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

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

[Out]

-(a + b*x)^8/(8*a*x^8)

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Rubi [A]  time = 0.0016742, antiderivative size = 17, normalized size of antiderivative = 1., 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} \[ -\frac{(a+b x)^8}{8 a x^8} \]

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*}

Mathematica [B]  time = 0.0069037, size = 87, normalized size = 5.12 \[ -\frac{7 a^5 b^2}{2 x^6}-\frac{7 a^4 b^3}{x^5}-\frac{35 a^3 b^4}{4 x^4}-\frac{7 a^2 b^5}{x^3}-\frac{a^6 b}{x^7}-\frac{a^7}{8 x^8}-\frac{7 a b^6}{2 x^2}-\frac{b^7}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^9,x]

[Out]

-a^7/(8*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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Maple [B]  time = 0.007, size = 80, normalized size = 4.7 \begin{align*} -7\,{\frac{{a}^{2}{b}^{5}}{{x}^{3}}}-7\,{\frac{{a}^{4}{b}^{3}}{{x}^{5}}}-{\frac{35\,{a}^{3}{b}^{4}}{4\,{x}^{4}}}-{\frac{7\,{a}^{5}{b}^{2}}{2\,{x}^{6}}}-{\frac{{a}^{7}}{8\,{x}^{8}}}-{\frac{7\,a{b}^{6}}{2\,{x}^{2}}}-{\frac{{a}^{6}b}{{x}^{7}}}-{\frac{{b}^{7}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01849, size = 104, normalized size = 6.12 \begin{align*} -\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{align*}

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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Fricas [B]  time = 1.74426, size = 166, normalized size = 9.76 \begin{align*} -\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{align*}

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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Sympy [B]  time = 0.851885, size = 83, normalized size = 4.88 \begin{align*} - \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{align*}

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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Giac [B]  time = 1.21237, size = 104, normalized size = 6.12 \begin{align*} -\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{align*}

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