3.242 \(\int \frac{(a+b x)^9}{x^{10}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{36 a^7 b^2}{7 x^7}-\frac{14 a^6 b^3}{x^6}-\frac{126 a^5 b^4}{5 x^5}-\frac{63 a^4 b^5}{2 x^4}-\frac{28 a^3 b^6}{x^3}-\frac{18 a^2 b^7}{x^2}-\frac{9 a^8 b}{8 x^8}-\frac{a^9}{9 x^9}-\frac{9 a b^8}{x}+b^9 \log (x) \]

[Out]

-a^9/(9*x^9) - (9*a^8*b)/(8*x^8) - (36*a^7*b^2)/(7*x^7) - (14*a^6*b^3)/x^6 - (126*a^5*b^4)/(5*x^5) - (63*a^4*b
^5)/(2*x^4) - (28*a^3*b^6)/x^3 - (18*a^2*b^7)/x^2 - (9*a*b^8)/x + b^9*Log[x]

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Rubi [A]  time = 0.0516384, antiderivative size = 109, normalized size of antiderivative = 1., 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} \[ -\frac{36 a^7 b^2}{7 x^7}-\frac{14 a^6 b^3}{x^6}-\frac{126 a^5 b^4}{5 x^5}-\frac{63 a^4 b^5}{2 x^4}-\frac{28 a^3 b^6}{x^3}-\frac{18 a^2 b^7}{x^2}-\frac{9 a^8 b}{8 x^8}-\frac{a^9}{9 x^9}-\frac{9 a b^8}{x}+b^9 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^9/x^10,x]

[Out]

-a^9/(9*x^9) - (9*a^8*b)/(8*x^8) - (36*a^7*b^2)/(7*x^7) - (14*a^6*b^3)/x^6 - (126*a^5*b^4)/(5*x^5) - (63*a^4*b
^5)/(2*x^4) - (28*a^3*b^6)/x^3 - (18*a^2*b^7)/x^2 - (9*a*b^8)/x + b^9*Log[x]

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)^9}{x^{10}} \, dx &=\int \left (\frac{a^9}{x^{10}}+\frac{9 a^8 b}{x^9}+\frac{36 a^7 b^2}{x^8}+\frac{84 a^6 b^3}{x^7}+\frac{126 a^5 b^4}{x^6}+\frac{126 a^4 b^5}{x^5}+\frac{84 a^3 b^6}{x^4}+\frac{36 a^2 b^7}{x^3}+\frac{9 a b^8}{x^2}+\frac{b^9}{x}\right ) \, dx\\ &=-\frac{a^9}{9 x^9}-\frac{9 a^8 b}{8 x^8}-\frac{36 a^7 b^2}{7 x^7}-\frac{14 a^6 b^3}{x^6}-\frac{126 a^5 b^4}{5 x^5}-\frac{63 a^4 b^5}{2 x^4}-\frac{28 a^3 b^6}{x^3}-\frac{18 a^2 b^7}{x^2}-\frac{9 a b^8}{x}+b^9 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.007086, size = 109, normalized size = 1. \[ -\frac{36 a^7 b^2}{7 x^7}-\frac{14 a^6 b^3}{x^6}-\frac{126 a^5 b^4}{5 x^5}-\frac{63 a^4 b^5}{2 x^4}-\frac{28 a^3 b^6}{x^3}-\frac{18 a^2 b^7}{x^2}-\frac{9 a^8 b}{8 x^8}-\frac{a^9}{9 x^9}-\frac{9 a b^8}{x}+b^9 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^9/x^10,x]

[Out]

-a^9/(9*x^9) - (9*a^8*b)/(8*x^8) - (36*a^7*b^2)/(7*x^7) - (14*a^6*b^3)/x^6 - (126*a^5*b^4)/(5*x^5) - (63*a^4*b
^5)/(2*x^4) - (28*a^3*b^6)/x^3 - (18*a^2*b^7)/x^2 - (9*a*b^8)/x + b^9*Log[x]

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Maple [A]  time = 0.008, size = 100, normalized size = 0.9 \begin{align*} -{\frac{{a}^{9}}{9\,{x}^{9}}}-{\frac{9\,{a}^{8}b}{8\,{x}^{8}}}-{\frac{36\,{a}^{7}{b}^{2}}{7\,{x}^{7}}}-14\,{\frac{{a}^{6}{b}^{3}}{{x}^{6}}}-{\frac{126\,{a}^{5}{b}^{4}}{5\,{x}^{5}}}-{\frac{63\,{a}^{4}{b}^{5}}{2\,{x}^{4}}}-28\,{\frac{{a}^{3}{b}^{6}}{{x}^{3}}}-18\,{\frac{{a}^{2}{b}^{7}}{{x}^{2}}}-9\,{\frac{a{b}^{8}}{x}}+{b}^{9}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^9/x^10,x)

