3.1.74 \(\int \frac {(a+b x)^3}{x^6} \, dx\)

Optimal. Leaf size=36 \[ \frac {b (a+b x)^4}{20 a^2 x^4}-\frac {(a+b x)^4}{5 a x^5} \]

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {45, 37} \begin {gather*} \frac {b (a+b x)^4}{20 a^2 x^4}-\frac {(a+b x)^4}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^6,x]

[Out]

-(a + b*x)^4/(5*a*x^5) + (b*(a + b*x)^4)/(20*a^2*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]

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {(a+b x)^3}{x^6} \, dx &=-\frac {(a+b x)^4}{5 a x^5}-\frac {b \int \frac {(a+b x)^3}{x^5} \, dx}{5 a}\\ &=-\frac {(a+b x)^4}{5 a x^5}+\frac {b (a+b x)^4}{20 a^2 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 1.14 \begin {gather*} -\frac {a^3}{5 x^5}-\frac {3 a^2 b}{4 x^4}-\frac {a b^2}{x^3}-\frac {b^3}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^6,x]

[Out]

-1/5*a^3/x^5 - (3*a^2*b)/(4*x^4) - (a*b^2)/x^3 - b^3/(2*x^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^3}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^3/x^6,x]

[Out]

IntegrateAlgebraic[(a + b*x)^3/x^6, x]

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fricas [A]  time = 1.48, size = 35, normalized size = 0.97 \begin {gather*} -\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^6,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*x^3 + 20*a*b^2*x^2 + 15*a^2*b*x + 4*a^3)/x^5

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giac [A]  time = 1.13, size = 35, normalized size = 0.97 \begin {gather*} -\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^6,x, algorithm="giac")

[Out]

-1/20*(10*b^3*x^3 + 20*a*b^2*x^2 + 15*a^2*b*x + 4*a^3)/x^5

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maple [A]  time = 0.00, size = 36, normalized size = 1.00 \begin {gather*} -\frac {b^{3}}{2 x^{2}}-\frac {a \,b^{2}}{x^{3}}-\frac {3 a^{2} b}{4 x^{4}}-\frac {a^{3}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^6,x)

[Out]

-1/5*a^3/x^5-a*b^2/x^3-3/4*a^2*b/x^4-1/2*b^3/x^2

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maxima [A]  time = 1.35, size = 35, normalized size = 0.97 \begin {gather*} -\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^6,x, algorithm="maxima")

[Out]

-1/20*(10*b^3*x^3 + 20*a*b^2*x^2 + 15*a^2*b*x + 4*a^3)/x^5

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mupad [B]  time = 0.03, size = 34, normalized size = 0.94 \begin {gather*} -\frac {\frac {a^3}{5}+\frac {3\,a^2\,b\,x}{4}+a\,b^2\,x^2+\frac {b^3\,x^3}{2}}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^6,x)

[Out]

-(a^3/5 + (b^3*x^3)/2 + a*b^2*x^2 + (3*a^2*b*x)/4)/x^5

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sympy [A]  time = 0.25, size = 37, normalized size = 1.03 \begin {gather*} \frac {- 4 a^{3} - 15 a^{2} b x - 20 a b^{2} x^{2} - 10 b^{3} x^{3}}{20 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**6,x)

[Out]

(-4*a**3 - 15*a**2*b*x - 20*a*b**2*x**2 - 10*b**3*x**3)/(20*x**5)

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