3.2519 \(\int x^{-1-4 n} \left (a+b x^n\right ) \, dx\)

Optimal. Leaf size=27 \[ -\frac{a x^{-4 n}}{4 n}-\frac{b x^{-3 n}}{3 n} \]

[Out]

-a/(4*n*x^(4*n)) - b/(3*n*x^(3*n))

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Rubi [A]  time = 0.0233111, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{a x^{-4 n}}{4 n}-\frac{b x^{-3 n}}{3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 4*n)*(a + b*x^n),x]

[Out]

-a/(4*n*x^(4*n)) - b/(3*n*x^(3*n))

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Rubi in Sympy [A]  time = 3.97649, size = 20, normalized size = 0.74 \[ - \frac{a x^{- 4 n}}{4 n} - \frac{b x^{- 3 n}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*x**(-4*n)/(4*n) - b*x**(-3*n)/(3*n)

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Mathematica [A]  time = 0.0115411, size = 22, normalized size = 0.81 \[ -\frac{x^{-4 n} \left (3 a+4 b x^n\right )}{12 n} \]

Antiderivative was successfully verified.

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

[Out]

-(3*a + 4*b*x^n)/(12*n*x^(4*n))

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Maple [A]  time = 0.026, size = 27, normalized size = 1. \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ( -{\frac{a}{4\,n}}-{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{3\,n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-4*n)*(a+b*x^n),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222717, size = 30, normalized size = 1.11 \[ -\frac{4 \, b x^{n} + 3 \, a}{12 \, n x^{4 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(4*b*x^n + 3*a)/(n*x^(4*n))

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Sympy [A]  time = 15.8574, size = 27, normalized size = 1. \[ \begin{cases} - \frac{a x^{- 4 n}}{4 n} - \frac{b x^{- 3 n}}{3 n} & \text{for}\: n \neq 0 \\\left (a + b\right ) \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a*x**(-4*n)/(4*n) - b*x**(-3*n)/(3*n), Ne(n, 0)), ((a + b)*log(x), T
rue))

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GIAC/XCAS [A]  time = 0.215324, size = 31, normalized size = 1.15 \[ -\frac{{\left (4 \, b e^{\left (n{\rm ln}\left (x\right )\right )} + 3 \, a\right )} e^{\left (-4 \, n{\rm ln}\left (x\right )\right )}}{12 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(4*b*e^(n*ln(x)) + 3*a)*e^(-4*n*ln(x))/n