3.245 \(\int \frac{(a+b x)^6}{x^{10}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{b^2 (a+b x)^7}{252 a^3 x^7}+\frac{b (a+b x)^7}{36 a^2 x^8}-\frac{(a+b x)^7}{9 a x^9} \]

[Out]

-(a + b*x)^7/(9*a*x^9) + (b*(a + b*x)^7)/(36*a^2*x^8) - (b^2*(a + b*x)^7)/(252*a
^3*x^7)

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Rubi [A]  time = 0.0415242, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^2 (a+b x)^7}{252 a^3 x^7}+\frac{b (a+b x)^7}{36 a^2 x^8}-\frac{(a+b x)^7}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^6/x^10,x]

[Out]

-(a + b*x)^7/(9*a*x^9) + (b*(a + b*x)^7)/(36*a^2*x^8) - (b^2*(a + b*x)^7)/(252*a
^3*x^7)

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Rubi in Sympy [A]  time = 5.80555, size = 48, normalized size = 0.86 \[ - \frac{\left (a + b x\right )^{7}}{9 a x^{9}} + \frac{b \left (a + b x\right )^{7}}{36 a^{2} x^{8}} - \frac{b^{2} \left (a + b x\right )^{7}}{252 a^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6/x**10,x)

[Out]

-(a + b*x)**7/(9*a*x**9) + b*(a + b*x)**7/(36*a**2*x**8) - b**2*(a + b*x)**7/(25
2*a**3*x**7)

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Mathematica [A]  time = 0.0133087, size = 80, normalized size = 1.43 \[ -\frac{a^6}{9 x^9}-\frac{3 a^5 b}{4 x^8}-\frac{15 a^4 b^2}{7 x^7}-\frac{10 a^3 b^3}{3 x^6}-\frac{3 a^2 b^4}{x^5}-\frac{3 a b^5}{2 x^4}-\frac{b^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^6/x^10,x]

[Out]

-a^6/(9*x^9) - (3*a^5*b)/(4*x^8) - (15*a^4*b^2)/(7*x^7) - (10*a^3*b^3)/(3*x^6) -
(3*a^2*b^4)/x^5 - (3*a*b^5)/(2*x^4) - b^6/(3*x^3)

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Maple [A]  time = 0.01, size = 69, normalized size = 1.2 \[ -{\frac{3\,{a}^{5}b}{4\,{x}^{8}}}-{\frac{15\,{a}^{4}{b}^{2}}{7\,{x}^{7}}}-{\frac{{a}^{6}}{9\,{x}^{9}}}-3\,{\frac{{a}^{2}{b}^{4}}{{x}^{5}}}-{\frac{{b}^{6}}{3\,{x}^{3}}}-{\frac{3\,a{b}^{5}}{2\,{x}^{4}}}-{\frac{10\,{a}^{3}{b}^{3}}{3\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6/x^10,x)

[Out]

-3/4*a^5*b/x^8-15/7*a^4*b^2/x^7-1/9*a^6/x^9-3*a^2*b^4/x^5-1/3*b^6/x^3-3/2*a*b^5/
x^4-10/3*a^3*b^3/x^6

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Maxima [A]  time = 1.34302, size = 92, normalized size = 1.64 \[ -\frac{84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/x^10,x, algorithm="maxima")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4
*b^2*x^2 + 189*a^5*b*x + 28*a^6)/x^9

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Fricas [A]  time = 0.199422, size = 92, normalized size = 1.64 \[ -\frac{84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/x^10,x, algorithm="fricas")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4
*b^2*x^2 + 189*a^5*b*x + 28*a^6)/x^9

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Sympy [A]  time = 2.39614, size = 73, normalized size = 1.3 \[ - \frac{28 a^{6} + 189 a^{5} b x + 540 a^{4} b^{2} x^{2} + 840 a^{3} b^{3} x^{3} + 756 a^{2} b^{4} x^{4} + 378 a b^{5} x^{5} + 84 b^{6} x^{6}}{252 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**6/x**10,x)

[Out]

-(28*a**6 + 189*a**5*b*x + 540*a**4*b**2*x**2 + 840*a**3*b**3*x**3 + 756*a**2*b*
*4*x**4 + 378*a*b**5*x**5 + 84*b**6*x**6)/(252*x**9)

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GIAC/XCAS [A]  time = 0.20192, size = 92, normalized size = 1.64 \[ -\frac{84 \, b^{6} x^{6} + 378 \, a b^{5} x^{5} + 756 \, a^{2} b^{4} x^{4} + 840 \, a^{3} b^{3} x^{3} + 540 \, a^{4} b^{2} x^{2} + 189 \, a^{5} b x + 28 \, a^{6}}{252 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^6/x^10,x, algorithm="giac")

[Out]

-1/252*(84*b^6*x^6 + 378*a*b^5*x^5 + 756*a^2*b^4*x^4 + 840*a^3*b^3*x^3 + 540*a^4
*b^2*x^2 + 189*a^5*b*x + 28*a^6)/x^9