3.3.12 \(\int \frac {a+b x^3}{x^3} \, dx\) [212]

Optimal. Leaf size=12 \[ -\frac {a}{2 x^2}+b x \]

[Out]

-1/2*a/x^2+b*x

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Rubi [A]
time = 0.00, antiderivative size = 12, 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} \begin {gather*} b x-\frac {a}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)/x^3,x]

[Out]

-1/2*a/x^2 + b*x

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {a}{2 x^2}+b x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)/x^3,x]

[Out]

-1/2*a/x^2 + b*x

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Maple [A]
time = 0.01, size = 11, normalized size = 0.92

method	result	size
default	\(-\frac {a}{2 x^{2}}+b x\)	\(11\)
risch	\(-\frac {a}{2 x^{2}}+b x\)	\(11\)
gosper	\(-\frac {-2 b \,x^{3}+a}{2 x^{2}}\)	\(14\)
norman	\(\frac {b \,x^{3}-\frac {a}{2}}{x^{2}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a/x^2+b*x

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Maxima [A]
time = 0.30, size = 10, normalized size = 0.83 \begin {gather*} b x - \frac {a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)/x^3,x, algorithm="maxima")

[Out]

b*x - 1/2*a/x^2

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Fricas [A]
time = 0.34, size = 15, normalized size = 1.25 \begin {gather*} \frac {2 \, b x^{3} - a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)/x^3,x, algorithm="fricas")

[Out]

1/2*(2*b*x^3 - a)/x^2

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Sympy [A]
time = 0.02, size = 8, normalized size = 0.67 \begin {gather*} - \frac {a}{2 x^{2}} + b x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)/x**3,x)

[Out]

-a/(2*x**2) + b*x

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Giac [A]
time = 2.24, size = 10, normalized size = 0.83 \begin {gather*} b x - \frac {a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)/x^3,x, algorithm="giac")

[Out]

b*x - 1/2*a/x^2

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Mupad [B]
time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} b\,x-\frac {a}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)/x^3,x)

[Out]

b*x - a/(2*x^2)

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