3.109 \(\int \frac{(a+b x)^7}{x^3} \, dx\)

Optimal. Leaf size=84 \[ -\frac{a^7}{2 x^2}-\frac{7 a^6 b}{x}+21 a^5 b^2 \log (x)+35 a^4 b^3 x+\frac{35}{2} a^3 b^4 x^2+7 a^2 b^5 x^3+\frac{7}{4} a b^6 x^4+\frac{b^7 x^5}{5} \]

[Out]

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

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Rubi [A]  time = 0.0712833, antiderivative size = 84, 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 \[ -\frac{a^7}{2 x^2}-\frac{7 a^6 b}{x}+21 a^5 b^2 \log (x)+35 a^4 b^3 x+\frac{35}{2} a^3 b^4 x^2+7 a^2 b^5 x^3+\frac{7}{4} a b^6 x^4+\frac{b^7 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.00638366, size = 84, normalized size = 1. \[ -\frac{a^7}{2 x^2}-\frac{7 a^6 b}{x}+21 a^5 b^2 \log (x)+35 a^4 b^3 x+\frac{35}{2} a^3 b^4 x^2+7 a^2 b^5 x^3+\frac{7}{4} a b^6 x^4+\frac{b^7 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/x^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 77, normalized size = 0.9 \[ -{\frac{{a}^{7}}{2\,{x}^{2}}}-7\,{\frac{{a}^{6}b}{x}}+35\,{a}^{4}{b}^{3}x+{\frac{35\,{a}^{3}{b}^{4}{x}^{2}}{2}}+7\,{a}^{2}{b}^{5}{x}^{3}+{\frac{7\,a{b}^{6}{x}^{4}}{4}}+{\frac{{b}^{7}{x}^{5}}{5}}+21\,{a}^{5}{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/x^3,x)

[Out]

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

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Maxima [A]  time = 1.34749, size = 101, normalized size = 1.2 \[ \frac{1}{5} \, b^{7} x^{5} + \frac{7}{4} \, a b^{6} x^{4} + 7 \, a^{2} b^{5} x^{3} + \frac{35}{2} \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x + 21 \, a^{5} b^{2} \log \left (x\right ) - \frac{14 \, a^{6} b x + a^{7}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^3,x, algorithm="maxima")

[Out]

1/5*b^7*x^5 + 7/4*a*b^6*x^4 + 7*a^2*b^5*x^3 + 35/2*a^3*b^4*x^2 + 35*a^4*b^3*x +
21*a^5*b^2*log(x) - 1/2*(14*a^6*b*x + a^7)/x^2

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Fricas [A]  time = 0.202818, size = 109, normalized size = 1.3 \[ \frac{4 \, b^{7} x^{7} + 35 \, a b^{6} x^{6} + 140 \, a^{2} b^{5} x^{5} + 350 \, a^{3} b^{4} x^{4} + 700 \, a^{4} b^{3} x^{3} + 420 \, a^{5} b^{2} x^{2} \log \left (x\right ) - 140 \, a^{6} b x - 10 \, a^{7}}{20 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^3,x, algorithm="fricas")

[Out]

1/20*(4*b^7*x^7 + 35*a*b^6*x^6 + 140*a^2*b^5*x^5 + 350*a^3*b^4*x^4 + 700*a^4*b^3
*x^3 + 420*a^5*b^2*x^2*log(x) - 140*a^6*b*x - 10*a^7)/x^2

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Sympy [A]  time = 1.49877, size = 83, normalized size = 0.99 \[ 21 a^{5} b^{2} \log{\left (x \right )} + 35 a^{4} b^{3} x + \frac{35 a^{3} b^{4} x^{2}}{2} + 7 a^{2} b^{5} x^{3} + \frac{7 a b^{6} x^{4}}{4} + \frac{b^{7} x^{5}}{5} - \frac{a^{7} + 14 a^{6} b x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/x**3,x)

[Out]

21*a**5*b**2*log(x) + 35*a**4*b**3*x + 35*a**3*b**4*x**2/2 + 7*a**2*b**5*x**3 +
7*a*b**6*x**4/4 + b**7*x**5/5 - (a**7 + 14*a**6*b*x)/(2*x**2)

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GIAC/XCAS [A]  time = 0.216558, size = 103, normalized size = 1.23 \[ \frac{1}{5} \, b^{7} x^{5} + \frac{7}{4} \, a b^{6} x^{4} + 7 \, a^{2} b^{5} x^{3} + \frac{35}{2} \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x + 21 \, a^{5} b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{14 \, a^{6} b x + a^{7}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/x^3,x, algorithm="giac")

[Out]

1/5*b^7*x^5 + 7/4*a*b^6*x^4 + 7*a^2*b^5*x^3 + 35/2*a^3*b^4*x^2 + 35*a^4*b^3*x +
21*a^5*b^2*ln(abs(x)) - 1/2*(14*a^6*b*x + a^7)/x^2