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

Optimal. Leaf size=124 \[ -\frac{45 a^8 b^2}{8 x^8}-\frac{120 a^7 b^3}{7 x^7}-\frac{35 a^6 b^4}{x^6}-\frac{252 a^5 b^5}{5 x^5}-\frac{105 a^4 b^6}{2 x^4}-\frac{40 a^3 b^7}{x^3}-\frac{45 a^2 b^8}{2 x^2}-\frac{10 a^9 b}{9 x^9}-\frac{a^{10}}{10 x^{10}}-\frac{10 a b^9}{x}+b^{10} \log (x) \]

[Out]

-a^10/(10*x^10) - (10*a^9*b)/(9*x^9) - (45*a^8*b^2)/(8*x^8) - (120*a^7*b^3)/(7*x^7) - (35*a^6*b^4)/x^6 - (252*
a^5*b^5)/(5*x^5) - (105*a^4*b^6)/(2*x^4) - (40*a^3*b^7)/x^3 - (45*a^2*b^8)/(2*x^2) - (10*a*b^9)/x + b^10*Log[x
]

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Rubi [A]  time = 0.0484612, antiderivative size = 124, 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} \[ -\frac{45 a^8 b^2}{8 x^8}-\frac{120 a^7 b^3}{7 x^7}-\frac{35 a^6 b^4}{x^6}-\frac{252 a^5 b^5}{5 x^5}-\frac{105 a^4 b^6}{2 x^4}-\frac{40 a^3 b^7}{x^3}-\frac{45 a^2 b^8}{2 x^2}-\frac{10 a^9 b}{9 x^9}-\frac{a^{10}}{10 x^{10}}-\frac{10 a b^9}{x}+b^{10} \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^11,x]

[Out]

-a^10/(10*x^10) - (10*a^9*b)/(9*x^9) - (45*a^8*b^2)/(8*x^8) - (120*a^7*b^3)/(7*x^7) - (35*a^6*b^4)/x^6 - (252*
a^5*b^5)/(5*x^5) - (105*a^4*b^6)/(2*x^4) - (40*a^3*b^7)/x^3 - (45*a^2*b^8)/(2*x^2) - (10*a*b^9)/x + b^10*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)^{10}}{x^{11}} \, dx &=\int \left (\frac{a^{10}}{x^{11}}+\frac{10 a^9 b}{x^{10}}+\frac{45 a^8 b^2}{x^9}+\frac{120 a^7 b^3}{x^8}+\frac{210 a^6 b^4}{x^7}+\frac{252 a^5 b^5}{x^6}+\frac{210 a^4 b^6}{x^5}+\frac{120 a^3 b^7}{x^4}+\frac{45 a^2 b^8}{x^3}+\frac{10 a b^9}{x^2}+\frac{b^{10}}{x}\right ) \, dx\\ &=-\frac{a^{10}}{10 x^{10}}-\frac{10 a^9 b}{9 x^9}-\frac{45 a^8 b^2}{8 x^8}-\frac{120 a^7 b^3}{7 x^7}-\frac{35 a^6 b^4}{x^6}-\frac{252 a^5 b^5}{5 x^5}-\frac{105 a^4 b^6}{2 x^4}-\frac{40 a^3 b^7}{x^3}-\frac{45 a^2 b^8}{2 x^2}-\frac{10 a b^9}{x}+b^{10} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0082178, size = 124, normalized size = 1. \[ -\frac{45 a^8 b^2}{8 x^8}-\frac{120 a^7 b^3}{7 x^7}-\frac{35 a^6 b^4}{x^6}-\frac{252 a^5 b^5}{5 x^5}-\frac{105 a^4 b^6}{2 x^4}-\frac{40 a^3 b^7}{x^3}-\frac{45 a^2 b^8}{2 x^2}-\frac{10 a^9 b}{9 x^9}-\frac{a^{10}}{10 x^{10}}-\frac{10 a b^9}{x}+b^{10} \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^11,x]

[Out]

-a^10/(10*x^10) - (10*a^9*b)/(9*x^9) - (45*a^8*b^2)/(8*x^8) - (120*a^7*b^3)/(7*x^7) - (35*a^6*b^4)/x^6 - (252*
a^5*b^5)/(5*x^5) - (105*a^4*b^6)/(2*x^4) - (40*a^3*b^7)/x^3 - (45*a^2*b^8)/(2*x^2) - (10*a*b^9)/x + b^10*Log[x
]

