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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.06, size = 57, normalized size = 0.80 \[ \frac {35 a^3 x^{m+1}+35 a^2 b x^{3 m+3}+21 a b^2 x^{5 m+5}+5 b^3 x^{7 m+7}}{35 m+35} \]

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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fricas [A]  time = 0.66, size = 58, normalized size = 0.82 \[ \frac {5 \, b^{3} x^{7} x^{7 \, m} + 21 \, a b^{2} x^{5} x^{5 \, m} + 35 \, a^{2} b x^{3} x^{3 \, m} + 35 \, a^{3} x x^{m}}{35 \, {\left (m + 1\right )}} \]

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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giac [A]  time = 0.24, size = 58, normalized size = 0.82 \[ \frac {5 \, b^{3} x^{7} x^{7 \, m} + 21 \, a b^{2} x^{5} x^{5 \, m} + 35 \, a^{2} b x^{3} x^{3 \, m} + 35 \, a^{3} x x^{m}}{35 \, {\left (m + 1\right )}} \]

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)

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maple [A]  time = 0.02, size = 70, normalized size = 0.99 \[ \frac {b^{3} x^{7} x^{7 m}}{7 m +7}+\frac {3 a \,b^{2} x^{5} x^{5 m}}{5 \left (m +1\right )}+\frac {a^{2} b \,x^{3} x^{3 m}}{m +1}+\frac {a^{3} x \,x^{m}}{m +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 67, normalized size = 0.94 \[ \frac {b^{3} x^{7 \, m + 7}}{7 \, {\left (m + 1\right )}} + \frac {3 \, a b^{2} x^{5 \, m + 5}}{5 \, {\left (m + 1\right )}} + \frac {a^{2} b x^{3 \, m + 3}}{m + 1} + \frac {a^{3} x^{m + 1}}{m + 1} \]

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]

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

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mupad [B]  time = 1.45, size = 72, normalized size = 1.01 \[ \frac {a^3\,x\,x^m}{m+1}+\frac {b^3\,x^{7\,m}\,x^7}{7\,m+7}+\frac {a^2\,b\,x^{3\,m}\,x^3}{m+1}+\frac {3\,a\,b^2\,x^{5\,m}\,x^5}{5\,m+5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^(2*m + 2))^3,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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