3.240 \(\int \frac{(a+b x)^{11}}{x^{10}} \, dx\)

Optimal. Leaf size=132 \[ -\frac{55 a^9 b^2}{7 x^7}-\frac{55 a^8 b^3}{2 x^6}-\frac{66 a^7 b^4}{x^5}-\frac{231 a^6 b^5}{2 x^4}-\frac{154 a^5 b^6}{x^3}-\frac{165 a^4 b^7}{x^2}-\frac{165 a^3 b^8}{x}+55 a^2 b^9 \log (x)-\frac{11 a^{10} b}{8 x^8}-\frac{a^{11}}{9 x^9}+11 a b^{10} x+\frac{b^{11} x^2}{2} \]

[Out]

-a^11/(9*x^9) - (11*a^10*b)/(8*x^8) - (55*a^9*b^2)/(7*x^7) - (55*a^8*b^3)/(2*x^6) - (66*a^7*b^4)/x^5 - (231*a^
6*b^5)/(2*x^4) - (154*a^5*b^6)/x^3 - (165*a^4*b^7)/x^2 - (165*a^3*b^8)/x + 11*a*b^10*x + (b^11*x^2)/2 + 55*a^2
*b^9*Log[x]

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Rubi [A]  time = 0.0695941, antiderivative size = 132, 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} \[ -\frac{55 a^9 b^2}{7 x^7}-\frac{55 a^8 b^3}{2 x^6}-\frac{66 a^7 b^4}{x^5}-\frac{231 a^6 b^5}{2 x^4}-\frac{154 a^5 b^6}{x^3}-\frac{165 a^4 b^7}{x^2}-\frac{165 a^3 b^8}{x}+55 a^2 b^9 \log (x)-\frac{11 a^{10} b}{8 x^8}-\frac{a^{11}}{9 x^9}+11 a b^{10} x+\frac{b^{11} x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^11/x^10,x]

[Out]

-a^11/(9*x^9) - (11*a^10*b)/(8*x^8) - (55*a^9*b^2)/(7*x^7) - (55*a^8*b^3)/(2*x^6) - (66*a^7*b^4)/x^5 - (231*a^
6*b^5)/(2*x^4) - (154*a^5*b^6)/x^3 - (165*a^4*b^7)/x^2 - (165*a^3*b^8)/x + 11*a*b^10*x + (b^11*x^2)/2 + 55*a^2
*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)^{11}}{x^{10}} \, dx &=\int \left (11 a b^{10}+\frac{a^{11}}{x^{10}}+\frac{11 a^{10} b}{x^9}+\frac{55 a^9 b^2}{x^8}+\frac{165 a^8 b^3}{x^7}+\frac{330 a^7 b^4}{x^6}+\frac{462 a^6 b^5}{x^5}+\frac{462 a^5 b^6}{x^4}+\frac{330 a^4 b^7}{x^3}+\frac{165 a^3 b^8}{x^2}+\frac{55 a^2 b^9}{x}+b^{11} x\right ) \, dx\\ &=-\frac{a^{11}}{9 x^9}-\frac{11 a^{10} b}{8 x^8}-\frac{55 a^9 b^2}{7 x^7}-\frac{55 a^8 b^3}{2 x^6}-\frac{66 a^7 b^4}{x^5}-\frac{231 a^6 b^5}{2 x^4}-\frac{154 a^5 b^6}{x^3}-\frac{165 a^4 b^7}{x^2}-\frac{165 a^3 b^8}{x}+11 a b^{10} x+\frac{b^{11} x^2}{2}+55 a^2 b^9 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0068515, size = 132, normalized size = 1. \[ -\frac{55 a^9 b^2}{7 x^7}-\frac{55 a^8 b^3}{2 x^6}-\frac{66 a^7 b^4}{x^5}-\frac{231 a^6 b^5}{2 x^4}-\frac{154 a^5 b^6}{x^3}-\frac{165 a^4 b^7}{x^2}-\frac{165 a^3 b^8}{x}+55 a^2 b^9 \log (x)-\frac{11 a^{10} b}{8 x^8}-\frac{a^{11}}{9 x^9}+11 a b^{10} x+\frac{b^{11} x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^11/x^10,x]

[Out]

-a^11/(9*x^9) - (11*a^10*b)/(8*x^8) - (55*a^9*b^2)/(7*x^7) - (55*a^8*b^3)/(2*x^6) - (66*a^7*b^4)/x^5 - (231*a^
6*b^5)/(2*x^4) - (154*a^5*b^6)/x^3 - (165*a^4*b^7)/x^2 - (165*a^3*b^8)/x + 11*a*b^10*x + (b^11*x^2)/2 + 55*a^2
*b^9*Log[x]

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Maple [A]  time = 0.009, size = 121, normalized size = 0.9 \begin{align*} -{\frac{{a}^{11}}{9\,{x}^{9}}}-{\frac{11\,{a}^{10}b}{8\,{x}^{8}}}-{\frac{55\,{a}^{9}{b}^{2}}{7\,{x}^{7}}}-{\frac{55\,{a}^{8}{b}^{3}}{2\,{x}^{6}}}-66\,{\frac{{a}^{7}{b}^{4}}{{x}^{5}}}-{\frac{231\,{a}^{6}{b}^{5}}{2\,{x}^{4}}}-154\,{\frac{{a}^{5}{b}^{6}}{{x}^{3}}}-165\,{\frac{{a}^{4}{b}^{7}}{{x}^{2}}}-165\,{\frac{{a}^{3}{b}^{8}}{x}}+11\,a{b}^{10}x+{\frac{{b}^{11}{x}^{2}}{2}}+55\,{a}^{2}{b}^{9}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^11/x^10,x)

