3.138 \(\int \frac{(a+b x)^{10}}{x^4} \, dx\)

Optimal. Leaf size=115 \[ -\frac{a^{10}}{3 x^3}-\frac{5 a^9 b}{x^2}-\frac{45 a^8 b^2}{x}+120 a^7 b^3 \log (x)+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac{5}{3} a b^9 x^6+\frac{b^{10} x^7}{7} \]

[Out]

-a^10/(3*x^3) - (5*a^9*b)/x^2 - (45*a^8*b^2)/x + 210*a^6*b^4*x + 126*a^5*b^5*x^2
+ 70*a^4*b^6*x^3 + 30*a^3*b^7*x^4 + 9*a^2*b^8*x^5 + (5*a*b^9*x^6)/3 + (b^10*x^7
)/7 + 120*a^7*b^3*Log[x]

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Rubi [A]  time = 0.117369, antiderivative size = 115, 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 \[ -\frac{a^{10}}{3 x^3}-\frac{5 a^9 b}{x^2}-\frac{45 a^8 b^2}{x}+120 a^7 b^3 \log (x)+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac{5}{3} a b^9 x^6+\frac{b^{10} x^7}{7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^4,x]

[Out]

-a^10/(3*x^3) - (5*a^9*b)/x^2 - (45*a^8*b^2)/x + 210*a^6*b^4*x + 126*a^5*b^5*x^2
+ 70*a^4*b^6*x^3 + 30*a^3*b^7*x^4 + 9*a^2*b^8*x^5 + (5*a*b^9*x^6)/3 + (b^10*x^7
)/7 + 120*a^7*b^3*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{10}}{3 x^{3}} - \frac{5 a^{9} b}{x^{2}} - \frac{45 a^{8} b^{2}}{x} + 120 a^{7} b^{3} \log{\left (x \right )} + 210 a^{6} b^{4} x + 252 a^{5} b^{5} \int x\, dx + 70 a^{4} b^{6} x^{3} + 30 a^{3} b^{7} x^{4} + 9 a^{2} b^{8} x^{5} + \frac{5 a b^{9} x^{6}}{3} + \frac{b^{10} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**4,x)

[Out]

-a**10/(3*x**3) - 5*a**9*b/x**2 - 45*a**8*b**2/x + 120*a**7*b**3*log(x) + 210*a*
*6*b**4*x + 252*a**5*b**5*Integral(x, x) + 70*a**4*b**6*x**3 + 30*a**3*b**7*x**4
+ 9*a**2*b**8*x**5 + 5*a*b**9*x**6/3 + b**10*x**7/7

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Mathematica [A]  time = 0.0214888, size = 115, normalized size = 1. \[ -\frac{a^{10}}{3 x^3}-\frac{5 a^9 b}{x^2}-\frac{45 a^8 b^2}{x}+120 a^7 b^3 \log (x)+210 a^6 b^4 x+126 a^5 b^5 x^2+70 a^4 b^6 x^3+30 a^3 b^7 x^4+9 a^2 b^8 x^5+\frac{5}{3} a b^9 x^6+\frac{b^{10} x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^4,x]

[Out]

-a^10/(3*x^3) - (5*a^9*b)/x^2 - (45*a^8*b^2)/x + 210*a^6*b^4*x + 126*a^5*b^5*x^2
+ 70*a^4*b^6*x^3 + 30*a^3*b^7*x^4 + 9*a^2*b^8*x^5 + (5*a*b^9*x^6)/3 + (b^10*x^7
)/7 + 120*a^7*b^3*Log[x]

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Maple [A]  time = 0.008, size = 110, normalized size = 1. \[ -{\frac{{a}^{10}}{3\,{x}^{3}}}-5\,{\frac{{a}^{9}b}{{x}^{2}}}-45\,{\frac{{a}^{8}{b}^{2}}{x}}+210\,{a}^{6}{b}^{4}x+126\,{a}^{5}{b}^{5}{x}^{2}+70\,{a}^{4}{b}^{6}{x}^{3}+30\,{a}^{3}{b}^{7}{x}^{4}+9\,{a}^{2}{b}^{8}{x}^{5}+{\frac{5\,a{b}^{9}{x}^{6}}{3}}+{\frac{{b}^{10}{x}^{7}}{7}}+120\,{a}^{7}{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^4,x)

[Out]

