} \frac{5 \, b^{3} x^{7} x^{7 \, m} + 21 \, a b^{2} x^{5} x^{5 \, m} + 35 \, a^{2} b x^{3} x^{3 \, m} + 35 \, a^{3} x x^{m}}{35 \,{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(2+2*m))^3,x, algorithm="fricas")

[Out]

1/35*(5*b^3*x^7*x^(7*m) + 21*a*b^2*x^5*x^(5*m) + 35*a^2*b*x^3*x^(3*m) + 35*a^3*x*x^m)/(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**(2+2*m))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.13911, size = 78, normalized size = 1.1 \begin{align*} \frac{5 \, b^{3} x^{7} x^{7 \, m} + 21 \, a b^{2} x^{5} x^{5 \, m} + 35 \, a^{2} b x^{3} x^{3 \, m} + 35 \, a^{3} x x^{m}}{35 \,{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(2+2*m))^3,x, algorithm="giac")

[Out]

1/35*(5*b^3*x^7*x^(7*m) + 21*a*b^2*x^5*x^(5*m) + 35*a^2*b*x^3*x^(3*m) + 35*a^3*x*x^m)/(m + 1)