3.76 \(\int \frac {(a+b x)^3}{x^8} \, dx\)

Optimal. Leaf size=43 \[ -\frac {a^3}{7 x^7}-\frac {a^2 b}{2 x^6}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{4 x^4} \]

[Out]

-1/7*a^3/x^7-1/2*a^2*b/x^6-3/5*a*b^2/x^5-1/4*b^3/x^4

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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} \[ -\frac {a^2 b}{2 x^6}-\frac {a^3}{7 x^7}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^8,x]

[Out]

-a^3/(7*x^7) - (a^2*b)/(2*x^6) - (3*a*b^2)/(5*x^5) - b^3/(4*x^4)

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^8} \, dx &=\int \left (\frac {a^3}{x^8}+\frac {3 a^2 b}{x^7}+\frac {3 a b^2}{x^6}+\frac {b^3}{x^5}\right ) \, dx\\ &=-\frac {a^3}{7 x^7}-\frac {a^2 b}{2 x^6}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \[ -\frac {a^3}{7 x^7}-\frac {a^2 b}{2 x^6}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^8,x]

[Out]

-1/7*a^3/x^7 - (a^2*b)/(2*x^6) - (3*a*b^2)/(5*x^5) - b^3/(4*x^4)

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fricas [A]  time = 0.43, size = 35, normalized size = 0.81 \[ -\frac {35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7

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giac [A]  time = 1.12, size = 35, normalized size = 0.81 \[ -\frac {35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \[ -\frac {b^{3}}{4 x^{4}}-\frac {3 a \,b^{2}}{5 x^{5}}-\frac {a^{2} b}{2 x^{6}}-\frac {a^{3}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^8,x)

[Out]

-1/7*a^3/x^7-1/2*a^2*b/x^6-3/5*a*b^2/x^5-1/4*b^3/x^4

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maxima [A]  time = 1.35, size = 35, normalized size = 0.81 \[ -\frac {35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^8,x, algorithm="maxima")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7

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mupad [B]  time = 0.03, size = 35, normalized size = 0.81 \[ -\frac {\frac {a^3}{7}+\frac {a^2\,b\,x}{2}+\frac {3\,a\,b^2\,x^2}{5}+\frac {b^3\,x^3}{4}}{x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^8,x)

[Out]

-(a^3/7 + (b^3*x^3)/4 + (3*a*b^2*x^2)/5 + (a^2*b*x)/2)/x^7

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sympy [A]  time = 0.29, size = 37, normalized size = 0.86 \[ \frac {- 20 a^{3} - 70 a^{2} b x - 84 a b^{2} x^{2} - 35 b^{3} x^{3}}{140 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**8,x)

[Out]

(-20*a**3 - 70*a**2*b*x - 84*a*b**2*x**2 - 35*b**3*x**3)/(140*x**7)

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