3.611 \(\int \frac {a+b x^4}{x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac {b x^3}{3}-\frac {a}{x} \]

[Out]

-a/x+1/3*b*x^3

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Rubi [A]  time = 0.00, antiderivative size = 15, 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} \[ \frac {b x^3}{3}-\frac {a}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)/x^2,x]

[Out]

-(a/x) + (b*x^3)/3

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 \frac {a+b x^4}{x^2} \, dx &=\int \left (\frac {a}{x^2}+b x^2\right ) \, dx\\ &=-\frac {a}{x}+\frac {b x^3}{3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {b x^3}{3}-\frac {a}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)/x^2,x]

[Out]

-(a/x) + (b*x^3)/3

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fricas [A]  time = 0.65, size = 14, normalized size = 0.93 \[ \frac {b x^{4} - 3 \, a}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^2,x, algorithm="fricas")

[Out]

1/3*(b*x^4 - 3*a)/x

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giac [A]  time = 0.15, size = 13, normalized size = 0.87 \[ \frac {1}{3} \, b x^{3} - \frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^2,x, algorithm="giac")

[Out]

1/3*b*x^3 - a/x

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \frac {b \,x^{3}}{3}-\frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)/x^2,x)

[Out]

-a/x+1/3*b*x^3

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maxima [A]  time = 1.37, size = 13, normalized size = 0.87 \[ \frac {1}{3} \, b x^{3} - \frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^2,x, algorithm="maxima")

[Out]

1/3*b*x^3 - a/x

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mupad [B]  time = 0.02, size = 13, normalized size = 0.87 \[ \frac {b\,x^3}{3}-\frac {a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)/x^2,x)

[Out]

(b*x^3)/3 - a/x

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sympy [A]  time = 0.14, size = 8, normalized size = 0.53 \[ - \frac {a}{x} + \frac {b x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)/x**2,x)

[Out]

-a/x + b*x**3/3

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