3.119 \(\int \frac{(a+b x)^7}{x^{13}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{b^4 (a+b x)^8}{3960 a^5 x^8}+\frac{b^3 (a+b x)^8}{495 a^4 x^9}-\frac{b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac{b (a+b x)^8}{33 a^2 x^{11}}-\frac{(a+b x)^8}{12 a x^{12}} \]

[Out]

-(a + b*x)^8/(12*a*x^12) + (b*(a + b*x)^8)/(33*a^2*x^11) - (b^2*(a + b*x)^8)/(11
0*a^3*x^10) + (b^3*(a + b*x)^8)/(495*a^4*x^9) - (b^4*(a + b*x)^8)/(3960*a^5*x^8)

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Rubi [A]  time = 0.0757435, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^4 (a+b x)^8}{3960 a^5 x^8}+\frac{b^3 (a+b x)^8}{495 a^4 x^9}-\frac{b^2 (a+b x)^8}{110 a^3 x^{10}}+\frac{b (a+b x)^8}{33 a^2 x^{11}}-\frac{(a+b x)^8}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/x^13,x]

[Out]

-(a + b*x)^8/(12*a*x^12) + (b*(a + b*x)^8)/(33*a^2*x^11) - (b^2*(a + b*x)^8)/(11
0*a^3*x^10) + (b^3*(a + b*x)^8)/(495*a^4*x^9) - (b^4*(a + b*x)^8)/(3960*a^5*x^8)

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Rubi in Sympy [A]  time = 15.3769, size = 94, normalized size = 0.98 \[ - \frac{a^{7}}{12 x^{12}} - \frac{7 a^{6} b}{11 x^{11}} - \frac{21 a^{5} b^{2}}{10 x^{10}} - \frac{35 a^{4} b^{3}}{9 x^{9}} - \frac{35 a^{3} b^{4}}{8 x^{8}} - \frac{3 a^{2} b^{5}}{x^{7}} - \frac{7 a b^{6}}{6 x^{6}} - \frac{b^{7}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/x**13,x)

[Out]

-a**7/(12*x**12) - 7*a**6*b/(11*x**11) - 21*a**5*b**2/(10*x**10) - 35*a**4*b**3/
(9*x**9) - 35*a**3*b**4/(8*x**8) - 3*a**2*b**5/x**7 - 7*a*b**6/(6*x**6) - b**7/(
5*x**5)

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Mathematica [A]  time = 0.00557122, size = 93, normalized size = 0.97 \[ -\frac{a^7}{12 x^{12}}-\frac{7 a^6 b}{11 x^{11}}-\frac{21 a^5 b^2}{10 x^{10}}-\frac{35 a^4 b^3}{9 x^9}-\frac{35 a^3 b^4}{8 x^8}-\frac{3 a^2 b^5}{x^7}-\frac{7 a b^6}{6 x^6}-\frac{b^7}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/x^13,x]

[Out]

-a^7/(12*x^12) - (7*a^6*b)/(11*x^11) - (21*a^5*b^2)/(10*x^10) - (35*a^4*b^3)/(9*
x^9) - (35*a^3*b^4)/(8*x^8) - (3*a^2*b^5)/x^7 - (7*a*b^6)/(6*x^6) - b^7/(5*x^5)

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Maple [A]  time = 0.01, size = 80, normalized size = 0.8 \[ -{\frac{35\,{a}^{4}{b}^{3}}{9\,{x}^{9}}}-3\,{\frac{{a}^{2}{b}^{5}}{{x}^{7}}}-{\frac{{b}^{7}}{5\,{x}^{5}}}-{\frac{7\,{a}^{6}b}{11\,{x}^{11}}}-{\frac{35\,{a}^{3}{b}^{4}}{8\,{x}^{8}}}-{\frac{21\,{a}^{5}{b}^{2}}{10\,{x}^{10}}}-{\frac{{a}^{7}}{12\,{x}^{12}}}-{\frac{7\,a{b}^{6}}{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/x^13,x)

[Out]

-35/9*a^4*b^3/x^9-3*a^2*b^5/x^7-1/5*b^7/x^5-7/11*a^6*b/x^11-35/8*a^3*b^4/x^8-21/
10*a^5*b^2/x^10-1/12*a^7/x^12-7/6*a*b^6/x^6

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Maxima [A]  time = 1.34628, size = 107, normalized size = 1.11 \[ -\frac{792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^13,x, algorithm="maxima")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 +
15400*a^4*b^3*x^3 + 8316*a^5*b^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12

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Fricas [A]  time = 0.186837, size = 107, normalized size = 1.11 \[ -\frac{792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^13,x, algorithm="fricas")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 +
15400*a^4*b^3*x^3 + 8316*a^5*b^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12

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Sympy [A]  time = 3.09676, size = 85, normalized size = 0.89 \[ - \frac{330 a^{7} + 2520 a^{6} b x + 8316 a^{5} b^{2} x^{2} + 15400 a^{4} b^{3} x^{3} + 17325 a^{3} b^{4} x^{4} + 11880 a^{2} b^{5} x^{5} + 4620 a b^{6} x^{6} + 792 b^{7} x^{7}}{3960 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/x**13,x)

[Out]

-(330*a**7 + 2520*a**6*b*x + 8316*a**5*b**2*x**2 + 15400*a**4*b**3*x**3 + 17325*
a**3*b**4*x**4 + 11880*a**2*b**5*x**5 + 4620*a*b**6*x**6 + 792*b**7*x**7)/(3960*
x**12)

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GIAC/XCAS [A]  time = 0.201741, size = 107, normalized size = 1.11 \[ -\frac{792 \, b^{7} x^{7} + 4620 \, a b^{6} x^{6} + 11880 \, a^{2} b^{5} x^{5} + 17325 \, a^{3} b^{4} x^{4} + 15400 \, a^{4} b^{3} x^{3} + 8316 \, a^{5} b^{2} x^{2} + 2520 \, a^{6} b x + 330 \, a^{7}}{3960 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^13,x, algorithm="giac")

[Out]

-1/3960*(792*b^7*x^7 + 4620*a*b^6*x^6 + 11880*a^2*b^5*x^5 + 17325*a^3*b^4*x^4 +
15400*a^4*b^3*x^3 + 8316*a^5*b^2*x^2 + 2520*a^6*b*x + 330*a^7)/x^12