3.139 \(\int \frac{(a+b x)^{10}}{x^5} \, dx\)

Optimal. Leaf size=119 \[ -\frac{a^{10}}{4 x^4}-\frac{10 a^9 b}{3 x^3}-\frac{45 a^8 b^2}{2 x^2}-\frac{120 a^7 b^3}{x}+210 a^6 b^4 \log (x)+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac{45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac{b^{10} x^6}{6} \]

[Out]

-a^10/(4*x^4) - (10*a^9*b)/(3*x^3) - (45*a^8*b^2)/(2*x^2) - (120*a^7*b^3)/x + 25
2*a^5*b^5*x + 105*a^4*b^6*x^2 + 40*a^3*b^7*x^3 + (45*a^2*b^8*x^4)/4 + 2*a*b^9*x^
5 + (b^10*x^6)/6 + 210*a^6*b^4*Log[x]

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Rubi [A]  time = 0.118007, 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 \[ -\frac{a^{10}}{4 x^4}-\frac{10 a^9 b}{3 x^3}-\frac{45 a^8 b^2}{2 x^2}-\frac{120 a^7 b^3}{x}+210 a^6 b^4 \log (x)+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac{45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac{b^{10} x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^5,x]

[Out]

-a^10/(4*x^4) - (10*a^9*b)/(3*x^3) - (45*a^8*b^2)/(2*x^2) - (120*a^7*b^3)/x + 25
2*a^5*b^5*x + 105*a^4*b^6*x^2 + 40*a^3*b^7*x^3 + (45*a^2*b^8*x^4)/4 + 2*a*b^9*x^
5 + (b^10*x^6)/6 + 210*a^6*b^4*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{4 x^{4}} - \frac{10 a^{9} b}{3 x^{3}} - \frac{45 a^{8} b^{2}}{2 x^{2}} - \frac{120 a^{7} b^{3}}{x} + 210 a^{6} b^{4} \log{\left (x \right )} + 252 a^{5} b^{5} x + 210 a^{4} b^{6} \int x\, dx + 40 a^{3} b^{7} x^{3} + \frac{45 a^{2} b^{8} x^{4}}{4} + 2 a b^{9} x^{5} + \frac{b^{10} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**5,x)

[Out]

-a**10/(4*x**4) - 10*a**9*b/(3*x**3) - 45*a**8*b**2/(2*x**2) - 120*a**7*b**3/x +
210*a**6*b**4*log(x) + 252*a**5*b**5*x + 210*a**4*b**6*Integral(x, x) + 40*a**3
*b**7*x**3 + 45*a**2*b**8*x**4/4 + 2*a*b**9*x**5 + b**10*x**6/6

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Mathematica [A]  time = 0.0177834, size = 119, normalized size = 1. \[ -\frac{a^{10}}{4 x^4}-\frac{10 a^9 b}{3 x^3}-\frac{45 a^8 b^2}{2 x^2}-\frac{120 a^7 b^3}{x}+210 a^6 b^4 \log (x)+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac{45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac{b^{10} x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^5,x]

[Out]

-a^10/(4*x^4) - (10*a^9*b)/(3*x^3) - (45*a^8*b^2)/(2*x^2) - (120*a^7*b^3)/x + 25
2*a^5*b^5*x + 105*a^4*b^6*x^2 + 40*a^3*b^7*x^3 + (45*a^2*b^8*x^4)/4 + 2*a*b^9*x^
5 + (b^10*x^6)/6 + 210*a^6*b^4*Log[x]

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Maple [A]  time = 0.01, size = 110, normalized size = 0.9 \[ -{\frac{{a}^{10}}{4\,{x}^{4}}}-{\frac{10\,{a}^{9}b}{3\,{x}^{3}}}-{\frac{45\,{a}^{8}{b}^{2}}{2\,{x}^{2}}}-120\,{\frac{{a}^{7}{b}^{3}}{x}}+252\,{a}^{5}{b}^{5}x+105\,{a}^{4}{b}^{6}{x}^{2}+40\,{a}^{3}{b}^{7}{x}^{3}+{\frac{45\,{a}^{2}{b}^{8}{x}^{4}}{4}}+2\,a{b}^{9}{x}^{5}+{\frac{{b}^{10}{x}^{6}}{6}}+210\,{a}^{6}{b}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^5,x)

[Out]

-1/4*a^10/x^4-10/3*a^9*b/x^3-45/2*a^8*b^2/x^2-120*a^7*b^3/x+252*a^5*b^5*x+105*a^
4*b^6*x^2+40*a^3*b^7*x^3+45/4*a^2*b^8*x^4+2*a*b^9*x^5+1/6*b^10*x^6+210*a^6*b^4*l
n(x)

