3.2532 \(\int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=63 \[ \frac{a^3 x^{4 n}}{4 n}+\frac{3 a^2 b x^{5 n}}{5 n}+\frac{a b^2 x^{6 n}}{2 n}+\frac{b^3 x^{7 n}}{7 n} \]

[Out]

(a^3*x^(4*n))/(4*n) + (3*a^2*b*x^(5*n))/(5*n) + (a*b^2*x^(6*n))/(2*n) + (b^3*x^(
7*n))/(7*n)

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Rubi [A]  time = 0.0774832, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a^3 x^{4 n}}{4 n}+\frac{3 a^2 b x^{5 n}}{5 n}+\frac{a b^2 x^{6 n}}{2 n}+\frac{b^3 x^{7 n}}{7 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 4*n)*(a + b*x^n)^3,x]

[Out]

(a^3*x^(4*n))/(4*n) + (3*a^2*b*x^(5*n))/(5*n) + (a*b^2*x^(6*n))/(2*n) + (b^3*x^(
7*n))/(7*n)

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Rubi in Sympy [A]  time = 11.8985, size = 51, normalized size = 0.81 \[ \frac{a^{3} x^{4 n}}{4 n} + \frac{3 a^{2} b x^{5 n}}{5 n} + \frac{a b^{2} x^{6 n}}{2 n} + \frac{b^{3} x^{7 n}}{7 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+4*n)*(a+b*x**n)**3,x)

[Out]

a**3*x**(4*n)/(4*n) + 3*a**2*b*x**(5*n)/(5*n) + a*b**2*x**(6*n)/(2*n) + b**3*x**
(7*n)/(7*n)

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Mathematica [A]  time = 0.0272536, size = 48, normalized size = 0.76 \[ \frac{x^{4 n} \left (35 a^3+84 a^2 b x^n+70 a b^2 x^{2 n}+20 b^3 x^{3 n}\right )}{140 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 4*n)*(a + b*x^n)^3,x]

[Out]

(x^(4*n)*(35*a^3 + 84*a^2*b*x^n + 70*a*b^2*x^(2*n) + 20*b^3*x^(3*n)))/(140*n)

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Maple [A]  time = 0.031, size = 56, normalized size = 0.9 \[{\frac{{b}^{3} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{a{b}^{2} \left ({x}^{n} \right ) ^{6}}{2\,n}}+{\frac{3\,{a}^{2}b \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{{a}^{3} \left ({x}^{n} \right ) ^{4}}{4\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+4*n)*(a+b*x^n)^3,x)

[Out]

1/7*b^3/n*(x^n)^7+1/2*a*b^2/n*(x^n)^6+3/5*a^2*b/n*(x^n)^5+1/4*a^3/n*(x^n)^4

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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^n + a)^3*x^(4*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224153, size = 65, normalized size = 1.03 \[ \frac{20 \, b^{3} x^{7 \, n} + 70 \, a b^{2} x^{6 \, n} + 84 \, a^{2} b x^{5 \, n} + 35 \, a^{3} x^{4 \, n}}{140 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(4*n - 1),x, algorithm="fricas")

[Out]

1/140*(20*b^3*x^(7*n) + 70*a*b^2*x^(6*n) + 84*a^2*b*x^(5*n) + 35*a^3*x^(4*n))/n

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Sympy [A]  time = 115.963, size = 60, normalized size = 0.95 \[ \begin{cases} \frac{a^{3} x^{4 n}}{4 n} + \frac{3 a^{2} b x^{5 n}}{5 n} + \frac{a b^{2} x^{6 n}}{2 n} + \frac{b^{3} x^{7 n}}{7 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+4*n)*(a+b*x**n)**3,x)

[Out]

Piecewise((a**3*x**(4*n)/(4*n) + 3*a**2*b*x**(5*n)/(5*n) + a*b**2*x**(6*n)/(2*n)
+ b**3*x**(7*n)/(7*n), Ne(n, 0)), ((a + b)**3*log(x), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{3} x^{4 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(4*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^3*x^(4*n - 1), x)