3.2534 \(\int x^{-1+2 n} \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (a+b x^n\right )^5}{5 b^2 n}-\frac{a \left (a+b x^n\right )^4}{4 b^2 n} \]

[Out]

-(a*(a + b*x^n)^4)/(4*b^2*n) + (a + b*x^n)^5/(5*b^2*n)

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Rubi [A]  time = 0.0577, antiderivative size = 40, 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 \[ \frac{\left (a+b x^n\right )^5}{5 b^2 n}-\frac{a \left (a+b x^n\right )^4}{4 b^2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)*(a + b*x^n)^3,x]

[Out]

-(a*(a + b*x^n)^4)/(4*b^2*n) + (a + b*x^n)^5/(5*b^2*n)

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Rubi in Sympy [A]  time = 9.04643, size = 31, normalized size = 0.78 \[ - \frac{a \left (a + b x^{n}\right )^{4}}{4 b^{2} n} + \frac{\left (a + b x^{n}\right )^{5}}{5 b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)*(a+b*x**n)**3,x)

[Out]

-a*(a + b*x**n)**4/(4*b**2*n) + (a + b*x**n)**5/(5*b**2*n)

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Mathematica [A]  time = 0.023384, size = 48, normalized size = 1.2 \[ \frac{x^{2 n} \left (10 a^3+20 a^2 b x^n+15 a b^2 x^{2 n}+4 b^3 x^{3 n}\right )}{20 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 2*n)*(a + b*x^n)^3,x]

[Out]

(x^(2*n)*(10*a^3 + 20*a^2*b*x^n + 15*a*b^2*x^(2*n) + 4*b^3*x^(3*n)))/(20*n)

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Maple [A]  time = 0.026, size = 63, normalized size = 1.6 \[{\frac{{a}^{2}b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+{\frac{{a}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{5\,n}}+{\frac{3\,a{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)*(a+b*x^n)^3,x)

[Out]

a^2*b/n*exp(n*ln(x))^3+1/2*a^3/n*exp(n*ln(x))^2+1/5*b^3/n*exp(n*ln(x))^5+3/4*a*b
^2/n*exp(n*ln(x))^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^(2*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224449, size = 65, normalized size = 1.62 \[ \frac{4 \, b^{3} x^{5 \, n} + 15 \, a b^{2} x^{4 \, n} + 20 \, a^{2} b x^{3 \, n} + 10 \, a^{3} x^{2 \, n}}{20 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(2*n - 1),x, algorithm="fricas")

[Out]

1/20*(4*b^3*x^(5*n) + 15*a*b^2*x^(4*n) + 20*a^2*b*x^(3*n) + 10*a^3*x^(2*n))/n

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Sympy [A]  time = 116.127, size = 58, normalized size = 1.45 \[ \begin{cases} \frac{a^{3} x^{2 n}}{2 n} + \frac{a^{2} b x^{3 n}}{n} + \frac{3 a b^{2} x^{4 n}}{4 n} + \frac{b^{3} x^{5 n}}{5 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+2*n)*(a+b*x**n)**3,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{3} x^{2 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(2*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^3*x^(2*n - 1), x)