3.2.15 \(\int \frac {(a+b x)^7}{x^9} \, dx\)

Optimal. Leaf size=17 \[ -\frac {(a+b x)^8}{8 a x^8} \]

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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)^8}{8 a x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^9,x]

[Out]

-(a + b*x)^8/(8*a*x^8)

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)^7}{x^9} \, dx &=-\frac {(a+b x)^8}{8 a x^8}\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 87, normalized size = 5.12 \begin {gather*} -\frac {a^7}{8 x^8}-\frac {a^6 b}{x^7}-\frac {7 a^5 b^2}{2 x^6}-\frac {7 a^4 b^3}{x^5}-\frac {35 a^3 b^4}{4 x^4}-\frac {7 a^2 b^5}{x^3}-\frac {7 a b^6}{2 x^2}-\frac {b^7}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^9,x]

[Out]

-1/8*a^7/x^8 - (a^6*b)/x^7 - (7*a^5*b^2)/(2*x^6) - (7*a^4*b^3)/x^5 - (35*a^3*b^4)/(4*x^4) - (7*a^2*b^5)/x^3 -
(7*a*b^6)/(2*x^2) - b^7/x

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^7}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^7/x^9,x]

[Out]

IntegrateAlgebraic[(a + b*x)^7/x^9, x]

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fricas [B]  time = 1.41, size = 77, normalized size = 4.53 \begin {gather*} -\frac {8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^9,x, algorithm="fricas")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^3 + 28*a^5*b^2*x^2 + 8*a^6*b*x
+ a^7)/x^8

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giac [B]  time = 0.92, size = 77, normalized size = 4.53 \begin {gather*} -\frac {8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^9,x, algorithm="giac")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^3 + 28*a^5*b^2*x^2 + 8*a^6*b*x
+ a^7)/x^8

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maple [B]  time = 0.00, size = 80, normalized size = 4.71 \begin {gather*} -\frac {b^{7}}{x}-\frac {7 a \,b^{6}}{2 x^{2}}-\frac {7 a^{2} b^{5}}{x^{3}}-\frac {35 a^{3} b^{4}}{4 x^{4}}-\frac {7 a^{4} b^{3}}{x^{5}}-\frac {7 a^{5} b^{2}}{2 x^{6}}-\frac {a^{6} b}{x^{7}}-\frac {a^{7}}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^9,x)

[Out]

-7*a^4*b^3/x^5-1/8*a^7/x^8-7/2*a^5*b^2/x^6-35/4*a^3*b^4/x^4-7*a^2*b^5/x^3-b^7/x-a^6*b/x^7-7/2*a*b^6/x^2

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maxima [B]  time = 1.29, size = 77, normalized size = 4.53 \begin {gather*} -\frac {8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^9,x, algorithm="maxima")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^3 + 28*a^5*b^2*x^2 + 8*a^6*b*x
+ a^7)/x^8

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mupad [B]  time = 0.07, size = 77, normalized size = 4.53 \begin {gather*} -\frac {\frac {a^7}{8}+a^6\,b\,x+\frac {7\,a^5\,b^2\,x^2}{2}+7\,a^4\,b^3\,x^3+\frac {35\,a^3\,b^4\,x^4}{4}+7\,a^2\,b^5\,x^5+\frac {7\,a\,b^6\,x^6}{2}+b^7\,x^7}{x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/x^9,x)

[Out]

-(a^7/8 + b^7*x^7 + (7*a*b^6*x^6)/2 + (7*a^5*b^2*x^2)/2 + 7*a^4*b^3*x^3 + (35*a^3*b^4*x^4)/4 + 7*a^2*b^5*x^5 +
a^6*b*x)/x^8

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sympy [B]  time = 0.60, size = 83, normalized size = 4.88 \begin {gather*} \frac {- a^{7} - 8 a^{6} b x - 28 a^{5} b^{2} x^{2} - 56 a^{4} b^{3} x^{3} - 70 a^{3} b^{4} x^{4} - 56 a^{2} b^{5} x^{5} - 28 a b^{6} x^{6} - 8 b^{7} x^{7}}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**9,x)

[Out]

(-a**7 - 8*a**6*b*x - 28*a**5*b**2*x**2 - 56*a**4*b**3*x**3 - 70*a**3*b**4*x**4 - 56*a**2*b**5*x**5 - 28*a*b**
6*x**6 - 8*b**7*x**7)/(8*x**8)

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