3.243 \(\int \frac{(a+b x)^8}{x^{10}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{(a+b x)^9}{9 a x^9} \]

[Out]

-(a + b*x)^9/(9*a*x^9)

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Rubi [A]  time = 0.0016689, 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} \[ -\frac{(a+b x)^9}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^8/x^10,x]

[Out]

-(a + b*x)^9/(9*a*x^9)

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)^8}{x^{10}} \, dx &=-\frac{(a+b x)^9}{9 a x^9}\\ \end{align*}

Mathematica [B]  time = 0.0154863, size = 96, normalized size = 5.65 \[ -\frac{4 a^6 b^2}{x^7}-\frac{28 a^5 b^3}{3 x^6}-\frac{14 a^4 b^4}{x^5}-\frac{14 a^3 b^5}{x^4}-\frac{28 a^2 b^6}{3 x^3}-\frac{a^7 b}{x^8}-\frac{a^8}{9 x^9}-\frac{4 a b^7}{x^2}-\frac{b^8}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^8/x^10,x]

[Out]

-a^8/(9*x^9) - (a^7*b)/x^8 - (4*a^6*b^2)/x^7 - (28*a^5*b^3)/(3*x^6) - (14*a^4*b^4)/x^5 - (14*a^3*b^5)/x^4 - (2
8*a^2*b^6)/(3*x^3) - (4*a*b^7)/x^2 - b^8/x

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Maple [B]  time = 0.005, size = 91, normalized size = 5.4 \begin{align*} -{\frac{28\,{a}^{2}{b}^{6}}{3\,{x}^{3}}}-14\,{\frac{{a}^{4}{b}^{4}}{{x}^{5}}}-14\,{\frac{{a}^{3}{b}^{5}}{{x}^{4}}}-{\frac{28\,{a}^{5}{b}^{3}}{3\,{x}^{6}}}-{\frac{{a}^{7}b}{{x}^{8}}}-4\,{\frac{a{b}^{7}}{{x}^{2}}}-4\,{\frac{{a}^{6}{b}^{2}}{{x}^{7}}}-{\frac{{b}^{8}}{x}}-{\frac{{a}^{8}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^8/x^10,x)

[Out]

-28/3*a^2*b^6/x^3-14*a^4*b^4/x^5-14*a^3*b^5/x^4-28/3*a^5*b^3/x^6-a^7*b/x^8-4*a*b^7/x^2-4*a^6*b^2/x^7-b^8/x-1/9
*a^8/x^9

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Maxima [B]  time = 1.06012, size = 119, normalized size = 7. \begin{align*} -\frac{9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/x^10,x, algorithm="maxima")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9

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Fricas [B]  time = 1.45661, size = 192, normalized size = 11.29 \begin{align*} -\frac{9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/x^10,x, algorithm="fricas")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9

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Sympy [B]  time = 1.09683, size = 95, normalized size = 5.59 \begin{align*} - \frac{a^{8} + 9 a^{7} b x + 36 a^{6} b^{2} x^{2} + 84 a^{5} b^{3} x^{3} + 126 a^{4} b^{4} x^{4} + 126 a^{3} b^{5} x^{5} + 84 a^{2} b^{6} x^{6} + 36 a b^{7} x^{7} + 9 b^{8} x^{8}}{9 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**8/x**10,x)

[Out]

-(a**8 + 9*a**7*b*x + 36*a**6*b**2*x**2 + 84*a**5*b**3*x**3 + 126*a**4*b**4*x**4 + 126*a**3*b**5*x**5 + 84*a**
2*b**6*x**6 + 36*a*b**7*x**7 + 9*b**8*x**8)/(9*x**9)

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Giac [B]  time = 1.19798, size = 119, normalized size = 7. \begin{align*} -\frac{9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/x^10,x, algorithm="giac")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9