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

Optimal. Leaf size=119 \[ -\frac{a^{10}}{8 x^8}-\frac{10 a^9 b}{7 x^7}-\frac{15 a^8 b^2}{2 x^6}-\frac{24 a^7 b^3}{x^5}-\frac{105 a^6 b^4}{2 x^4}-\frac{84 a^5 b^5}{x^3}-\frac{105 a^4 b^6}{x^2}-\frac{120 a^3 b^7}{x}+45 a^2 b^8 \log (x)+10 a b^9 x+\frac{b^{10} x^2}{2} \]

[Out]

-a^10/(8*x^8) - (10*a^9*b)/(7*x^7) - (15*a^8*b^2)/(2*x^6) - (24*a^7*b^3)/x^5 - (
105*a^6*b^4)/(2*x^4) - (84*a^5*b^5)/x^3 - (105*a^4*b^6)/x^2 - (120*a^3*b^7)/x +
10*a*b^9*x + (b^10*x^2)/2 + 45*a^2*b^8*Log[x]

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Rubi [A]  time = 0.117922, antiderivative size = 119, 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 \[ -\frac{a^{10}}{8 x^8}-\frac{10 a^9 b}{7 x^7}-\frac{15 a^8 b^2}{2 x^6}-\frac{24 a^7 b^3}{x^5}-\frac{105 a^6 b^4}{2 x^4}-\frac{84 a^5 b^5}{x^3}-\frac{105 a^4 b^6}{x^2}-\frac{120 a^3 b^7}{x}+45 a^2 b^8 \log (x)+10 a b^9 x+\frac{b^{10} x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^9,x]

[Out]

-a^10/(8*x^8) - (10*a^9*b)/(7*x^7) - (15*a^8*b^2)/(2*x^6) - (24*a^7*b^3)/x^5 - (
105*a^6*b^4)/(2*x^4) - (84*a^5*b^5)/x^3 - (105*a^4*b^6)/x^2 - (120*a^3*b^7)/x +
10*a*b^9*x + (b^10*x^2)/2 + 45*a^2*b^8*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{8 x^{8}} - \frac{10 a^{9} b}{7 x^{7}} - \frac{15 a^{8} b^{2}}{2 x^{6}} - \frac{24 a^{7} b^{3}}{x^{5}} - \frac{105 a^{6} b^{4}}{2 x^{4}} - \frac{84 a^{5} b^{5}}{x^{3}} - \frac{105 a^{4} b^{6}}{x^{2}} - \frac{120 a^{3} b^{7}}{x} + 45 a^{2} b^{8} \log{\left (x \right )} + 10 a b^{9} x + b^{10} \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**9,x)

[Out]

-a**10/(8*x**8) - 10*a**9*b/(7*x**7) - 15*a**8*b**2/(2*x**6) - 24*a**7*b**3/x**5
- 105*a**6*b**4/(2*x**4) - 84*a**5*b**5/x**3 - 105*a**4*b**6/x**2 - 120*a**3*b*
*7/x + 45*a**2*b**8*log(x) + 10*a*b**9*x + b**10*Integral(x, x)

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Mathematica [A]  time = 0.00727417, size = 119, normalized size = 1. \[ -\frac{a^{10}}{8 x^8}-\frac{10 a^9 b}{7 x^7}-\frac{15 a^8 b^2}{2 x^6}-\frac{24 a^7 b^3}{x^5}-\frac{105 a^6 b^4}{2 x^4}-\frac{84 a^5 b^5}{x^3}-\frac{105 a^4 b^6}{x^2}-\frac{120 a^3 b^7}{x}+45 a^2 b^8 \log (x)+10 a b^9 x+\frac{b^{10} x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^9,x]

[Out]

-a^10/(8*x^8) - (10*a^9*b)/(7*x^7) - (15*a^8*b^2)/(2*x^6) - (24*a^7*b^3)/x^5 - (
105*a^6*b^4)/(2*x^4) - (84*a^5*b^5)/x^3 - (105*a^4*b^6)/x^2 - (120*a^3*b^7)/x +
10*a*b^9*x + (b^10*x^2)/2 + 45*a^2*b^8*Log[x]

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Maple [A]  time = 0.011, size = 110, normalized size = 0.9 \[ -{\frac{{a}^{10}}{8\,{x}^{8}}}-{\frac{10\,{a}^{9}b}{7\,{x}^{7}}}-{\frac{15\,{a}^{8}{b}^{2}}{2\,{x}^{6}}}-24\,{\frac{{a}^{7}{b}^{3}}{{x}^{5}}}-{\frac{105\,{a}^{6}{b}^{4}}{2\,{x}^{4}}}-84\,{\frac{{a}^{5}{b}^{5}}{{x}^{3}}}-105\,{\frac{{a}^{4}{b}^{6}}{{x}^{2}}}-120\,{\frac{{a}^{3}{b}^{7}}{x}}+10\,a{b}^{9}x+{\frac{{b}^{10}{x}^{2}}{2}}+45\,{a}^{2}{b}^{8}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^9,x)

