3.120 \(\int \frac{(a+b x)^7}{x^{14}} \, dx\)

Optimal. Leaf size=93 \[ -\frac{a^7}{13 x^{13}}-\frac{7 a^6 b}{12 x^{12}}-\frac{21 a^5 b^2}{11 x^{11}}-\frac{7 a^4 b^3}{2 x^{10}}-\frac{35 a^3 b^4}{9 x^9}-\frac{21 a^2 b^5}{8 x^8}-\frac{a b^6}{x^7}-\frac{b^7}{6 x^6} \]

[Out]

-a^7/(13*x^13) - (7*a^6*b)/(12*x^12) - (21*a^5*b^2)/(11*x^11) - (7*a^4*b^3)/(2*x
^10) - (35*a^3*b^4)/(9*x^9) - (21*a^2*b^5)/(8*x^8) - (a*b^6)/x^7 - b^7/(6*x^6)

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Rubi [A]  time = 0.0771543, antiderivative size = 93, 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 \[ -\frac{a^7}{13 x^{13}}-\frac{7 a^6 b}{12 x^{12}}-\frac{21 a^5 b^2}{11 x^{11}}-\frac{7 a^4 b^3}{2 x^{10}}-\frac{35 a^3 b^4}{9 x^9}-\frac{21 a^2 b^5}{8 x^8}-\frac{a b^6}{x^7}-\frac{b^7}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/x^14,x]

[Out]

-a^7/(13*x^13) - (7*a^6*b)/(12*x^12) - (21*a^5*b^2)/(11*x^11) - (7*a^4*b^3)/(2*x
^10) - (35*a^3*b^4)/(9*x^9) - (21*a^2*b^5)/(8*x^8) - (a*b^6)/x^7 - b^7/(6*x^6)

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Rubi in Sympy [A]  time = 15.4623, size = 92, normalized size = 0.99 \[ - \frac{a^{7}}{13 x^{13}} - \frac{7 a^{6} b}{12 x^{12}} - \frac{21 a^{5} b^{2}}{11 x^{11}} - \frac{7 a^{4} b^{3}}{2 x^{10}} - \frac{35 a^{3} b^{4}}{9 x^{9}} - \frac{21 a^{2} b^{5}}{8 x^{8}} - \frac{a b^{6}}{x^{7}} - \frac{b^{7}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/x**14,x)

[Out]

-a**7/(13*x**13) - 7*a**6*b/(12*x**12) - 21*a**5*b**2/(11*x**11) - 7*a**4*b**3/(
2*x**10) - 35*a**3*b**4/(9*x**9) - 21*a**2*b**5/(8*x**8) - a*b**6/x**7 - b**7/(6
*x**6)

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Mathematica [A]  time = 0.00944462, size = 93, normalized size = 1. \[ -\frac{a^7}{13 x^{13}}-\frac{7 a^6 b}{12 x^{12}}-\frac{21 a^5 b^2}{11 x^{11}}-\frac{7 a^4 b^3}{2 x^{10}}-\frac{35 a^3 b^4}{9 x^9}-\frac{21 a^2 b^5}{8 x^8}-\frac{a b^6}{x^7}-\frac{b^7}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/x^14,x]

[Out]

-a^7/(13*x^13) - (7*a^6*b)/(12*x^12) - (21*a^5*b^2)/(11*x^11) - (7*a^4*b^3)/(2*x
^10) - (35*a^3*b^4)/(9*x^9) - (21*a^2*b^5)/(8*x^8) - (a*b^6)/x^7 - b^7/(6*x^6)

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Maple [A]  time = 0.01, size = 80, normalized size = 0.9 \[ -{\frac{{a}^{7}}{13\,{x}^{13}}}-{\frac{7\,{a}^{6}b}{12\,{x}^{12}}}-{\frac{21\,{a}^{5}{b}^{2}}{11\,{x}^{11}}}-{\frac{7\,{a}^{4}{b}^{3}}{2\,{x}^{10}}}-{\frac{35\,{a}^{3}{b}^{4}}{9\,{x}^{9}}}-{\frac{21\,{a}^{2}{b}^{5}}{8\,{x}^{8}}}-{\frac{a{b}^{6}}{{x}^{7}}}-{\frac{{b}^{7}}{6\,{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/x^14,x)

[Out]

-1/13*a^7/x^13-7/12*a^6*b/x^12-21/11*a^5*b^2/x^11-7/2*a^4*b^3/x^10-35/9*a^3*b^4/
x^9-21/8*a^2*b^5/x^8-a*b^6/x^7-1/6*b^7/x^6

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Maxima [A]  time = 1.34893, size = 107, normalized size = 1.15 \[ -\frac{1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^14,x, algorithm="maxima")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4
+ 36036*a^4*b^3*x^3 + 19656*a^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13

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Fricas [A]  time = 0.194475, size = 107, normalized size = 1.15 \[ -\frac{1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^14,x, algorithm="fricas")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4
+ 36036*a^4*b^3*x^3 + 19656*a^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13

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Sympy [A]  time = 3.19113, size = 85, normalized size = 0.91 \[ - \frac{792 a^{7} + 6006 a^{6} b x + 19656 a^{5} b^{2} x^{2} + 36036 a^{4} b^{3} x^{3} + 40040 a^{3} b^{4} x^{4} + 27027 a^{2} b^{5} x^{5} + 10296 a b^{6} x^{6} + 1716 b^{7} x^{7}}{10296 x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/x**14,x)

[Out]

-(792*a**7 + 6006*a**6*b*x + 19656*a**5*b**2*x**2 + 36036*a**4*b**3*x**3 + 40040
*a**3*b**4*x**4 + 27027*a**2*b**5*x**5 + 10296*a*b**6*x**6 + 1716*b**7*x**7)/(10
296*x**13)

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GIAC/XCAS [A]  time = 0.203175, size = 107, normalized size = 1.15 \[ -\frac{1716 \, b^{7} x^{7} + 10296 \, a b^{6} x^{6} + 27027 \, a^{2} b^{5} x^{5} + 40040 \, a^{3} b^{4} x^{4} + 36036 \, a^{4} b^{3} x^{3} + 19656 \, a^{5} b^{2} x^{2} + 6006 \, a^{6} b x + 792 \, a^{7}}{10296 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^14,x, algorithm="giac")

[Out]

-1/10296*(1716*b^7*x^7 + 10296*a*b^6*x^6 + 27027*a^2*b^5*x^5 + 40040*a^3*b^4*x^4
+ 36036*a^4*b^3*x^3 + 19656*a^5*b^2*x^2 + 6006*a^6*b*x + 792*a^7)/x^13