3.147 \(\int \frac{(a+b x)^{10}}{x^{13}} \, dx\)

Optimal. Leaf size=36 \[ \frac{b (a+b x)^{11}}{132 a^2 x^{11}}-\frac{(a+b x)^{11}}{12 a x^{12}} \]

[Out]

-(a + b*x)^11/(12*a*x^12) + (b*(a + b*x)^11)/(132*a^2*x^11)

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Rubi [A]  time = 0.025199, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b (a+b x)^{11}}{132 a^2 x^{11}}-\frac{(a+b x)^{11}}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^13,x]

[Out]

-(a + b*x)^11/(12*a*x^12) + (b*(a + b*x)^11)/(132*a^2*x^11)

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Rubi in Sympy [A]  time = 3.57084, size = 29, normalized size = 0.81 \[ - \frac{\left (a + b x\right )^{11}}{12 a x^{12}} + \frac{b \left (a + b x\right )^{11}}{132 a^{2} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**13,x)

[Out]

-(a + b*x)**11/(12*a*x**12) + b*(a + b*x)**11/(132*a**2*x**11)

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Mathematica [B]  time = 0.00648062, size = 128, normalized size = 3.56 \[ -\frac{a^{10}}{12 x^{12}}-\frac{10 a^9 b}{11 x^{11}}-\frac{9 a^8 b^2}{2 x^{10}}-\frac{40 a^7 b^3}{3 x^9}-\frac{105 a^6 b^4}{4 x^8}-\frac{36 a^5 b^5}{x^7}-\frac{35 a^4 b^6}{x^6}-\frac{24 a^3 b^7}{x^5}-\frac{45 a^2 b^8}{4 x^4}-\frac{10 a b^9}{3 x^3}-\frac{b^{10}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^13,x]

[Out]

-a^10/(12*x^12) - (10*a^9*b)/(11*x^11) - (9*a^8*b^2)/(2*x^10) - (40*a^7*b^3)/(3*
x^9) - (105*a^6*b^4)/(4*x^8) - (36*a^5*b^5)/x^7 - (35*a^4*b^6)/x^6 - (24*a^3*b^7
)/x^5 - (45*a^2*b^8)/(4*x^4) - (10*a*b^9)/(3*x^3) - b^10/(2*x^2)

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Maple [B]  time = 0.008, size = 113, normalized size = 3.1 \[ -{\frac{105\,{a}^{6}{b}^{4}}{4\,{x}^{8}}}-36\,{\frac{{a}^{5}{b}^{5}}{{x}^{7}}}-{\frac{40\,{a}^{7}{b}^{3}}{3\,{x}^{9}}}-{\frac{{b}^{10}}{2\,{x}^{2}}}-{\frac{{a}^{10}}{12\,{x}^{12}}}-{\frac{9\,{a}^{8}{b}^{2}}{2\,{x}^{10}}}-24\,{\frac{{a}^{3}{b}^{7}}{{x}^{5}}}-{\frac{10\,a{b}^{9}}{3\,{x}^{3}}}-{\frac{10\,{a}^{9}b}{11\,{x}^{11}}}-{\frac{45\,{a}^{2}{b}^{8}}{4\,{x}^{4}}}-35\,{\frac{{a}^{4}{b}^{6}}{{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^13,x)

[Out]

-105/4*a^6*b^4/x^8-36*a^5*b^5/x^7-40/3*a^7*b^3/x^9-1/2*b^10/x^2-1/12*a^10/x^12-9
/2*a^8*b^2/x^10-24*a^3*b^7/x^5-10/3*a*b^9/x^3-10/11*a^9*b/x^11-45/4*a^2*b^8/x^4-
35*a^4*b^6/x^6

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Maxima [A]  time = 1.35778, size = 151, normalized size = 4.19 \[ -\frac{66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^13,x, algorithm="maxima")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 462
0*a^4*b^6*x^6 + 4752*a^5*b^5*x^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8
*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12

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Fricas [A]  time = 0.192872, size = 151, normalized size = 4.19 \[ -\frac{66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^13,x, algorithm="fricas")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 462
0*a^4*b^6*x^6 + 4752*a^5*b^5*x^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8
*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12

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Sympy [A]  time = 3.9634, size = 121, normalized size = 3.36 \[ - \frac{11 a^{10} + 120 a^{9} b x + 594 a^{8} b^{2} x^{2} + 1760 a^{7} b^{3} x^{3} + 3465 a^{6} b^{4} x^{4} + 4752 a^{5} b^{5} x^{5} + 4620 a^{4} b^{6} x^{6} + 3168 a^{3} b^{7} x^{7} + 1485 a^{2} b^{8} x^{8} + 440 a b^{9} x^{9} + 66 b^{10} x^{10}}{132 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**13,x)

[Out]

-(11*a**10 + 120*a**9*b*x + 594*a**8*b**2*x**2 + 1760*a**7*b**3*x**3 + 3465*a**6
*b**4*x**4 + 4752*a**5*b**5*x**5 + 4620*a**4*b**6*x**6 + 3168*a**3*b**7*x**7 + 1
485*a**2*b**8*x**8 + 440*a*b**9*x**9 + 66*b**10*x**10)/(132*x**12)

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GIAC/XCAS [A]  time = 0.213768, size = 151, normalized size = 4.19 \[ -\frac{66 \, b^{10} x^{10} + 440 \, a b^{9} x^{9} + 1485 \, a^{2} b^{8} x^{8} + 3168 \, a^{3} b^{7} x^{7} + 4620 \, a^{4} b^{6} x^{6} + 4752 \, a^{5} b^{5} x^{5} + 3465 \, a^{6} b^{4} x^{4} + 1760 \, a^{7} b^{3} x^{3} + 594 \, a^{8} b^{2} x^{2} + 120 \, a^{9} b x + 11 \, a^{10}}{132 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^13,x, algorithm="giac")

[Out]

-1/132*(66*b^10*x^10 + 440*a*b^9*x^9 + 1485*a^2*b^8*x^8 + 3168*a^3*b^7*x^7 + 462
0*a^4*b^6*x^6 + 4752*a^5*b^5*x^5 + 3465*a^6*b^4*x^4 + 1760*a^7*b^3*x^3 + 594*a^8
*b^2*x^2 + 120*a^9*b*x + 11*a^10)/x^12