3.1.7 \(\int \frac {a+b x^2}{x^2} \, dx\) [7]

Optimal. Leaf size=10 \[ -\frac {a}{x}+b x \]

[Out]

-a/x+b*x

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Rubi [A]
time = 0.00, antiderivative size = 10, 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}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^2,x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} -\frac {a}{x}+b x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^2,x]

[Out]

-(a/x) + b*x

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Maple [A]
time = 0.01, size = 11, normalized size = 1.10

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/x^2,x,method=_RETURNVERBOSE)

[Out]

-a/x+b*x

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Maxima [A]
time = 0.29, size = 10, normalized size = 1.00 \begin {gather*} b x - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x - a/x

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Fricas [A]
time = 1.37, size = 13, normalized size = 1.30 \begin {gather*} \frac {b x^{2} - a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^2 - a)/x

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Sympy [A]
time = 0.02, size = 5, normalized size = 0.50 \begin {gather*} - \frac {a}{x} + b x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x + b*x

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Giac [A]
time = 1.73, size = 10, normalized size = 1.00 \begin {gather*} b x - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x - a/x

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Mupad [B]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} b\,x-\frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/x^2,x)

[Out]

b*x - a/x

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