3.26.26 \(\int x^{-1-4 n} (a+b x^n) \, dx\) [2526]

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

[Out]

-1/4*a/n/(x^(4*n))-1/3*b/n/(x^(3*n))

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \begin {gather*} -\frac {a x^{-4 n}}{4 n}-\frac {b x^{-3 n}}{3 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 4*n)*(a + b*x^n),x]

[Out]

-1/4*a/(n*x^(4*n)) - b/(3*n*x^(3*n))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int x^{-1-4 n} \left (a+b x^n\right ) \, dx &=\int \left (a x^{-1-4 n}+b x^{-1-3 n}\right ) \, dx\\ &=-\frac {a x^{-4 n}}{4 n}-\frac {b x^{-3 n}}{3 n}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {x^{-4 n} \left (-3 a-4 b x^n\right )}{12 n} \end {gather*}

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))

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 24, normalized size = 0.89

method	result	size
risch	\(-\frac {b \,x^{-3 n}}{3 n}-\frac {a \,x^{-4 n}}{4 n}\)	\(24\)
norman	\(\left (-\frac {a}{4 n}-\frac {b \,{\mathrm e}^{n \ln \left (x \right )}}{3 n}\right ) {\mathrm e}^{-4 n \ln \left (x \right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-4*n)*(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

-1/3*b/n/(x^n)^3-1/4*a/n/(x^n)^4

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 27, normalized size = 1.00 \begin {gather*} -\frac {a}{4 \, n x^{4 \, n}} - \frac {b}{3 \, n x^{3 \, n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a/(n*x^(4*n)) - 1/3*b/(n*x^(3*n))

________________________________________________________________________________________

Fricas [A]
time = 0.49, size = 22, normalized size = 0.81 \begin {gather*} -\frac {4 \, b x^{n} + 3 \, a}{12 \, n x^{4 \, n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*b*x^n + 3*a)/(n*x^(4*n))

________________________________________________________________________________________

Sympy [A]
time = 0.62, size = 27, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 2.28, size = 22, normalized size = 0.81 \begin {gather*} -\frac {4 \, b x^{n} + 3 \, a}{12 \, n x^{4 \, n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*b*x^n + 3*a)/(n*x^(4*n))

________________________________________________________________________________________

Mupad [B]
time = 1.23, size = 22, normalized size = 0.81 \begin {gather*} -\frac {3\,a+4\,b\,x^n}{12\,n\,x^{4\,n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)/x^(4*n + 1),x)

[Out]

-(3*a + 4*b*x^n)/(12*n*x^(4*n))

________________________________________________________________________________________