[Out]

-1/9*a^11/x^9-11/8*a^10*b/x^8-55/7*a^9*b^2/x^7-55/2*a^8*b^3/x^6-66*a^7*b^4/x^5-231/2*a^6*b^5/x^4-154*a^5*b^6/x
^3-165*a^4*b^7/x^2-165*a^3*b^8/x+11*a*b^10*x+1/2*b^11*x^2+55*a^2*b^9*ln(x)

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Maxima [A]  time = 1.06214, size = 163, normalized size = 1.23 \begin{align*} \frac{1}{2} \, b^{11} x^{2} + 11 \, a b^{10} x + 55 \, a^{2} b^{9} \log \left (x\right ) - \frac{83160 \, a^{3} b^{8} x^{8} + 83160 \, a^{4} b^{7} x^{7} + 77616 \, a^{5} b^{6} x^{6} + 58212 \, a^{6} b^{5} x^{5} + 33264 \, a^{7} b^{4} x^{4} + 13860 \, a^{8} b^{3} x^{3} + 3960 \, a^{9} b^{2} x^{2} + 693 \, a^{10} b x + 56 \, a^{11}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^11/x^10,x, algorithm="maxima")

[Out]

1/2*b^11*x^2 + 11*a*b^10*x + 55*a^2*b^9*log(x) - 1/504*(83160*a^3*b^8*x^8 + 83160*a^4*b^7*x^7 + 77616*a^5*b^6*
x^6 + 58212*a^6*b^5*x^5 + 33264*a^7*b^4*x^4 + 13860*a^8*b^3*x^3 + 3960*a^9*b^2*x^2 + 693*a^10*b*x + 56*a^11)/x
^9

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Fricas [A]  time = 1.49566, size = 320, normalized size = 2.42 \begin{align*} \frac{252 \, b^{11} x^{11} + 5544 \, a b^{10} x^{10} + 27720 \, a^{2} b^{9} x^{9} \log \left (x\right ) - 83160 \, a^{3} b^{8} x^{8} - 83160 \, a^{4} b^{7} x^{7} - 77616 \, a^{5} b^{6} x^{6} - 58212 \, a^{6} b^{5} x^{5} - 33264 \, a^{7} b^{4} x^{4} - 13860 \, a^{8} b^{3} x^{3} - 3960 \, a^{9} b^{2} x^{2} - 693 \, a^{10} b x - 56 \, a^{11}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^11/x^10,x, algorithm="fricas")

[Out]

1/504*(252*b^11*x^11 + 5544*a*b^10*x^10 + 27720*a^2*b^9*x^9*log(x) - 83160*a^3*b^8*x^8 - 83160*a^4*b^7*x^7 - 7
7616*a^5*b^6*x^6 - 58212*a^6*b^5*x^5 - 33264*a^7*b^4*x^4 - 13860*a^8*b^3*x^3 - 3960*a^9*b^2*x^2 - 693*a^10*b*x
- 56*a^11)/x^9

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Sympy [A]  time = 1.2748, size = 129, normalized size = 0.98 \begin{align*} 55 a^{2} b^{9} \log{\left (x \right )} + 11 a b^{10} x + \frac{b^{11} x^{2}}{2} - \frac{56 a^{11} + 693 a^{10} b x + 3960 a^{9} b^{2} x^{2} + 13860 a^{8} b^{3} x^{3} + 33264 a^{7} b^{4} x^{4} + 58212 a^{6} b^{5} x^{5} + 77616 a^{5} b^{6} x^{6} + 83160 a^{4} b^{7} x^{7} + 83160 a^{3} b^{8} x^{8}}{504 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**11/x**10,x)

[Out]

55*a**2*b**9*log(x) + 11*a*b**10*x + b**11*x**2/2 - (56*a**11 + 693*a**10*b*x + 3960*a**9*b**2*x**2 + 13860*a*
*8*b**3*x**3 + 33264*a**7*b**4*x**4 + 58212*a**6*b**5*x**5 + 77616*a**5*b**6*x**6 + 83160*a**4*b**7*x**7 + 831
60*a**3*b**8*x**8)/(504*x**9)

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Giac [A]  time = 1.22444, size = 165, normalized size = 1.25 \begin{align*} \frac{1}{2} \, b^{11} x^{2} + 11 \, a b^{10} x + 55 \, a^{2} b^{9} \log \left ({\left | x \right |}\right ) - \frac{83160 \, a^{3} b^{8} x^{8} + 83160 \, a^{4} b^{7} x^{7} + 77616 \, a^{5} b^{6} x^{6} + 58212 \, a^{6} b^{5} x^{5} + 33264 \, a^{7} b^{4} x^{4} + 13860 \, a^{8} b^{3} x^{3} + 3960 \, a^{9} b^{2} x^{2} + 693 \, a^{10} b x + 56 \, a^{11}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^11/x^10,x, algorithm="giac")

[Out]

1/2*b^11*x^2 + 11*a*b^10*x + 55*a^2*b^9*log(abs(x)) - 1/504*(83160*a^3*b^8*x^8 + 83160*a^4*b^7*x^7 + 77616*a^5
*b^6*x^6 + 58212*a^6*b^5*x^5 + 33264*a^7*b^4*x^4 + 13860*a^8*b^3*x^3 + 3960*a^9*b^2*x^2 + 693*a^10*b*x + 56*a^
11)/x^9