3.1.73 \(\int \frac {(a+b x)^3}{x^5} \, dx\)

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

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37} \begin {gather*} -\frac {(a+b x)^4}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^5,x]

[Out]

-(a + b*x)^4/(4*a*x^4)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

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Mathematica [B]  time = 0.00, size = 39, normalized size = 2.29 \begin {gather*} -\frac {a^3}{4 x^4}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{2 x^2}-\frac {b^3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^5,x]

[Out]

-1/4*a^3/x^4 - (a^2*b)/x^3 - (3*a*b^2)/(2*x^2) - b^3/x

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^3}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^3/x^5,x]

[Out]

IntegrateAlgebraic[(a + b*x)^3/x^5, x]

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fricas [B]  time = 1.30, size = 33, normalized size = 1.94 \begin {gather*} -\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^5,x, algorithm="fricas")

[Out]

-1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)/x^4

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giac [B]  time = 1.29, size = 33, normalized size = 1.94 \begin {gather*} -\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^5,x, algorithm="giac")

[Out]

-1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)/x^4

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maple [B]  time = 0.01, size = 36, normalized size = 2.12 \begin {gather*} -\frac {b^{3}}{x}-\frac {3 a \,b^{2}}{2 x^{2}}-\frac {a^{2} b}{x^{3}}-\frac {a^{3}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^5,x)

[Out]

-a^2*b/x^3-1/4*a^3/x^4-b^3/x-3/2*a*b^2/x^2

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maxima [B]  time = 1.33, size = 33, normalized size = 1.94 \begin {gather*} -\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^5,x, algorithm="maxima")

[Out]

-1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)/x^4

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mupad [B]  time = 0.03, size = 33, normalized size = 1.94 \begin {gather*} -\frac {\frac {a^3}{4}+a^2\,b\,x+\frac {3\,a\,b^2\,x^2}{2}+b^3\,x^3}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^5,x)

[Out]

-(a^3/4 + b^3*x^3 + (3*a*b^2*x^2)/2 + a^2*b*x)/x^4

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sympy [B]  time = 0.26, size = 36, normalized size = 2.12 \begin {gather*} \frac {- a^{3} - 4 a^{2} b x - 6 a b^{2} x^{2} - 4 b^{3} x^{3}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**5,x)

[Out]

(-a**3 - 4*a**2*b*x - 6*a*b**2*x**2 - 4*b**3*x**3)/(4*x**4)

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