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

Optimal. Leaf size=87 \[ a^7 \log (x)+7 a^6 b x+\frac{21}{2} a^5 b^2 x^2+\frac{35}{3} a^4 b^3 x^3+\frac{35}{4} a^3 b^4 x^4+\frac{21}{5} a^2 b^5 x^5+\frac{7}{6} a b^6 x^6+\frac{b^7 x^7}{7} \]

[Out]

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

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Rubi [A]  time = 0.0614451, antiderivative size = 87, 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 \[ a^7 \log (x)+7 a^6 b x+\frac{21}{2} a^5 b^2 x^2+\frac{35}{3} a^4 b^3 x^3+\frac{35}{4} a^3 b^4 x^4+\frac{21}{5} a^2 b^5 x^5+\frac{7}{6} a b^6 x^6+\frac{b^7 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/x,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/x,x)

[Out]

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

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Mathematica [A]  time = 0.00469767, size = 87, normalized size = 1. \[ a^7 \log (x)+7 a^6 b x+\frac{21}{2} a^5 b^2 x^2+\frac{35}{3} a^4 b^3 x^3+\frac{35}{4} a^3 b^4 x^4+\frac{21}{5} a^2 b^5 x^5+\frac{7}{6} a b^6 x^6+\frac{b^7 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/x,x]

[Out]

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

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Maple [A]  time = 0.003, size = 76, normalized size = 0.9 \[ 7\,{a}^{6}bx+{\frac{21\,{a}^{5}{b}^{2}{x}^{2}}{2}}+{\frac{35\,{a}^{4}{b}^{3}{x}^{3}}{3}}+{\frac{35\,{a}^{3}{b}^{4}{x}^{4}}{4}}+{\frac{21\,{a}^{2}{b}^{5}{x}^{5}}{5}}+{\frac{7\,a{b}^{6}{x}^{6}}{6}}+{\frac{{b}^{7}{x}^{7}}{7}}+{a}^{7}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/x,x)

[Out]

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

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Maxima [A]  time = 1.33816, size = 101, normalized size = 1.16 \[ \frac{1}{7} \, b^{7} x^{7} + \frac{7}{6} \, a b^{6} x^{6} + \frac{21}{5} \, a^{2} b^{5} x^{5} + \frac{35}{4} \, a^{3} b^{4} x^{4} + \frac{35}{3} \, a^{4} b^{3} x^{3} + \frac{21}{2} \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.200846, size = 101, normalized size = 1.16 \[ \frac{1}{7} \, b^{7} x^{7} + \frac{7}{6} \, a b^{6} x^{6} + \frac{21}{5} \, a^{2} b^{5} x^{5} + \frac{35}{4} \, a^{3} b^{4} x^{4} + \frac{35}{3} \, a^{4} b^{3} x^{3} + \frac{21}{2} \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.24442, size = 88, normalized size = 1.01 \[ a^{7} \log{\left (x \right )} + 7 a^{6} b x + \frac{21 a^{5} b^{2} x^{2}}{2} + \frac{35 a^{4} b^{3} x^{3}}{3} + \frac{35 a^{3} b^{4} x^{4}}{4} + \frac{21 a^{2} b^{5} x^{5}}{5} + \frac{7 a b^{6} x^{6}}{6} + \frac{b^{7} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212537, size = 103, normalized size = 1.18 \[ \frac{1}{7} \, b^{7} x^{7} + \frac{7}{6} \, a b^{6} x^{6} + \frac{21}{5} \, a^{2} b^{5} x^{5} + \frac{35}{4} \, a^{3} b^{4} x^{4} + \frac{35}{3} \, a^{4} b^{3} x^{3} + \frac{21}{2} \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x + a^{7}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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