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Maple [A]  time = 0.008, size = 111, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{10\,{x}^{10}}}-{\frac{10\,{a}^{9}b}{9\,{x}^{9}}}-{\frac{45\,{a}^{8}{b}^{2}}{8\,{x}^{8}}}-{\frac{120\,{a}^{7}{b}^{3}}{7\,{x}^{7}}}-35\,{\frac{{a}^{6}{b}^{4}}{{x}^{6}}}-{\frac{252\,{a}^{5}{b}^{5}}{5\,{x}^{5}}}-{\frac{105\,{a}^{4}{b}^{6}}{2\,{x}^{4}}}-40\,{\frac{{a}^{3}{b}^{7}}{{x}^{3}}}-{\frac{45\,{a}^{2}{b}^{8}}{2\,{x}^{2}}}-10\,{\frac{a{b}^{9}}{x}}+{b}^{10}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*a^10/x^10-10/9*a^9*b/x^9-45/8*a^8*b^2/x^8-120/7*a^7*b^3/x^7-35*a^6*b^4/x^6-252/5*a^5*b^5/x^5-105/2*a^4*b
^6/x^4-40*a^3*b^7/x^3-45/2*a^2*b^8/x^2-10*a*b^9/x+b^10*ln(x)

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Maxima [A]  time = 1.0649, size = 150, normalized size = 1.21 \begin{align*} b^{10} \log \left (x\right ) - \frac{25200 \, a b^{9} x^{9} + 56700 \, a^{2} b^{8} x^{8} + 100800 \, a^{3} b^{7} x^{7} + 132300 \, a^{4} b^{6} x^{6} + 127008 \, a^{5} b^{5} x^{5} + 88200 \, a^{6} b^{4} x^{4} + 43200 \, a^{7} b^{3} x^{3} + 14175 \, a^{8} b^{2} x^{2} + 2800 \, a^{9} b x + 252 \, a^{10}}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^10*log(x) - 1/2520*(25200*a*b^9*x^9 + 56700*a^2*b^8*x^8 + 100800*a^3*b^7*x^7 + 132300*a^4*b^6*x^6 + 127008*a
^5*b^5*x^5 + 88200*a^6*b^4*x^4 + 43200*a^7*b^3*x^3 + 14175*a^8*b^2*x^2 + 2800*a^9*b*x + 252*a^10)/x^10

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Fricas [A]  time = 1.79562, size = 302, normalized size = 2.44 \begin{align*} \frac{2520 \, b^{10} x^{10} \log \left (x\right ) - 25200 \, a b^{9} x^{9} - 56700 \, a^{2} b^{8} x^{8} - 100800 \, a^{3} b^{7} x^{7} - 132300 \, a^{4} b^{6} x^{6} - 127008 \, a^{5} b^{5} x^{5} - 88200 \, a^{6} b^{4} x^{4} - 43200 \, a^{7} b^{3} x^{3} - 14175 \, a^{8} b^{2} x^{2} - 2800 \, a^{9} b x - 252 \, a^{10}}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(2520*b^10*x^10*log(x) - 25200*a*b^9*x^9 - 56700*a^2*b^8*x^8 - 100800*a^3*b^7*x^7 - 132300*a^4*b^6*x^6
- 127008*a^5*b^5*x^5 - 88200*a^6*b^4*x^4 - 43200*a^7*b^3*x^3 - 14175*a^8*b^2*x^2 - 2800*a^9*b*x - 252*a^10)/x^
10

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Sympy [A]  time = 1.17134, size = 117, normalized size = 0.94 \begin{align*} b^{10} \log{\left (x \right )} - \frac{252 a^{10} + 2800 a^{9} b x + 14175 a^{8} b^{2} x^{2} + 43200 a^{7} b^{3} x^{3} + 88200 a^{6} b^{4} x^{4} + 127008 a^{5} b^{5} x^{5} + 132300 a^{4} b^{6} x^{6} + 100800 a^{3} b^{7} x^{7} + 56700 a^{2} b^{8} x^{8} + 25200 a b^{9} x^{9}}{2520 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**10*log(x) - (252*a**10 + 2800*a**9*b*x + 14175*a**8*b**2*x**2 + 43200*a**7*b**3*x**3 + 88200*a**6*b**4*x**4
+ 127008*a**5*b**5*x**5 + 132300*a**4*b**6*x**6 + 100800*a**3*b**7*x**7 + 56700*a**2*b**8*x**8 + 25200*a*b**9
*x**9)/(2520*x**10)

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Giac [A]  time = 1.17817, size = 151, normalized size = 1.22 \begin{align*} b^{10} \log \left ({\left | x \right |}\right ) - \frac{25200 \, a b^{9} x^{9} + 56700 \, a^{2} b^{8} x^{8} + 100800 \, a^{3} b^{7} x^{7} + 132300 \, a^{4} b^{6} x^{6} + 127008 \, a^{5} b^{5} x^{5} + 88200 \, a^{6} b^{4} x^{4} + 43200 \, a^{7} b^{3} x^{3} + 14175 \, a^{8} b^{2} x^{2} + 2800 \, a^{9} b x + 252 \, a^{10}}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^10*log(abs(x)) - 1/2520*(25200*a*b^9*x^9 + 56700*a^2*b^8*x^8 + 100800*a^3*b^7*x^7 + 132300*a^4*b^6*x^6 + 127
008*a^5*b^5*x^5 + 88200*a^6*b^4*x^4 + 43200*a^7*b^3*x^3 + 14175*a^8*b^2*x^2 + 2800*a^9*b*x + 252*a^10)/x^10