3.88 \(\int \frac{(a+b x)^5}{x^5} \, dx\)

Optimal. Leaf size=57 \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{3 x^3}-\frac{5 a^3 b^2}{x^2}-\frac{10 a^2 b^3}{x}+5 a b^4 \log (x)+b^5 x \]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/(3*x^3) - (5*a^3*b^2)/x^2 - (10*a^2*b^3)/x + b^5*x + 5*
a*b^4*Log[x]

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Rubi [A]  time = 0.0507979, antiderivative size = 57, 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 \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{3 x^3}-\frac{5 a^3 b^2}{x^2}-\frac{10 a^2 b^3}{x}+5 a b^4 \log (x)+b^5 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^5,x]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/(3*x^3) - (5*a^3*b^2)/x^2 - (10*a^2*b^3)/x + b^5*x + 5*
a*b^4*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5}}{4 x^{4}} - \frac{5 a^{4} b}{3 x^{3}} - \frac{5 a^{3} b^{2}}{x^{2}} - \frac{10 a^{2} b^{3}}{x} + 5 a b^{4} \log{\left (x \right )} + \int b^{5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**5,x)

[Out]

-a**5/(4*x**4) - 5*a**4*b/(3*x**3) - 5*a**3*b**2/x**2 - 10*a**2*b**3/x + 5*a*b**
4*log(x) + Integral(b**5, x)

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Mathematica [A]  time = 0.00828468, size = 57, normalized size = 1. \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{3 x^3}-\frac{5 a^3 b^2}{x^2}-\frac{10 a^2 b^3}{x}+5 a b^4 \log (x)+b^5 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^5,x]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/(3*x^3) - (5*a^3*b^2)/x^2 - (10*a^2*b^3)/x + b^5*x + 5*
a*b^4*Log[x]

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Maple [A]  time = 0.01, size = 54, normalized size = 1. \[ -{\frac{{a}^{5}}{4\,{x}^{4}}}-{\frac{5\,{a}^{4}b}{3\,{x}^{3}}}-5\,{\frac{{a}^{3}{b}^{2}}{{x}^{2}}}-10\,{\frac{{a}^{2}{b}^{3}}{x}}+{b}^{5}x+5\,a{b}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^5,x)

[Out]

-1/4*a^5/x^4-5/3*a^4*b/x^3-5*a^3*b^2/x^2-10*a^2*b^3/x+b^5*x+5*a*b^4*ln(x)

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Maxima [A]  time = 1.34581, size = 73, normalized size = 1.28 \[ b^{5} x + 5 \, a b^{4} \log \left (x\right ) - \frac{120 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} + 20 \, a^{4} b x + 3 \, a^{5}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^5,x, algorithm="maxima")

[Out]

b^5*x + 5*a*b^4*log(x) - 1/12*(120*a^2*b^3*x^3 + 60*a^3*b^2*x^2 + 20*a^4*b*x + 3
*a^5)/x^4

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Fricas [A]  time = 0.201102, size = 80, normalized size = 1.4 \[ \frac{12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^5,x, algorithm="fricas")

[Out]

1/12*(12*b^5*x^5 + 60*a*b^4*x^4*log(x) - 120*a^2*b^3*x^3 - 60*a^3*b^2*x^2 - 20*a
^4*b*x - 3*a^5)/x^4

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Sympy [A]  time = 1.83429, size = 56, normalized size = 0.98 \[ 5 a b^{4} \log{\left (x \right )} + b^{5} x - \frac{3 a^{5} + 20 a^{4} b x + 60 a^{3} b^{2} x^{2} + 120 a^{2} b^{3} x^{3}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**5,x)

[Out]

5*a*b**4*log(x) + b**5*x - (3*a**5 + 20*a**4*b*x + 60*a**3*b**2*x**2 + 120*a**2*
b**3*x**3)/(12*x**4)

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GIAC/XCAS [A]  time = 0.212931, size = 74, normalized size = 1.3 \[ b^{5} x + 5 \, a b^{4}{\rm ln}\left ({\left | x \right |}\right ) - \frac{120 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} + 20 \, a^{4} b x + 3 \, a^{5}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^5,x, algorithm="giac")

[Out]

b^5*x + 5*a*b^4*ln(abs(x)) - 1/12*(120*a^2*b^3*x^3 + 60*a^3*b^2*x^2 + 20*a^4*b*x
+ 3*a^5)/x^4