3.7.12 \(\int \frac {a+b x^4}{x^3} \, dx\) [612]

Optimal. Leaf size=17 \[ -\frac {a}{2 x^2}+\frac {b x^2}{2} \]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)/x^3,x]

[Out]

-1/2*a/x^2 + (b*x^2)/2

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {a}{2 x^2}+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)/x^3,x]

[Out]

-1/2*a/x^2 + (b*x^2)/2

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Maple [A]
time = 0.01, size = 14, normalized size = 0.82

method	result	size
gosper	\(-\frac {-b \,x^{4}+a}{2 x^{2}}\)	\(14\)
default	\(-\frac {a}{2 x^{2}}+\frac {b \,x^{2}}{2}\)	\(14\)
risch	\(-\frac {a}{2 x^{2}}+\frac {b \,x^{2}}{2}\)	\(14\)
norman	\(\frac {\frac {b \,x^{4}}{2}-\frac {a}{2}}{x^{2}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, b x^{2} - \frac {a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^3,x, algorithm="maxima")

[Out]

1/2*b*x^2 - 1/2*a/x^2

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Fricas [A]
time = 0.36, size = 14, normalized size = 0.82 \begin {gather*} \frac {b x^{4} - a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^3,x, algorithm="fricas")

[Out]

1/2*(b*x^4 - a)/x^2

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Sympy [A]
time = 0.02, size = 12, normalized size = 0.71 \begin {gather*} - \frac {a}{2 x^{2}} + \frac {b x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)/x**3,x)

[Out]

-a/(2*x**2) + b*x**2/2

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Giac [A]
time = 1.34, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, b x^{2} - \frac {a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)/x^3,x, algorithm="giac")

[Out]

1/2*b*x^2 - 1/2*a/x^2

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} -\frac {a-b\,x^4}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)/x^3,x)

[Out]

-(a - b*x^4)/(2*x^2)

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