3.51 \(\int \frac {a+b x}{x^5} \, dx\)

Optimal. Leaf size=17 \[ -\frac {a}{4 x^4}-\frac {b}{3 x^3} \]

[Out]

-1/4*a/x^4-1/3*b/x^3

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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 = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \[ -\frac {a}{4 x^4}-\frac {b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^5,x]

[Out]

-a/(4*x^4) - b/(3*x^3)

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^5} \, dx &=\int \left (\frac {a}{x^5}+\frac {b}{x^4}\right ) \, dx\\ &=-\frac {a}{4 x^4}-\frac {b}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {a}{4 x^4}-\frac {b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^5,x]

[Out]

-1/4*a/x^4 - b/(3*x^3)

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fricas [A]  time = 0.38, size = 13, normalized size = 0.76 \[ -\frac {4 \, b x + 3 \, a}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*b*x + 3*a)/x^4

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giac [A]  time = 1.68, size = 13, normalized size = 0.76 \[ -\frac {4 \, b x + 3 \, a}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*b*x + 3*a)/x^4

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maple [A]  time = 0.01, size = 14, normalized size = 0.82 \[ -\frac {b}{3 x^{3}}-\frac {a}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^5,x)

[Out]

-1/4*a/x^4-1/3*b/x^3

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maxima [A]  time = 1.05, size = 13, normalized size = 0.76 \[ -\frac {4 \, b x + 3 \, a}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*b*x + 3*a)/x^4

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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \[ -\frac {3\,a+4\,b\,x}{12\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^5,x)

[Out]

-(3*a + 4*b*x)/(12*x^4)

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sympy [A]  time = 0.16, size = 14, normalized size = 0.82 \[ \frac {- 3 a - 4 b x}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a - 4*b*x)/(12*x**4)

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