3.108 \(\int \frac{(a+b x)^7}{x^2} \, dx\)

Optimal. Leaf size=86 \[ -\frac{a^7}{x}+7 a^6 b \log (x)+21 a^5 b^2 x+\frac{35}{2} a^4 b^3 x^2+\frac{35}{3} a^3 b^4 x^3+\frac{21}{4} a^2 b^5 x^4+\frac{7}{5} a b^6 x^5+\frac{b^7 x^6}{6} \]

[Out]

-(a^7/x) + 21*a^5*b^2*x + (35*a^4*b^3*x^2)/2 + (35*a^3*b^4*x^3)/3 + (21*a^2*b^5*
x^4)/4 + (7*a*b^6*x^5)/5 + (b^7*x^6)/6 + 7*a^6*b*Log[x]

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Rubi [A]  time = 0.0732908, antiderivative size = 86, 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 \[ -\frac{a^7}{x}+7 a^6 b \log (x)+21 a^5 b^2 x+\frac{35}{2} a^4 b^3 x^2+\frac{35}{3} a^3 b^4 x^3+\frac{21}{4} a^2 b^5 x^4+\frac{7}{5} a b^6 x^5+\frac{b^7 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/x^2,x]

[Out]

-(a^7/x) + 21*a^5*b^2*x + (35*a^4*b^3*x^2)/2 + (35*a^3*b^4*x^3)/3 + (21*a^2*b^5*
x^4)/4 + (7*a*b^6*x^5)/5 + (b^7*x^6)/6 + 7*a^6*b*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{7}}{x} + 7 a^{6} b \log{\left (x \right )} + 21 a^{5} b^{2} x + 35 a^{4} b^{3} \int x\, dx + \frac{35 a^{3} b^{4} x^{3}}{3} + \frac{21 a^{2} b^{5} x^{4}}{4} + \frac{7 a b^{6} x^{5}}{5} + \frac{b^{7} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/x**2,x)

[Out]

-a**7/x + 7*a**6*b*log(x) + 21*a**5*b**2*x + 35*a**4*b**3*Integral(x, x) + 35*a*
*3*b**4*x**3/3 + 21*a**2*b**5*x**4/4 + 7*a*b**6*x**5/5 + b**7*x**6/6

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Mathematica [A]  time = 0.00616383, size = 86, normalized size = 1. \[ -\frac{a^7}{x}+7 a^6 b \log (x)+21 a^5 b^2 x+\frac{35}{2} a^4 b^3 x^2+\frac{35}{3} a^3 b^4 x^3+\frac{21}{4} a^2 b^5 x^4+\frac{7}{5} a b^6 x^5+\frac{b^7 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/x^2,x]

[Out]

-(a^7/x) + 21*a^5*b^2*x + (35*a^4*b^3*x^2)/2 + (35*a^3*b^4*x^3)/3 + (21*a^2*b^5*
x^4)/4 + (7*a*b^6*x^5)/5 + (b^7*x^6)/6 + 7*a^6*b*Log[x]

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Maple [A]  time = 0.007, size = 77, normalized size = 0.9 \[ -{\frac{{a}^{7}}{x}}+21\,{a}^{5}{b}^{2}x+{\frac{35\,{a}^{4}{b}^{3}{x}^{2}}{2}}+{\frac{35\,{a}^{3}{b}^{4}{x}^{3}}{3}}+{\frac{21\,{a}^{2}{b}^{5}{x}^{4}}{4}}+{\frac{7\,a{b}^{6}{x}^{5}}{5}}+{\frac{{b}^{7}{x}^{6}}{6}}+7\,{a}^{6}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/x^2,x)

[Out]

-a^7/x+21*a^5*b^2*x+35/2*a^4*b^3*x^2+35/3*a^3*b^4*x^3+21/4*a^2*b^5*x^4+7/5*a*b^6
*x^5+1/6*b^7*x^6+7*a^6*b*ln(x)

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Maxima [A]  time = 1.34264, size = 103, normalized size = 1.2 \[ \frac{1}{6} \, b^{7} x^{6} + \frac{7}{5} \, a b^{6} x^{5} + \frac{21}{4} \, a^{2} b^{5} x^{4} + \frac{35}{3} \, a^{3} b^{4} x^{3} + \frac{35}{2} \, a^{4} b^{3} x^{2} + 21 \, a^{5} b^{2} x + 7 \, a^{6} b \log \left (x\right ) - \frac{a^{7}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^2,x, algorithm="maxima")

[Out]

1/6*b^7*x^6 + 7/5*a*b^6*x^5 + 21/4*a^2*b^5*x^4 + 35/3*a^3*b^4*x^3 + 35/2*a^4*b^3
*x^2 + 21*a^5*b^2*x + 7*a^6*b*log(x) - a^7/x

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Fricas [A]  time = 0.19518, size = 109, normalized size = 1.27 \[ \frac{10 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 315 \, a^{2} b^{5} x^{5} + 700 \, a^{3} b^{4} x^{4} + 1050 \, a^{4} b^{3} x^{3} + 1260 \, a^{5} b^{2} x^{2} + 420 \, a^{6} b x \log \left (x\right ) - 60 \, a^{7}}{60 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^2,x, algorithm="fricas")

[Out]

1/60*(10*b^7*x^7 + 84*a*b^6*x^6 + 315*a^2*b^5*x^5 + 700*a^3*b^4*x^4 + 1050*a^4*b
^3*x^3 + 1260*a^5*b^2*x^2 + 420*a^6*b*x*log(x) - 60*a^7)/x

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Sympy [A]  time = 1.38357, size = 85, normalized size = 0.99 \[ - \frac{a^{7}}{x} + 7 a^{6} b \log{\left (x \right )} + 21 a^{5} b^{2} x + \frac{35 a^{4} b^{3} x^{2}}{2} + \frac{35 a^{3} b^{4} x^{3}}{3} + \frac{21 a^{2} b^{5} x^{4}}{4} + \frac{7 a b^{6} x^{5}}{5} + \frac{b^{7} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/x**2,x)

[Out]

-a**7/x + 7*a**6*b*log(x) + 21*a**5*b**2*x + 35*a**4*b**3*x**2/2 + 35*a**3*b**4*
x**3/3 + 21*a**2*b**5*x**4/4 + 7*a*b**6*x**5/5 + b**7*x**6/6

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GIAC/XCAS [A]  time = 0.210409, size = 104, normalized size = 1.21 \[ \frac{1}{6} \, b^{7} x^{6} + \frac{7}{5} \, a b^{6} x^{5} + \frac{21}{4} \, a^{2} b^{5} x^{4} + \frac{35}{3} \, a^{3} b^{4} x^{3} + \frac{35}{2} \, a^{4} b^{3} x^{2} + 21 \, a^{5} b^{2} x + 7 \, a^{6} b{\rm ln}\left ({\left | x \right |}\right ) - \frac{a^{7}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^2,x, algorithm="giac")

[Out]

1/6*b^7*x^6 + 7/5*a*b^6*x^5 + 21/4*a^2*b^5*x^4 + 35/3*a^3*b^4*x^3 + 35/2*a^4*b^3
*x^2 + 21*a^5*b^2*x + 7*a^6*b*ln(abs(x)) - a^7/x