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Maxima [A]  time = 1.34043, size = 149, normalized size = 1.25 \[ \frac{1}{6} \, b^{10} x^{6} + 2 \, a b^{9} x^{5} + \frac{45}{4} \, a^{2} b^{8} x^{4} + 40 \, a^{3} b^{7} x^{3} + 105 \, a^{4} b^{6} x^{2} + 252 \, a^{5} b^{5} x + 210 \, a^{6} b^{4} \log \left (x\right ) - \frac{1440 \, a^{7} b^{3} x^{3} + 270 \, a^{8} b^{2} x^{2} + 40 \, a^{9} b x + 3 \, a^{10}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^5,x, algorithm="maxima")

[Out]

1/6*b^10*x^6 + 2*a*b^9*x^5 + 45/4*a^2*b^8*x^4 + 40*a^3*b^7*x^3 + 105*a^4*b^6*x^2
+ 252*a^5*b^5*x + 210*a^6*b^4*log(x) - 1/12*(1440*a^7*b^3*x^3 + 270*a^8*b^2*x^2
+ 40*a^9*b*x + 3*a^10)/x^4

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Fricas [A]  time = 0.194369, size = 154, normalized size = 1.29 \[ \frac{2 \, b^{10} x^{10} + 24 \, a b^{9} x^{9} + 135 \, a^{2} b^{8} x^{8} + 480 \, a^{3} b^{7} x^{7} + 1260 \, a^{4} b^{6} x^{6} + 3024 \, a^{5} b^{5} x^{5} + 2520 \, a^{6} b^{4} x^{4} \log \left (x\right ) - 1440 \, a^{7} b^{3} x^{3} - 270 \, a^{8} b^{2} x^{2} - 40 \, a^{9} b x - 3 \, a^{10}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^5,x, algorithm="fricas")

[Out]

1/12*(2*b^10*x^10 + 24*a*b^9*x^9 + 135*a^2*b^8*x^8 + 480*a^3*b^7*x^7 + 1260*a^4*
b^6*x^6 + 3024*a^5*b^5*x^5 + 2520*a^6*b^4*x^4*log(x) - 1440*a^7*b^3*x^3 - 270*a^
8*b^2*x^2 - 40*a^9*b*x - 3*a^10)/x^4

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Sympy [A]  time = 2.02006, size = 119, normalized size = 1. \[ 210 a^{6} b^{4} \log{\left (x \right )} + 252 a^{5} b^{5} x + 105 a^{4} b^{6} x^{2} + 40 a^{3} b^{7} x^{3} + \frac{45 a^{2} b^{8} x^{4}}{4} + 2 a b^{9} x^{5} + \frac{b^{10} x^{6}}{6} - \frac{3 a^{10} + 40 a^{9} b x + 270 a^{8} b^{2} x^{2} + 1440 a^{7} b^{3} x^{3}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**5,x)

[Out]

210*a**6*b**4*log(x) + 252*a**5*b**5*x + 105*a**4*b**6*x**2 + 40*a**3*b**7*x**3
+ 45*a**2*b**8*x**4/4 + 2*a*b**9*x**5 + b**10*x**6/6 - (3*a**10 + 40*a**9*b*x +
270*a**8*b**2*x**2 + 1440*a**7*b**3*x**3)/(12*x**4)

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GIAC/XCAS [A]  time = 0.205331, size = 150, normalized size = 1.26 \[ \frac{1}{6} \, b^{10} x^{6} + 2 \, a b^{9} x^{5} + \frac{45}{4} \, a^{2} b^{8} x^{4} + 40 \, a^{3} b^{7} x^{3} + 105 \, a^{4} b^{6} x^{2} + 252 \, a^{5} b^{5} x + 210 \, a^{6} b^{4}{\rm ln}\left ({\left | x \right |}\right ) - \frac{1440 \, a^{7} b^{3} x^{3} + 270 \, a^{8} b^{2} x^{2} + 40 \, a^{9} b x + 3 \, a^{10}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^5,x, algorithm="giac")

[Out]

1/6*b^10*x^6 + 2*a*b^9*x^5 + 45/4*a^2*b^8*x^4 + 40*a^3*b^7*x^3 + 105*a^4*b^6*x^2
+ 252*a^5*b^5*x + 210*a^6*b^4*ln(abs(x)) - 1/12*(1440*a^7*b^3*x^3 + 270*a^8*b^2
*x^2 + 40*a^9*b*x + 3*a^10)/x^4