3.1547 \(\int (a+\frac {b}{x}) x^5 \, dx\)

Optimal. Leaf size=17 \[ \frac {a x^6}{6}+\frac {b x^5}{5} \]

[Out]

1/5*b*x^5+1/6*a*x^6

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \frac {a x^6}{6}+\frac {b x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)*x^5,x]

[Out]

(b*x^5)/5 + (a*x^6)/6

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 \left (a+\frac {b}{x}\right ) x^5 \, dx &=\int \left (b x^4+a x^5\right ) \, dx\\ &=\frac {b x^5}{5}+\frac {a x^6}{6}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {a x^6}{6}+\frac {b x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)*x^5,x]

[Out]

(b*x^5)/5 + (a*x^6)/6

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fricas [A]  time = 0.94, size = 13, normalized size = 0.76 \[ \frac {1}{6} \, a x^{6} + \frac {1}{5} \, b x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)*x^5,x, algorithm="fricas")

[Out]

1/6*a*x^6 + 1/5*b*x^5

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giac [A]  time = 0.19, size = 13, normalized size = 0.76 \[ \frac {1}{6} \, a x^{6} + \frac {1}{5} \, b x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)*x^5,x, algorithm="giac")

[Out]

1/6*a*x^6 + 1/5*b*x^5

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ \frac {1}{6} a \,x^{6}+\frac {1}{5} b \,x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)*x^5,x)

[Out]

1/5*b*x^5+1/6*a*x^6

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maxima [A]  time = 1.06, size = 13, normalized size = 0.76 \[ \frac {1}{6} \, a x^{6} + \frac {1}{5} \, b x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)*x^5,x, algorithm="maxima")

[Out]

1/6*a*x^6 + 1/5*b*x^5

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \[ \frac {x^5\,\left (6\,b+5\,a\,x\right )}{30} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b/x),x)

[Out]

(x^5*(6*b + 5*a*x))/30

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \[ \frac {a x^{6}}{6} + \frac {b x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)*x**5,x)

[Out]

a*x**6/6 + b*x**5/5

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