3.2733 \(\int x^m \left (a+b x^{2+2 m}\right )^3 \, dx\)

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*(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 = 0.0882571, 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 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{a^2 b x^{3 (m+1)}}{m+1}+\frac{3 a b^2 x^{5 (m+1)}}{5 (m+1)}+\frac{b^3 x^{7 (m+1)}}{7 (m+1)} \]

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))

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Rubi in Sympy [A]  time = 13.1738, size = 60, normalized size = 0.85 \[ \frac{a^{3} x^{m + 1}}{m + 1} + \frac{a^{2} b x^{3 m + 3}}{m + 1} + \frac{3 a b^{2} x^{5 m + 5}}{5 \left (m + 1\right )} + \frac{b^{3} x^{7 m + 7}}{7 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(a+b*x**(2+2*m))**3,x)

[Out]

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

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Mathematica [A]  time = 0.0544144, size = 57, normalized size = 0.8 \[ \frac{35 a^3 x^{m+1}+35 a^2 b x^{3 m+3}+21 a b^2 x^{5 m+5}+5 b^3 x^{7 m+7}}{35 m+35} \]

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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Maple [A]  time = 0.031, size = 70, normalized size = 1. \[{\frac{{b}^{3}{x}^{7} \left ({x}^{m} \right ) ^{7}}{7+7\,m}}+{\frac{3\,a{b}^{2}{x}^{5} \left ({x}^{m} \right ) ^{5}}{5+5\,m}}+{\frac{{a}^{2}b{x}^{3} \left ({x}^{m} \right ) ^{3}}{1+m}}+{\frac{{a}^{3}x{x}^{m}}{1+m}} \]

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*a*b^2*x^5/(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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(2*m + 2) + a)^3*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23596, size = 78, normalized size = 1.1 \[ \frac{5 \, b^{3} x^{7} x^{7 \, m} + 21 \, a b^{2} x^{5} x^{5 \, m} + 35 \, a^{2} b x^{3} x^{3 \, m} + 35 \, a^{3} x x^{m}}{35 \,{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(2*m + 2) + a)^3*x^m,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. \[ \text{Timed out} \]

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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GIAC/XCAS [A]  time = 0.225, size = 85, normalized size = 1.2 \[ \frac{5 \, b^{3} x^{7} e^{\left (7 \, m{\rm ln}\left (x\right )\right )} + 21 \, a b^{2} x^{5} e^{\left (5 \, m{\rm ln}\left (x\right )\right )} + 35 \, a^{2} b x^{3} e^{\left (3 \, m{\rm ln}\left (x\right )\right )} + 35 \, a^{3} x e^{\left (m{\rm ln}\left (x\right )\right )}}{35 \,{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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