3.121 \(\int \frac{(a+b x)^7}{x^{15}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{a^7}{14 x^{14}}-\frac{7 a^6 b}{13 x^{13}}-\frac{7 a^5 b^2}{4 x^{12}}-\frac{35 a^4 b^3}{11 x^{11}}-\frac{7 a^3 b^4}{2 x^{10}}-\frac{7 a^2 b^5}{3 x^9}-\frac{7 a b^6}{8 x^8}-\frac{b^7}{7 x^7} \]

[Out]

-a^7/(14*x^14) - (7*a^6*b)/(13*x^13) - (7*a^5*b^2)/(4*x^12) - (35*a^4*b^3)/(11*x
^11) - (7*a^3*b^4)/(2*x^10) - (7*a^2*b^5)/(3*x^9) - (7*a*b^6)/(8*x^8) - b^7/(7*x
^7)

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Rubi [A]  time = 0.0748104, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^7}{14 x^{14}}-\frac{7 a^6 b}{13 x^{13}}-\frac{7 a^5 b^2}{4 x^{12}}-\frac{35 a^4 b^3}{11 x^{11}}-\frac{7 a^3 b^4}{2 x^{10}}-\frac{7 a^2 b^5}{3 x^9}-\frac{7 a b^6}{8 x^8}-\frac{b^7}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/x^15,x]

[Out]

-a^7/(14*x^14) - (7*a^6*b)/(13*x^13) - (7*a^5*b^2)/(4*x^12) - (35*a^4*b^3)/(11*x
^11) - (7*a^3*b^4)/(2*x^10) - (7*a^2*b^5)/(3*x^9) - (7*a*b^6)/(8*x^8) - b^7/(7*x
^7)

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Rubi in Sympy [A]  time = 15.5083, size = 95, normalized size = 1. \[ - \frac{a^{7}}{14 x^{14}} - \frac{7 a^{6} b}{13 x^{13}} - \frac{7 a^{5} b^{2}}{4 x^{12}} - \frac{35 a^{4} b^{3}}{11 x^{11}} - \frac{7 a^{3} b^{4}}{2 x^{10}} - \frac{7 a^{2} b^{5}}{3 x^{9}} - \frac{7 a b^{6}}{8 x^{8}} - \frac{b^{7}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/x**15,x)

[Out]

-a**7/(14*x**14) - 7*a**6*b/(13*x**13) - 7*a**5*b**2/(4*x**12) - 35*a**4*b**3/(1
1*x**11) - 7*a**3*b**4/(2*x**10) - 7*a**2*b**5/(3*x**9) - 7*a*b**6/(8*x**8) - b*
*7/(7*x**7)

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Mathematica [A]  time = 0.00551619, size = 95, normalized size = 1. \[ -\frac{a^7}{14 x^{14}}-\frac{7 a^6 b}{13 x^{13}}-\frac{7 a^5 b^2}{4 x^{12}}-\frac{35 a^4 b^3}{11 x^{11}}-\frac{7 a^3 b^4}{2 x^{10}}-\frac{7 a^2 b^5}{3 x^9}-\frac{7 a b^6}{8 x^8}-\frac{b^7}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/x^15,x]

[Out]

-a^7/(14*x^14) - (7*a^6*b)/(13*x^13) - (7*a^5*b^2)/(4*x^12) - (35*a^4*b^3)/(11*x
^11) - (7*a^3*b^4)/(2*x^10) - (7*a^2*b^5)/(3*x^9) - (7*a*b^6)/(8*x^8) - b^7/(7*x
^7)

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Maple [A]  time = 0.008, size = 80, normalized size = 0.8 \[ -{\frac{{a}^{7}}{14\,{x}^{14}}}-{\frac{7\,{a}^{6}b}{13\,{x}^{13}}}-{\frac{7\,{a}^{5}{b}^{2}}{4\,{x}^{12}}}-{\frac{35\,{a}^{4}{b}^{3}}{11\,{x}^{11}}}-{\frac{7\,{a}^{3}{b}^{4}}{2\,{x}^{10}}}-{\frac{7\,{a}^{2}{b}^{5}}{3\,{x}^{9}}}-{\frac{7\,a{b}^{6}}{8\,{x}^{8}}}-{\frac{{b}^{7}}{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/x^15,x)

