3.3.13 \(\int \frac {a+b x^3}{x^4} \, dx\) [213]

Optimal. Leaf size=13 \[ -\frac {a}{3 x^3}+b \log (x) \]

[Out]

-1/3*a/x^3+b*ln(x)

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Rubi [A]
time = 0.00, antiderivative size = 13, 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 \log (x)-\frac {a}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)/x^4,x]

[Out]

-1/3*a/x^3 + b*Log[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^4} \, dx &=\int \left (\frac {a}{x^4}+\frac {b}{x}\right ) \, dx\\ &=-\frac {a}{3 x^3}+b \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {a}{3 x^3}+b \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)/x^4,x]

[Out]

-1/3*a/x^3 + b*Log[x]

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Maple [A]
time = 0.02, size = 12, normalized size = 0.92

method	result	size
default	\(-\frac {a}{3 x^{3}}+b \ln \left (x \right )\)	\(12\)
norman	\(-\frac {a}{3 x^{3}}+b \ln \left (x \right )\)	\(12\)
risch	\(-\frac {a}{3 x^{3}}+b \ln \left (x \right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a/x^3+b*ln(x)

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Maxima [A]
time = 0.31, size = 14, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, b \log \left (x^{3}\right ) - \frac {a}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*log(x^3) - 1/3*a/x^3

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Fricas [A]
time = 0.34, size = 17, normalized size = 1.31 \begin {gather*} \frac {3 \, b x^{3} \log \left (x\right ) - a}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b*x^3*log(x) - a)/x^3

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.77 \begin {gather*} - \frac {a}{3 x^{3}} + b \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(3*x**3) + b*log(x)

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Giac [A]
time = 2.33, size = 18, normalized size = 1.38 \begin {gather*} b \log \left ({\left | x \right |}\right ) - \frac {b x^{3} + a}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(abs(x)) - 1/3*(b*x^3 + a)/x^3

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Mupad [B]
time = 0.95, size = 11, normalized size = 0.85 \begin {gather*} b\,\ln \left (x\right )-\frac {a}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)/x^4,x)

[Out]

b*log(x) - a/(3*x^3)

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