3.50 \(\int \frac{a+b x}{x^4} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{3 x^3}-\frac{b}{2 x^2} \]

[Out]

-a/(3*x^3) - b/(2*x^2)

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Rubi [A]  time = 0.0047191, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ -\frac{a}{3 x^3}-\frac{b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^4,x]

[Out]

-a/(3*x^3) - b/(2*x^2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b x}{x^4} \, dx &=\int \left (\frac{a}{x^4}+\frac{b}{x^3}\right ) \, dx\\ &=-\frac{a}{3 x^3}-\frac{b}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0020062, size = 17, normalized size = 1. \[ -\frac{a}{3 x^3}-\frac{b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^4,x]

[Out]

-a/(3*x^3) - b/(2*x^2)

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Maple [A]  time = 0.005, size = 14, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^4,x)

[Out]

-1/3*a/x^3-1/2/x^2*b

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Maxima [A]  time = 1.0356, size = 18, normalized size = 1.06 \begin{align*} -\frac{3 \, b x + 2 \, a}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*b*x + 2*a)/x^3

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Fricas [A]  time = 1.62695, size = 32, normalized size = 1.88 \begin{align*} -\frac{3 \, b x + 2 \, a}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*b*x + 2*a)/x^3

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Sympy [A]  time = 0.424286, size = 14, normalized size = 0.82 \begin{align*} - \frac{2 a + 3 b x}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a + 3*b*x)/(6*x**3)

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Giac [A]  time = 1.17997, size = 18, normalized size = 1.06 \begin{align*} -\frac{3 \, b x + 2 \, a}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*b*x + 2*a)/x^3