3.137 \(\int \frac{(a+b x)^{10}}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.115209, 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 \[ -\frac{a^{10}}{2 x^2}-\frac{10 a^9 b}{x}+45 a^8 b^2 \log (x)+120 a^7 b^3 x+105 a^6 b^4 x^2+84 a^5 b^5 x^3+\frac{105}{2} a^4 b^6 x^4+24 a^3 b^7 x^5+\frac{15}{2} a^2 b^8 x^6+\frac{10}{7} a b^9 x^7+\frac{b^{10} x^8}{8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{2 x^{2}} - \frac{10 a^{9} b}{x} + 45 a^{8} b^{2} \log{\left (x \right )} + 120 a^{7} b^{3} x + 210 a^{6} b^{4} \int x\, dx + 84 a^{5} b^{5} x^{3} + \frac{105 a^{4} b^{6} x^{4}}{2} + 24 a^{3} b^{7} x^{5} + \frac{15 a^{2} b^{8} x^{6}}{2} + \frac{10 a b^{9} x^{7}}{7} + \frac{b^{10} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**3,x)

[Out]

-a**10/(2*x**2) - 10*a**9*b/x + 45*a**8*b**2*log(x) + 120*a**7*b**3*x + 210*a**6
*b**4*Integral(x, x) + 84*a**5*b**5*x**3 + 105*a**4*b**6*x**4/2 + 24*a**3*b**7*x
**5 + 15*a**2*b**8*x**6/2 + 10*a*b**9*x**7/7 + b**10*x**8/8

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Mathematica [A]  time = 0.0072313, size = 119, normalized size = 1. \[ -\frac{a^{10}}{2 x^2}-\frac{10 a^9 b}{x}+45 a^8 b^2 \log (x)+120 a^7 b^3 x+105 a^6 b^4 x^2+84 a^5 b^5 x^3+\frac{105}{2} a^4 b^6 x^4+24 a^3 b^7 x^5+\frac{15}{2} a^2 b^8 x^6+\frac{10}{7} a b^9 x^7+\frac{b^{10} x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 110, normalized size = 0.9 \[ -{\frac{{a}^{10}}{2\,{x}^{2}}}-10\,{\frac{{a}^{9}b}{x}}+120\,{a}^{7}{b}^{3}x+105\,{a}^{6}{b}^{4}{x}^{2}+84\,{a}^{5}{b}^{5}{x}^{3}+{\frac{105\,{a}^{4}{b}^{6}{x}^{4}}{2}}+24\,{a}^{3}{b}^{7}{x}^{5}+{\frac{15\,{a}^{2}{b}^{8}{x}^{6}}{2}}+{\frac{10\,a{b}^{9}{x}^{7}}{7}}+{\frac{{b}^{10}{x}^{8}}{8}}+45\,{a}^{8}{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^3,x)

[Out]

-1/2*a^10/x^2-10*a^9*b/x+120*a^7*b^3*x+105*a^6*b^4*x^2+84*a^5*b^5*x^3+105/2*a^4*
b^6*x^4+24*a^3*b^7*x^5+15/2*a^2*b^8*x^6+10/7*a*b^9*x^7+1/8*b^10*x^8+45*a^8*b^2*l
n(x)

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Maxima [A]  time = 1.33014, size = 146, normalized size = 1.23 \[ \frac{1}{8} \, b^{10} x^{8} + \frac{10}{7} \, a b^{9} x^{7} + \frac{15}{2} \, a^{2} b^{8} x^{6} + 24 \, a^{3} b^{7} x^{5} + \frac{105}{2} \, a^{4} b^{6} x^{4} + 84 \, a^{5} b^{5} x^{3} + 105 \, a^{6} b^{4} x^{2} + 120 \, a^{7} b^{3} x + 45 \, a^{8} b^{2} \log \left (x\right ) - \frac{20 \, a^{9} b x + a^{10}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^3,x, algorithm="maxima")

[Out]

1/8*b^10*x^8 + 10/7*a*b^9*x^7 + 15/2*a^2*b^8*x^6 + 24*a^3*b^7*x^5 + 105/2*a^4*b^
6*x^4 + 84*a^5*b^5*x^3 + 105*a^6*b^4*x^2 + 120*a^7*b^3*x + 45*a^8*b^2*log(x) - 1
/2*(20*a^9*b*x + a^10)/x^2

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Fricas [A]  time = 0.195307, size = 154, normalized size = 1.29 \[ \frac{7 \, b^{10} x^{10} + 80 \, a b^{9} x^{9} + 420 \, a^{2} b^{8} x^{8} + 1344 \, a^{3} b^{7} x^{7} + 2940 \, a^{4} b^{6} x^{6} + 4704 \, a^{5} b^{5} x^{5} + 5880 \, a^{6} b^{4} x^{4} + 6720 \, a^{7} b^{3} x^{3} + 2520 \, a^{8} b^{2} x^{2} \log \left (x\right ) - 560 \, a^{9} b x - 28 \, a^{10}}{56 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^3,x, algorithm="fricas")

[Out]

1/56*(7*b^10*x^10 + 80*a*b^9*x^9 + 420*a^2*b^8*x^8 + 1344*a^3*b^7*x^7 + 2940*a^4
*b^6*x^6 + 4704*a^5*b^5*x^5 + 5880*a^6*b^4*x^4 + 6720*a^7*b^3*x^3 + 2520*a^8*b^2
*x^2*log(x) - 560*a^9*b*x - 28*a^10)/x^2

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Sympy [A]  time = 1.66618, size = 121, normalized size = 1.02 \[ 45 a^{8} b^{2} \log{\left (x \right )} + 120 a^{7} b^{3} x + 105 a^{6} b^{4} x^{2} + 84 a^{5} b^{5} x^{3} + \frac{105 a^{4} b^{6} x^{4}}{2} + 24 a^{3} b^{7} x^{5} + \frac{15 a^{2} b^{8} x^{6}}{2} + \frac{10 a b^{9} x^{7}}{7} + \frac{b^{10} x^{8}}{8} - \frac{a^{10} + 20 a^{9} b x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**3,x)

[Out]

45*a**8*b**2*log(x) + 120*a**7*b**3*x + 105*a**6*b**4*x**2 + 84*a**5*b**5*x**3 +
105*a**4*b**6*x**4/2 + 24*a**3*b**7*x**5 + 15*a**2*b**8*x**6/2 + 10*a*b**9*x**7
/7 + b**10*x**8/8 - (a**10 + 20*a**9*b*x)/(2*x**2)

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GIAC/XCAS [A]  time = 0.205734, size = 147, normalized size = 1.24 \[ \frac{1}{8} \, b^{10} x^{8} + \frac{10}{7} \, a b^{9} x^{7} + \frac{15}{2} \, a^{2} b^{8} x^{6} + 24 \, a^{3} b^{7} x^{5} + \frac{105}{2} \, a^{4} b^{6} x^{4} + 84 \, a^{5} b^{5} x^{3} + 105 \, a^{6} b^{4} x^{2} + 120 \, a^{7} b^{3} x + 45 \, a^{8} b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{20 \, a^{9} b x + a^{10}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^3,x, algorithm="giac")

[Out]

1/8*b^10*x^8 + 10/7*a*b^9*x^7 + 15/2*a^2*b^8*x^6 + 24*a^3*b^7*x^5 + 105/2*a^4*b^
6*x^4 + 84*a^5*b^5*x^3 + 105*a^6*b^4*x^2 + 120*a^7*b^3*x + 45*a^8*b^2*ln(abs(x))
- 1/2*(20*a^9*b*x + a^10)/x^2