3.141 \(\int \frac{(a+b x)^{10}}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.117378, 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 \[ -\frac{a^{10}}{6 x^6}-\frac{2 a^9 b}{x^5}-\frac{45 a^8 b^2}{4 x^4}-\frac{40 a^7 b^3}{x^3}-\frac{105 a^6 b^4}{x^2}-\frac{252 a^5 b^5}{x}+210 a^4 b^6 \log (x)+120 a^3 b^7 x+\frac{45}{2} a^2 b^8 x^2+\frac{10}{3} a b^9 x^3+\frac{b^{10} x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^7,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{6 x^{6}} - \frac{2 a^{9} b}{x^{5}} - \frac{45 a^{8} b^{2}}{4 x^{4}} - \frac{40 a^{7} b^{3}}{x^{3}} - \frac{105 a^{6} b^{4}}{x^{2}} - \frac{252 a^{5} b^{5}}{x} + 210 a^{4} b^{6} \log{\left (x \right )} + 120 a^{3} b^{7} x + 45 a^{2} b^{8} \int x\, dx + \frac{10 a b^{9} x^{3}}{3} + \frac{b^{10} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.00694907, size = 119, normalized size = 1. \[ -\frac{a^{10}}{6 x^6}-\frac{2 a^9 b}{x^5}-\frac{45 a^8 b^2}{4 x^4}-\frac{40 a^7 b^3}{x^3}-\frac{105 a^6 b^4}{x^2}-\frac{252 a^5 b^5}{x}+210 a^4 b^6 \log (x)+120 a^3 b^7 x+\frac{45}{2} a^2 b^8 x^2+\frac{10}{3} a b^9 x^3+\frac{b^{10} x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^7,x)

[Out]

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

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Maxima [A]  time = 1.48716, size = 149, normalized size = 1.25 \[ \frac{1}{4} \, b^{10} x^{4} + \frac{10}{3} \, a b^{9} x^{3} + \frac{45}{2} \, a^{2} b^{8} x^{2} + 120 \, a^{3} b^{7} x + 210 \, a^{4} b^{6} \log \left (x\right ) - \frac{3024 \, a^{5} b^{5} x^{5} + 1260 \, a^{6} b^{4} x^{4} + 480 \, a^{7} b^{3} x^{3} + 135 \, a^{8} b^{2} x^{2} + 24 \, a^{9} b x + 2 \, a^{10}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^7,x, algorithm="maxima")

[Out]

1/4*b^10*x^4 + 10/3*a*b^9*x^3 + 45/2*a^2*b^8*x^2 + 120*a^3*b^7*x + 210*a^4*b^6*l
og(x) - 1/12*(3024*a^5*b^5*x^5 + 1260*a^6*b^4*x^4 + 480*a^7*b^3*x^3 + 135*a^8*b^
2*x^2 + 24*a^9*b*x + 2*a^10)/x^6

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Fricas [A]  time = 0.194578, size = 154, normalized size = 1.29 \[ \frac{3 \, b^{10} x^{10} + 40 \, a b^{9} x^{9} + 270 \, a^{2} b^{8} x^{8} + 1440 \, a^{3} b^{7} x^{7} + 2520 \, a^{4} b^{6} x^{6} \log \left (x\right ) - 3024 \, a^{5} b^{5} x^{5} - 1260 \, a^{6} b^{4} x^{4} - 480 \, a^{7} b^{3} x^{3} - 135 \, a^{8} b^{2} x^{2} - 24 \, a^{9} b x - 2 \, a^{10}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.47318, size = 121, normalized size = 1.02 \[ 210 a^{4} b^{6} \log{\left (x \right )} + 120 a^{3} b^{7} x + \frac{45 a^{2} b^{8} x^{2}}{2} + \frac{10 a b^{9} x^{3}}{3} + \frac{b^{10} x^{4}}{4} - \frac{2 a^{10} + 24 a^{9} b x + 135 a^{8} b^{2} x^{2} + 480 a^{7} b^{3} x^{3} + 1260 a^{6} b^{4} x^{4} + 3024 a^{5} b^{5} x^{5}}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**7,x)

[Out]

210*a**4*b**6*log(x) + 120*a**3*b**7*x + 45*a**2*b**8*x**2/2 + 10*a*b**9*x**3/3
+ b**10*x**4/4 - (2*a**10 + 24*a**9*b*x + 135*a**8*b**2*x**2 + 480*a**7*b**3*x**
3 + 1260*a**6*b**4*x**4 + 3024*a**5*b**5*x**5)/(12*x**6)

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GIAC/XCAS [A]  time = 0.203111, size = 150, normalized size = 1.26 \[ \frac{1}{4} \, b^{10} x^{4} + \frac{10}{3} \, a b^{9} x^{3} + \frac{45}{2} \, a^{2} b^{8} x^{2} + 120 \, a^{3} b^{7} x + 210 \, a^{4} b^{6}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3024 \, a^{5} b^{5} x^{5} + 1260 \, a^{6} b^{4} x^{4} + 480 \, a^{7} b^{3} x^{3} + 135 \, a^{8} b^{2} x^{2} + 24 \, a^{9} b x + 2 \, a^{10}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^7,x, algorithm="giac")

[Out]

1/4*b^10*x^4 + 10/3*a*b^9*x^3 + 45/2*a^2*b^8*x^2 + 120*a^3*b^7*x + 210*a^4*b^6*l
n(abs(x)) - 1/12*(3024*a^5*b^5*x^5 + 1260*a^6*b^4*x^4 + 480*a^7*b^3*x^3 + 135*a^
8*b^2*x^2 + 24*a^9*b*x + 2*a^10)/x^6