[Out]

-1/14*a^7/x^14-7/13*a^6*b/x^13-7/4*a^5*b^2/x^12-35/11*a^4*b^3/x^11-7/2*a^3*b^4/x
^10-7/3*a^2*b^5/x^9-7/8*a*b^6/x^8-1/7*b^7/x^7

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Maxima [A]  time = 1.34268, size = 107, normalized size = 1.13 \[ -\frac{3432 \, b^{7} x^{7} + 21021 \, a b^{6} x^{6} + 56056 \, a^{2} b^{5} x^{5} + 84084 \, a^{3} b^{4} x^{4} + 76440 \, a^{4} b^{3} x^{3} + 42042 \, a^{5} b^{2} x^{2} + 12936 \, a^{6} b x + 1716 \, a^{7}}{24024 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^15,x, algorithm="maxima")

[Out]

-1/24024*(3432*b^7*x^7 + 21021*a*b^6*x^6 + 56056*a^2*b^5*x^5 + 84084*a^3*b^4*x^4
+ 76440*a^4*b^3*x^3 + 42042*a^5*b^2*x^2 + 12936*a^6*b*x + 1716*a^7)/x^14

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Fricas [A]  time = 0.191817, size = 107, normalized size = 1.13 \[ -\frac{3432 \, b^{7} x^{7} + 21021 \, a b^{6} x^{6} + 56056 \, a^{2} b^{5} x^{5} + 84084 \, a^{3} b^{4} x^{4} + 76440 \, a^{4} b^{3} x^{3} + 42042 \, a^{5} b^{2} x^{2} + 12936 \, a^{6} b x + 1716 \, a^{7}}{24024 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^15,x, algorithm="fricas")

[Out]

-1/24024*(3432*b^7*x^7 + 21021*a*b^6*x^6 + 56056*a^2*b^5*x^5 + 84084*a^3*b^4*x^4
+ 76440*a^4*b^3*x^3 + 42042*a^5*b^2*x^2 + 12936*a^6*b*x + 1716*a^7)/x^14

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Sympy [A]  time = 3.3231, size = 85, normalized size = 0.89 \[ - \frac{1716 a^{7} + 12936 a^{6} b x + 42042 a^{5} b^{2} x^{2} + 76440 a^{4} b^{3} x^{3} + 84084 a^{3} b^{4} x^{4} + 56056 a^{2} b^{5} x^{5} + 21021 a b^{6} x^{6} + 3432 b^{7} x^{7}}{24024 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/x**15,x)

[Out]

-(1716*a**7 + 12936*a**6*b*x + 42042*a**5*b**2*x**2 + 76440*a**4*b**3*x**3 + 840
84*a**3*b**4*x**4 + 56056*a**2*b**5*x**5 + 21021*a*b**6*x**6 + 3432*b**7*x**7)/(
24024*x**14)

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GIAC/XCAS [A]  time = 0.210611, size = 107, normalized size = 1.13 \[ -\frac{3432 \, b^{7} x^{7} + 21021 \, a b^{6} x^{6} + 56056 \, a^{2} b^{5} x^{5} + 84084 \, a^{3} b^{4} x^{4} + 76440 \, a^{4} b^{3} x^{3} + 42042 \, a^{5} b^{2} x^{2} + 12936 \, a^{6} b x + 1716 \, a^{7}}{24024 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^15,x, algorithm="giac")

[Out]

-1/24024*(3432*b^7*x^7 + 21021*a*b^6*x^6 + 56056*a^2*b^5*x^5 + 84084*a^3*b^4*x^4
+ 76440*a^4*b^3*x^3 + 42042*a^5*b^2*x^2 + 12936*a^6*b*x + 1716*a^7)/x^14