3.1548 \(\int (a+\frac {b}{x}) x^4 \, dx\)

Optimal. Leaf size=17 \[ \frac {a x^5}{5}+\frac {b x^4}{4} \]

[Out]

1/4*b*x^4+1/5*a*x^5

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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} \[ \frac {a x^5}{5}+\frac {b x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)*x^4,x]

[Out]

(b*x^4)/4 + (a*x^5)/5

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^4 \, dx &=\int \left (b x^3+a x^4\right ) \, dx\\ &=\frac {b x^4}{4}+\frac {a x^5}{5}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {a x^5}{5}+\frac {b x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)*x^4,x]

[Out]

(b*x^4)/4 + (a*x^5)/5

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fricas [A]  time = 0.89, size = 13, normalized size = 0.76 \[ \frac {1}{5} \, a x^{5} + \frac {1}{4} \, b x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)*x^4,x, algorithm="fricas")

[Out]

1/5*a*x^5 + 1/4*b*x^4

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giac [A]  time = 0.15, size = 13, normalized size = 0.76 \[ \frac {1}{5} \, a x^{5} + \frac {1}{4} \, b x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)*x^4,x, algorithm="giac")

[Out]

1/5*a*x^5 + 1/4*b*x^4

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ \frac {1}{5} a \,x^{5}+\frac {1}{4} b \,x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)*x^4,x)

[Out]

1/4*b*x^4+1/5*a*x^5

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maxima [A]  time = 1.02, size = 13, normalized size = 0.76 \[ \frac {1}{5} \, a x^{5} + \frac {1}{4} \, b x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)*x^4,x, algorithm="maxima")

[Out]

1/5*a*x^5 + 1/4*b*x^4

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \[ \frac {x^4\,\left (5\,b+4\,a\,x\right )}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b/x),x)

[Out]

(x^4*(5*b + 4*a*x))/20

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \[ \frac {a x^{5}}{5} + \frac {b x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)*x**4,x)

[Out]

a*x**5/5 + b*x**4/4

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