3.118 \(\int \frac{(a+b x)^7}{x^{12}} \, dx\)

Optimal. Leaf size=76 \[ \frac{b^3 (a+b x)^8}{1320 a^4 x^8}-\frac{b^2 (a+b x)^8}{165 a^3 x^9}+\frac{3 b (a+b x)^8}{110 a^2 x^{10}}-\frac{(a+b x)^8}{11 a x^{11}} \]

[Out]

-(a + b*x)^8/(11*a*x^11) + (3*b*(a + b*x)^8)/(110*a^2*x^10) - (b^2*(a + b*x)^8)/
(165*a^3*x^9) + (b^3*(a + b*x)^8)/(1320*a^4*x^8)

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Rubi [A]  time = 0.0561874, 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 \[ \frac{b^3 (a+b x)^8}{1320 a^4 x^8}-\frac{b^2 (a+b x)^8}{165 a^3 x^9}+\frac{3 b (a+b x)^8}{110 a^2 x^{10}}-\frac{(a+b x)^8}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/x^12,x]

[Out]

-(a + b*x)^8/(11*a*x^11) + (3*b*(a + b*x)^8)/(110*a^2*x^10) - (b^2*(a + b*x)^8)/
(165*a^3*x^9) + (b^3*(a + b*x)^8)/(1320*a^4*x^8)

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Rubi in Sympy [A]  time = 8.65817, size = 68, normalized size = 0.89 \[ - \frac{\left (a + b x\right )^{8}}{11 a x^{11}} + \frac{3 b \left (a + b x\right )^{8}}{110 a^{2} x^{10}} - \frac{b^{2} \left (a + b x\right )^{8}}{165 a^{3} x^{9}} + \frac{b^{3} \left (a + b x\right )^{8}}{1320 a^{4} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/x**12,x)

[Out]

-(a + b*x)**8/(11*a*x**11) + 3*b*(a + b*x)**8/(110*a**2*x**10) - b**2*(a + b*x)*
*8/(165*a**3*x**9) + b**3*(a + b*x)**8/(1320*a**4*x**8)

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Mathematica [A]  time = 0.00543107, size = 93, normalized size = 1.22 \[ -\frac{a^7}{11 x^{11}}-\frac{7 a^6 b}{10 x^{10}}-\frac{7 a^5 b^2}{3 x^9}-\frac{35 a^4 b^3}{8 x^8}-\frac{5 a^3 b^4}{x^7}-\frac{7 a^2 b^5}{2 x^6}-\frac{7 a b^6}{5 x^5}-\frac{b^7}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/x^12,x]

[Out]

-a^7/(11*x^11) - (7*a^6*b)/(10*x^10) - (7*a^5*b^2)/(3*x^9) - (35*a^4*b^3)/(8*x^8
) - (5*a^3*b^4)/x^7 - (7*a^2*b^5)/(2*x^6) - (7*a*b^6)/(5*x^5) - b^7/(4*x^4)

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Maple [A]  time = 0.01, size = 80, normalized size = 1.1 \[ -{\frac{7\,a{b}^{6}}{5\,{x}^{5}}}-{\frac{7\,{a}^{6}b}{10\,{x}^{10}}}-{\frac{7\,{a}^{5}{b}^{2}}{3\,{x}^{9}}}-5\,{\frac{{a}^{3}{b}^{4}}{{x}^{7}}}-{\frac{35\,{a}^{4}{b}^{3}}{8\,{x}^{8}}}-{\frac{{a}^{7}}{11\,{x}^{11}}}-{\frac{{b}^{7}}{4\,{x}^{4}}}-{\frac{7\,{a}^{2}{b}^{5}}{2\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/x^12,x)

[Out]

-7/5*a*b^6/x^5-7/10*a^6*b/x^10-7/3*a^5*b^2/x^9-5*a^3*b^4/x^7-35/8*a^4*b^3/x^8-1/
11*a^7/x^11-1/4*b^7/x^4-7/2*a^2*b^5/x^6

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Maxima [A]  time = 1.33803, size = 107, normalized size = 1.41 \[ -\frac{330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^12,x, algorithm="maxima")

[Out]

-1/1320*(330*b^7*x^7 + 1848*a*b^6*x^6 + 4620*a^2*b^5*x^5 + 6600*a^3*b^4*x^4 + 57
75*a^4*b^3*x^3 + 3080*a^5*b^2*x^2 + 924*a^6*b*x + 120*a^7)/x^11

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Fricas [A]  time = 0.190174, size = 107, normalized size = 1.41 \[ -\frac{330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^12,x, algorithm="fricas")

[Out]

-1/1320*(330*b^7*x^7 + 1848*a*b^6*x^6 + 4620*a^2*b^5*x^5 + 6600*a^3*b^4*x^4 + 57
75*a^4*b^3*x^3 + 3080*a^5*b^2*x^2 + 924*a^6*b*x + 120*a^7)/x^11

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Sympy [A]  time = 2.98672, size = 85, normalized size = 1.12 \[ - \frac{120 a^{7} + 924 a^{6} b x + 3080 a^{5} b^{2} x^{2} + 5775 a^{4} b^{3} x^{3} + 6600 a^{3} b^{4} x^{4} + 4620 a^{2} b^{5} x^{5} + 1848 a b^{6} x^{6} + 330 b^{7} x^{7}}{1320 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/x**12,x)

[Out]

-(120*a**7 + 924*a**6*b*x + 3080*a**5*b**2*x**2 + 5775*a**4*b**3*x**3 + 6600*a**
3*b**4*x**4 + 4620*a**2*b**5*x**5 + 1848*a*b**6*x**6 + 330*b**7*x**7)/(1320*x**1
1)

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GIAC/XCAS [A]  time = 0.216855, size = 107, normalized size = 1.41 \[ -\frac{330 \, b^{7} x^{7} + 1848 \, a b^{6} x^{6} + 4620 \, a^{2} b^{5} x^{5} + 6600 \, a^{3} b^{4} x^{4} + 5775 \, a^{4} b^{3} x^{3} + 3080 \, a^{5} b^{2} x^{2} + 924 \, a^{6} b x + 120 \, a^{7}}{1320 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^12,x, algorithm="giac")

[Out]

-1/1320*(330*b^7*x^7 + 1848*a*b^6*x^6 + 4620*a^2*b^5*x^5 + 6600*a^3*b^4*x^4 + 57
75*a^4*b^3*x^3 + 3080*a^5*b^2*x^2 + 924*a^6*b*x + 120*a^7)/x^11