[Out]

-1/8*a^10/x^8-10/7*a^9*b/x^7-15/2*a^8*b^2/x^6-24*a^7*b^3/x^5-105/2*a^6*b^4/x^4-8
4*a^5*b^5/x^3-105*a^4*b^6/x^2-120*a^3*b^7/x+10*a*b^9*x+1/2*b^10*x^2+45*a^2*b^8*l
n(x)

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Maxima [A]  time = 1.35197, size = 149, normalized size = 1.25 \[ \frac{1}{2} \, b^{10} x^{2} + 10 \, a b^{9} x + 45 \, a^{2} b^{8} \log \left (x\right ) - \frac{6720 \, a^{3} b^{7} x^{7} + 5880 \, a^{4} b^{6} x^{6} + 4704 \, a^{5} b^{5} x^{5} + 2940 \, a^{6} b^{4} x^{4} + 1344 \, a^{7} b^{3} x^{3} + 420 \, a^{8} b^{2} x^{2} + 80 \, a^{9} b x + 7 \, a^{10}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^10*x^2 + 10*a*b^9*x + 45*a^2*b^8*log(x) - 1/56*(6720*a^3*b^7*x^7 + 5880*a^
4*b^6*x^6 + 4704*a^5*b^5*x^5 + 2940*a^6*b^4*x^4 + 1344*a^7*b^3*x^3 + 420*a^8*b^2
*x^2 + 80*a^9*b*x + 7*a^10)/x^8

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Fricas [A]  time = 0.195655, size = 154, normalized size = 1.29 \[ \frac{28 \, b^{10} x^{10} + 560 \, a b^{9} x^{9} + 2520 \, a^{2} b^{8} x^{8} \log \left (x\right ) - 6720 \, a^{3} b^{7} x^{7} - 5880 \, a^{4} b^{6} x^{6} - 4704 \, a^{5} b^{5} x^{5} - 2940 \, a^{6} b^{4} x^{4} - 1344 \, a^{7} b^{3} x^{3} - 420 \, a^{8} b^{2} x^{2} - 80 \, a^{9} b x - 7 \, a^{10}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/56*(28*b^10*x^10 + 560*a*b^9*x^9 + 2520*a^2*b^8*x^8*log(x) - 6720*a^3*b^7*x^7
- 5880*a^4*b^6*x^6 - 4704*a^5*b^5*x^5 - 2940*a^6*b^4*x^4 - 1344*a^7*b^3*x^3 - 42
0*a^8*b^2*x^2 - 80*a^9*b*x - 7*a^10)/x^8

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Sympy [A]  time = 3.03506, size = 117, normalized size = 0.98 \[ 45 a^{2} b^{8} \log{\left (x \right )} + 10 a b^{9} x + \frac{b^{10} x^{2}}{2} - \frac{7 a^{10} + 80 a^{9} b x + 420 a^{8} b^{2} x^{2} + 1344 a^{7} b^{3} x^{3} + 2940 a^{6} b^{4} x^{4} + 4704 a^{5} b^{5} x^{5} + 5880 a^{4} b^{6} x^{6} + 6720 a^{3} b^{7} x^{7}}{56 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**9,x)

[Out]

45*a**2*b**8*log(x) + 10*a*b**9*x + b**10*x**2/2 - (7*a**10 + 80*a**9*b*x + 420*
a**8*b**2*x**2 + 1344*a**7*b**3*x**3 + 2940*a**6*b**4*x**4 + 4704*a**5*b**5*x**5
+ 5880*a**4*b**6*x**6 + 6720*a**3*b**7*x**7)/(56*x**8)

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GIAC/XCAS [A]  time = 0.206213, size = 150, normalized size = 1.26 \[ \frac{1}{2} \, b^{10} x^{2} + 10 \, a b^{9} x + 45 \, a^{2} b^{8}{\rm ln}\left ({\left | x \right |}\right ) - \frac{6720 \, a^{3} b^{7} x^{7} + 5880 \, a^{4} b^{6} x^{6} + 4704 \, a^{5} b^{5} x^{5} + 2940 \, a^{6} b^{4} x^{4} + 1344 \, a^{7} b^{3} x^{3} + 420 \, a^{8} b^{2} x^{2} + 80 \, a^{9} b x + 7 \, a^{10}}{56 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^10*x^2 + 10*a*b^9*x + 45*a^2*b^8*ln(abs(x)) - 1/56*(6720*a^3*b^7*x^7 + 588
0*a^4*b^6*x^6 + 4704*a^5*b^5*x^5 + 2940*a^6*b^4*x^4 + 1344*a^7*b^3*x^3 + 420*a^8
*b^2*x^2 + 80*a^9*b*x + 7*a^10)/x^8