-1/3*a^10/x^3-5*a^9*b/x^2-45*a^8*b^2/x+210*a^6*b^4*x+126*a^5*b^5*x^2+70*a^4*b^6*
x^3+30*a^3*b^7*x^4+9*a^2*b^8*x^5+5/3*a*b^9*x^6+1/7*b^10*x^7+120*a^7*b^3*ln(x)

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Maxima [A]  time = 1.36017, size = 146, normalized size = 1.27 \[ \frac{1}{7} \, b^{10} x^{7} + \frac{5}{3} \, a b^{9} x^{6} + 9 \, a^{2} b^{8} x^{5} + 30 \, a^{3} b^{7} x^{4} + 70 \, a^{4} b^{6} x^{3} + 126 \, a^{5} b^{5} x^{2} + 210 \, a^{6} b^{4} x + 120 \, a^{7} b^{3} \log \left (x\right ) - \frac{135 \, a^{8} b^{2} x^{2} + 15 \, a^{9} b x + a^{10}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^4,x, algorithm="maxima")

[Out]

1/7*b^10*x^7 + 5/3*a*b^9*x^6 + 9*a^2*b^8*x^5 + 30*a^3*b^7*x^4 + 70*a^4*b^6*x^3 +
126*a^5*b^5*x^2 + 210*a^6*b^4*x + 120*a^7*b^3*log(x) - 1/3*(135*a^8*b^2*x^2 + 1
5*a^9*b*x + a^10)/x^3

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Fricas [A]  time = 0.196924, size = 154, normalized size = 1.34 \[ \frac{3 \, b^{10} x^{10} + 35 \, a b^{9} x^{9} + 189 \, a^{2} b^{8} x^{8} + 630 \, a^{3} b^{7} x^{7} + 1470 \, a^{4} b^{6} x^{6} + 2646 \, a^{5} b^{5} x^{5} + 4410 \, a^{6} b^{4} x^{4} + 2520 \, a^{7} b^{3} x^{3} \log \left (x\right ) - 945 \, a^{8} b^{2} x^{2} - 105 \, a^{9} b x - 7 \, a^{10}}{21 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^4,x, algorithm="fricas")

[Out]

1/21*(3*b^10*x^10 + 35*a*b^9*x^9 + 189*a^2*b^8*x^8 + 630*a^3*b^7*x^7 + 1470*a^4*
b^6*x^6 + 2646*a^5*b^5*x^5 + 4410*a^6*b^4*x^4 + 2520*a^7*b^3*x^3*log(x) - 945*a^
8*b^2*x^2 - 105*a^9*b*x - 7*a^10)/x^3

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Sympy [A]  time = 1.79672, size = 117, normalized size = 1.02 \[ 120 a^{7} b^{3} \log{\left (x \right )} + 210 a^{6} b^{4} x + 126 a^{5} b^{5} x^{2} + 70 a^{4} b^{6} x^{3} + 30 a^{3} b^{7} x^{4} + 9 a^{2} b^{8} x^{5} + \frac{5 a b^{9} x^{6}}{3} + \frac{b^{10} x^{7}}{7} - \frac{a^{10} + 15 a^{9} b x + 135 a^{8} b^{2} x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**4,x)

[Out]

120*a**7*b**3*log(x) + 210*a**6*b**4*x + 126*a**5*b**5*x**2 + 70*a**4*b**6*x**3
+ 30*a**3*b**7*x**4 + 9*a**2*b**8*x**5 + 5*a*b**9*x**6/3 + b**10*x**7/7 - (a**10
+ 15*a**9*b*x + 135*a**8*b**2*x**2)/(3*x**3)

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GIAC/XCAS [A]  time = 0.205351, size = 147, normalized size = 1.28 \[ \frac{1}{7} \, b^{10} x^{7} + \frac{5}{3} \, a b^{9} x^{6} + 9 \, a^{2} b^{8} x^{5} + 30 \, a^{3} b^{7} x^{4} + 70 \, a^{4} b^{6} x^{3} + 126 \, a^{5} b^{5} x^{2} + 210 \, a^{6} b^{4} x + 120 \, a^{7} b^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{135 \, a^{8} b^{2} x^{2} + 15 \, a^{9} b x + a^{10}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^4,x, algorithm="giac")

[Out]

1/7*b^10*x^7 + 5/3*a*b^9*x^6 + 9*a^2*b^8*x^5 + 30*a^3*b^7*x^4 + 70*a^4*b^6*x^3 +
126*a^5*b^5*x^2 + 210*a^6*b^4*x + 120*a^7*b^3*ln(abs(x)) - 1/3*(135*a^8*b^2*x^2
+ 15*a^9*b*x + a^10)/x^3