3.1.75 \(\int \frac {(a+b x)^3}{x^7} \, dx\)

Optimal. Leaf size=43 \[ -\frac {a^3}{6 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{3 x^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} -\frac {3 a^2 b}{5 x^5}-\frac {a^3}{6 x^6}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^7,x]

[Out]

-a^3/(6*x^6) - (3*a^2*b)/(5*x^5) - (3*a*b^2)/(4*x^4) - b^3/(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)^3}{x^7} \, dx &=\int \left (\frac {a^3}{x^7}+\frac {3 a^2 b}{x^6}+\frac {3 a b^2}{x^5}+\frac {b^3}{x^4}\right ) \, dx\\ &=-\frac {a^3}{6 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} -\frac {a^3}{6 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^7,x]

[Out]

-1/6*a^3/x^6 - (3*a^2*b)/(5*x^5) - (3*a*b^2)/(4*x^4) - b^3/(3*x^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^3}{x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^3/x^7,x]

[Out]

IntegrateAlgebraic[(a + b*x)^3/x^7, x]

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fricas [A]  time = 0.97, size = 35, normalized size = 0.81 \begin {gather*} -\frac {20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^7,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*x^3 + 45*a*b^2*x^2 + 36*a^2*b*x + 10*a^3)/x^6

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giac [A]  time = 1.15, size = 35, normalized size = 0.81 \begin {gather*} -\frac {20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^7,x, algorithm="giac")

[Out]

-1/60*(20*b^3*x^3 + 45*a*b^2*x^2 + 36*a^2*b*x + 10*a^3)/x^6

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maple [A]  time = 0.01, size = 36, normalized size = 0.84 \begin {gather*} -\frac {b^{3}}{3 x^{3}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {3 a^{2} b}{5 x^{5}}-\frac {a^{3}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^7,x)

[Out]

-1/6*a^3/x^6-3/5*a^2*b/x^5-3/4*a*b^2/x^4-1/3*b^3/x^3

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maxima [A]  time = 1.35, size = 35, normalized size = 0.81 \begin {gather*} -\frac {20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^7,x, algorithm="maxima")

[Out]

-1/60*(20*b^3*x^3 + 45*a*b^2*x^2 + 36*a^2*b*x + 10*a^3)/x^6

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mupad [B]  time = 0.03, size = 35, normalized size = 0.81 \begin {gather*} -\frac {\frac {a^3}{6}+\frac {3\,a^2\,b\,x}{5}+\frac {3\,a\,b^2\,x^2}{4}+\frac {b^3\,x^3}{3}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^7,x)

[Out]

-(a^3/6 + (b^3*x^3)/3 + (3*a*b^2*x^2)/4 + (3*a^2*b*x)/5)/x^6

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sympy [A]  time = 0.34, size = 37, normalized size = 0.86 \begin {gather*} \frac {- 10 a^{3} - 36 a^{2} b x - 45 a b^{2} x^{2} - 20 b^{3} x^{3}}{60 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**7,x)

[Out]

(-10*a**3 - 36*a**2*b*x - 45*a*b**2*x**2 - 20*b**3*x**3)/(60*x**6)

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