3.149 \(\int \frac{(a+b x)^{10}}{x^{15}} \, dx\)

Optimal. Leaf size=76 \[ \frac{b^3 (a+b x)^{11}}{4004 a^4 x^{11}}-\frac{b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac{3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac{(a+b x)^{11}}{14 a x^{14}} \]

[Out]

-(a + b*x)^11/(14*a*x^14) + (3*b*(a + b*x)^11)/(182*a^2*x^13) - (b^2*(a + b*x)^1
1)/(364*a^3*x^12) + (b^3*(a + b*x)^11)/(4004*a^4*x^11)

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Rubi [A]  time = 0.0567551, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b^3 (a+b x)^{11}}{4004 a^4 x^{11}}-\frac{b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac{3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac{(a+b x)^{11}}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^10/x^15,x]

[Out]

-(a + b*x)^11/(14*a*x^14) + (3*b*(a + b*x)^11)/(182*a^2*x^13) - (b^2*(a + b*x)^1
1)/(364*a^3*x^12) + (b^3*(a + b*x)^11)/(4004*a^4*x^11)

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Rubi in Sympy [A]  time = 8.65147, size = 68, normalized size = 0.89 \[ - \frac{\left (a + b x\right )^{11}}{14 a x^{14}} + \frac{3 b \left (a + b x\right )^{11}}{182 a^{2} x^{13}} - \frac{b^{2} \left (a + b x\right )^{11}}{364 a^{3} x^{12}} + \frac{b^{3} \left (a + b x\right )^{11}}{4004 a^{4} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10/x**15,x)

[Out]

-(a + b*x)**11/(14*a*x**14) + 3*b*(a + b*x)**11/(182*a**2*x**13) - b**2*(a + b*x
)**11/(364*a**3*x**12) + b**3*(a + b*x)**11/(4004*a**4*x**11)

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Mathematica [A]  time = 0.0118586, size = 128, normalized size = 1.68 \[ -\frac{a^{10}}{14 x^{14}}-\frac{10 a^9 b}{13 x^{13}}-\frac{15 a^8 b^2}{4 x^{12}}-\frac{120 a^7 b^3}{11 x^{11}}-\frac{21 a^6 b^4}{x^{10}}-\frac{28 a^5 b^5}{x^9}-\frac{105 a^4 b^6}{4 x^8}-\frac{120 a^3 b^7}{7 x^7}-\frac{15 a^2 b^8}{2 x^6}-\frac{2 a b^9}{x^5}-\frac{b^{10}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^10/x^15,x]

[Out]

-a^10/(14*x^14) - (10*a^9*b)/(13*x^13) - (15*a^8*b^2)/(4*x^12) - (120*a^7*b^3)/(
11*x^11) - (21*a^6*b^4)/x^10 - (28*a^5*b^5)/x^9 - (105*a^4*b^6)/(4*x^8) - (120*a
^3*b^7)/(7*x^7) - (15*a^2*b^8)/(2*x^6) - (2*a*b^9)/x^5 - b^10/(4*x^4)

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Maple [A]  time = 0.011, size = 113, normalized size = 1.5 \[ -{\frac{105\,{a}^{4}{b}^{6}}{4\,{x}^{8}}}-{\frac{120\,{a}^{7}{b}^{3}}{11\,{x}^{11}}}-{\frac{120\,{a}^{3}{b}^{7}}{7\,{x}^{7}}}-28\,{\frac{{a}^{5}{b}^{5}}{{x}^{9}}}-{\frac{{a}^{10}}{14\,{x}^{14}}}-2\,{\frac{a{b}^{9}}{{x}^{5}}}-{\frac{10\,{a}^{9}b}{13\,{x}^{13}}}-{\frac{{b}^{10}}{4\,{x}^{4}}}-{\frac{15\,{a}^{8}{b}^{2}}{4\,{x}^{12}}}-21\,{\frac{{a}^{6}{b}^{4}}{{x}^{10}}}-{\frac{15\,{a}^{2}{b}^{8}}{2\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10/x^15,x)

[Out]

-105/4*a^4*b^6/x^8-120/11*a^7*b^3/x^11-120/7*a^3*b^7/x^7-28*a^5*b^5/x^9-1/14*a^1
0/x^14-2*a*b^9/x^5-10/13*a^9*b/x^13-1/4*b^10/x^4-15/4*a^8*b^2/x^12-21*a^6*b^4/x^
10-15/2*a^2*b^8/x^6

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Maxima [A]  time = 1.34871, size = 151, normalized size = 1.99 \[ -\frac{1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^15,x, algorithm="maxima")

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7
+ 105105*a^4*b^6*x^6 + 112112*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x
^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14

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Fricas [A]  time = 0.184921, size = 151, normalized size = 1.99 \[ -\frac{1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^15,x, algorithm="fricas")

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7
+ 105105*a^4*b^6*x^6 + 112112*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x
^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14

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Sympy [A]  time = 4.36319, size = 121, normalized size = 1.59 \[ - \frac{286 a^{10} + 3080 a^{9} b x + 15015 a^{8} b^{2} x^{2} + 43680 a^{7} b^{3} x^{3} + 84084 a^{6} b^{4} x^{4} + 112112 a^{5} b^{5} x^{5} + 105105 a^{4} b^{6} x^{6} + 68640 a^{3} b^{7} x^{7} + 30030 a^{2} b^{8} x^{8} + 8008 a b^{9} x^{9} + 1001 b^{10} x^{10}}{4004 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10/x**15,x)

[Out]

-(286*a**10 + 3080*a**9*b*x + 15015*a**8*b**2*x**2 + 43680*a**7*b**3*x**3 + 8408
4*a**6*b**4*x**4 + 112112*a**5*b**5*x**5 + 105105*a**4*b**6*x**6 + 68640*a**3*b*
*7*x**7 + 30030*a**2*b**8*x**8 + 8008*a*b**9*x**9 + 1001*b**10*x**10)/(4004*x**1
4)

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GIAC/XCAS [A]  time = 0.214585, size = 151, normalized size = 1.99 \[ -\frac{1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10/x^15,x, algorithm="giac")

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7
+ 105105*a^4*b^6*x^6 + 112112*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x
^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14