3.3.48 \(\int \frac {(a+b x)^3}{x^{10}} \, dx\)

Optimal. Leaf size=43 \[ -\frac {a^3}{9 x^9}-\frac {3 a^2 b}{8 x^8}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{6 x^6} \]

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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}{8 x^8}-\frac {a^3}{9 x^9}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^10,x]

[Out]

-a^3/(9*x^9) - (3*a^2*b)/(8*x^8) - (3*a*b^2)/(7*x^7) - b^3/(6*x^6)

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^{10}} \, dx &=\int \left (\frac {a^3}{x^{10}}+\frac {3 a^2 b}{x^9}+\frac {3 a b^2}{x^8}+\frac {b^3}{x^7}\right ) \, dx\\ &=-\frac {a^3}{9 x^9}-\frac {3 a^2 b}{8 x^8}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{6 x^6}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} -\frac {a^3}{9 x^9}-\frac {3 a^2 b}{8 x^8}-\frac {3 a b^2}{7 x^7}-\frac {b^3}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^10,x]

[Out]

-1/9*a^3/x^9 - (3*a^2*b)/(8*x^8) - (3*a*b^2)/(7*x^7) - b^3/(6*x^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^3}{x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^3/x^10,x]

[Out]

IntegrateAlgebraic[(a + b*x)^3/x^10, x]

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fricas [A]  time = 0.99, size = 35, normalized size = 0.81 \begin {gather*} -\frac {84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^10,x, algorithm="fricas")

[Out]

-1/504*(84*b^3*x^3 + 216*a*b^2*x^2 + 189*a^2*b*x + 56*a^3)/x^9

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giac [A]  time = 1.09, size = 35, normalized size = 0.81 \begin {gather*} -\frac {84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^10,x, algorithm="giac")

[Out]

-1/504*(84*b^3*x^3 + 216*a*b^2*x^2 + 189*a^2*b*x + 56*a^3)/x^9

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \begin {gather*} -\frac {b^{3}}{6 x^{6}}-\frac {3 a \,b^{2}}{7 x^{7}}-\frac {3 a^{2} b}{8 x^{8}}-\frac {a^{3}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^10,x)

[Out]

-1/9*a^3/x^9-3/8*a^2*b/x^8-3/7*a*b^2/x^7-1/6*b^3/x^6

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maxima [A]  time = 1.35, size = 35, normalized size = 0.81 \begin {gather*} -\frac {84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^10,x, algorithm="maxima")

[Out]

-1/504*(84*b^3*x^3 + 216*a*b^2*x^2 + 189*a^2*b*x + 56*a^3)/x^9

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mupad [B]  time = 0.03, size = 35, normalized size = 0.81 \begin {gather*} -\frac {\frac {a^3}{9}+\frac {3\,a^2\,b\,x}{8}+\frac {3\,a\,b^2\,x^2}{7}+\frac {b^3\,x^3}{6}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3/x^10,x)

[Out]

-(a^3/9 + (b^3*x^3)/6 + (3*a*b^2*x^2)/7 + (3*a^2*b*x)/8)/x^9

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sympy [A]  time = 0.38, size = 37, normalized size = 0.86 \begin {gather*} \frac {- 56 a^{3} - 189 a^{2} b x - 216 a b^{2} x^{2} - 84 b^{3} x^{3}}{504 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**10,x)

[Out]

(-56*a**3 - 189*a**2*b*x - 216*a*b**2*x**2 - 84*b**3*x**3)/(504*x**9)

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