\(\int \frac {a+b x^4}{x^5} \, dx\) [614]

Optimal result

Integrand size = 11, antiderivative size = 13 \[ \int \frac {a+b x^4}{x^5} \, dx=-\frac {a}{4 x^4}+b \log (x) \]

[Out]

-1/4*a/x^4+b*ln(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \frac {a+b x^4}{x^5} \, dx=b \log (x)-\frac {a}{4 x^4} \]

[In]

Int[(a + b*x^4)/x^5,x]

[Out]

-1/4*a/x^4 + 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*} \text {integral}& = \int \left (\frac {a}{x^5}+\frac {b}{x}\right ) \, dx \\ & = -\frac {a}{4 x^4}+b \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^4}{x^5} \, dx=-\frac {a}{4 x^4}+b \log (x) \]

[In]

Integrate[(a + b*x^4)/x^5,x]

[Out]

-1/4*a/x^4 + b*Log[x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {a}{4 x^{4}}+b \ln \left (x \right )\)	\(12\)
norman	\(-\frac {a}{4 x^{4}}+b \ln \left (x \right )\)	\(12\)
risch	\(-\frac {a}{4 x^{4}}+b \ln \left (x \right )\)	\(12\)
parallelrisch	\(\frac {4 b \ln \left (x \right ) x^{4}-a}{4 x^{4}}\)	\(18\)

[In]

int((b*x^4+a)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*a/x^4+b*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {a+b x^4}{x^5} \, dx=\frac {4 \, b x^{4} \log \left (x\right ) - a}{4 \, x^{4}} \]

[In]

integrate((b*x^4+a)/x^5,x, algorithm="fricas")

[Out]

1/4*(4*b*x^4*log(x) - a)/x^4

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {a+b x^4}{x^5} \, dx=- \frac {a}{4 x^{4}} + b \log {\left (x \right )} \]

[In]

integrate((b*x**4+a)/x**5,x)

[Out]

-a/(4*x**4) + b*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x^4}{x^5} \, dx=\frac {1}{4} \, b \log \left (x^{4}\right ) - \frac {a}{4 \, x^{4}} \]

[In]

integrate((b*x^4+a)/x^5,x, algorithm="maxima")

[Out]

1/4*b*log(x^4) - 1/4*a/x^4

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \frac {a+b x^4}{x^5} \, dx=\frac {1}{4} \, b \log \left (x^{4}\right ) - \frac {b x^{4} + a}{4 \, x^{4}} \]

[In]

integrate((b*x^4+a)/x^5,x, algorithm="giac")

[Out]

1/4*b*log(x^4) - 1/4*(b*x^4 + a)/x^4

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {a+b x^4}{x^5} \, dx=b\,\ln \left (x\right )-\frac {a}{4\,x^4} \]

[In]

int((a + b*x^4)/x^5,x)

[Out]

b*log(x) - a/(4*x^4)