[Out]

-1/9*a^9/x^9-9/8*a^8*b/x^8-36/7*a^7*b^2/x^7-14*a^6*b^3/x^6-126/5*a^5*b^4/x^5-63/2*a^4*b^5/x^4-28*a^3*b^6/x^3-1
8*a^2*b^7/x^2-9*a*b^8/x+b^9*ln(x)

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Maxima [A]  time = 1.06486, size = 135, normalized size = 1.24 \begin{align*} b^{9} \log \left (x\right ) - \frac{22680 \, a b^{8} x^{8} + 45360 \, a^{2} b^{7} x^{7} + 70560 \, a^{3} b^{6} x^{6} + 79380 \, a^{4} b^{5} x^{5} + 63504 \, a^{5} b^{4} x^{4} + 35280 \, a^{6} b^{3} x^{3} + 12960 \, a^{7} b^{2} x^{2} + 2835 \, a^{8} b x + 280 \, a^{9}}{2520 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/x^10,x, algorithm="maxima")

[Out]

b^9*log(x) - 1/2520*(22680*a*b^8*x^8 + 45360*a^2*b^7*x^7 + 70560*a^3*b^6*x^6 + 79380*a^4*b^5*x^5 + 63504*a^5*b
^4*x^4 + 35280*a^6*b^3*x^3 + 12960*a^7*b^2*x^2 + 2835*a^8*b*x + 280*a^9)/x^9

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Fricas [A]  time = 1.46236, size = 266, normalized size = 2.44 \begin{align*} \frac{2520 \, b^{9} x^{9} \log \left (x\right ) - 22680 \, a b^{8} x^{8} - 45360 \, a^{2} b^{7} x^{7} - 70560 \, a^{3} b^{6} x^{6} - 79380 \, a^{4} b^{5} x^{5} - 63504 \, a^{5} b^{4} x^{4} - 35280 \, a^{6} b^{3} x^{3} - 12960 \, a^{7} b^{2} x^{2} - 2835 \, a^{8} b x - 280 \, a^{9}}{2520 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/x^10,x, algorithm="fricas")

[Out]

1/2520*(2520*b^9*x^9*log(x) - 22680*a*b^8*x^8 - 45360*a^2*b^7*x^7 - 70560*a^3*b^6*x^6 - 79380*a^4*b^5*x^5 - 63
504*a^5*b^4*x^4 - 35280*a^6*b^3*x^3 - 12960*a^7*b^2*x^2 - 2835*a^8*b*x - 280*a^9)/x^9

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Sympy [A]  time = 1.27001, size = 105, normalized size = 0.96 \begin{align*} b^{9} \log{\left (x \right )} - \frac{280 a^{9} + 2835 a^{8} b x + 12960 a^{7} b^{2} x^{2} + 35280 a^{6} b^{3} x^{3} + 63504 a^{5} b^{4} x^{4} + 79380 a^{4} b^{5} x^{5} + 70560 a^{3} b^{6} x^{6} + 45360 a^{2} b^{7} x^{7} + 22680 a b^{8} x^{8}}{2520 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**9/x**10,x)

[Out]

b**9*log(x) - (280*a**9 + 2835*a**8*b*x + 12960*a**7*b**2*x**2 + 35280*a**6*b**3*x**3 + 63504*a**5*b**4*x**4 +
79380*a**4*b**5*x**5 + 70560*a**3*b**6*x**6 + 45360*a**2*b**7*x**7 + 22680*a*b**8*x**8)/(2520*x**9)

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Giac [A]  time = 1.22176, size = 136, normalized size = 1.25 \begin{align*} b^{9} \log \left ({\left | x \right |}\right ) - \frac{22680 \, a b^{8} x^{8} + 45360 \, a^{2} b^{7} x^{7} + 70560 \, a^{3} b^{6} x^{6} + 79380 \, a^{4} b^{5} x^{5} + 63504 \, a^{5} b^{4} x^{4} + 35280 \, a^{6} b^{3} x^{3} + 12960 \, a^{7} b^{2} x^{2} + 2835 \, a^{8} b x + 280 \, a^{9}}{2520 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/x^10,x, algorithm="giac")

[Out]

b^9*log(abs(x)) - 1/2520*(22680*a*b^8*x^8 + 45360*a^2*b^7*x^7 + 70560*a^3*b^6*x^6 + 79380*a^4*b^5*x^5 + 63504*
a^5*b^4*x^4 + 35280*a^6*b^3*x^3 + 12960*a^7*b^2*x^2 + 2835*a^8*b*x + 280*a^9)/x^9