3.86 \(\int \frac{(a+b x)^5}{x^3} \, dx\)

Optimal. Leaf size=60 \[ -\frac{a^5}{2 x^2}-\frac{5 a^4 b}{x}+10 a^3 b^2 \log (x)+10 a^2 b^3 x+\frac{5}{2} a b^4 x^2+\frac{b^5 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0497974, antiderivative size = 60, 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 \[ -\frac{a^5}{2 x^2}-\frac{5 a^4 b}{x}+10 a^3 b^2 \log (x)+10 a^2 b^3 x+\frac{5}{2} a b^4 x^2+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0063235, size = 60, normalized size = 1. \[ -\frac{a^5}{2 x^2}-\frac{5 a^4 b}{x}+10 a^3 b^2 \log (x)+10 a^2 b^3 x+\frac{5}{2} a b^4 x^2+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^3,x]

[Out]

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

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Maple [A]  time = 0.01, size = 55, normalized size = 0.9 \[ -{\frac{{a}^{5}}{2\,{x}^{2}}}-5\,{\frac{{a}^{4}b}{x}}+10\,{a}^{2}{b}^{3}x+{\frac{5\,a{b}^{4}{x}^{2}}{2}}+{\frac{{b}^{5}{x}^{3}}{3}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^3,x)

[Out]

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

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Maxima [A]  time = 1.33464, size = 72, normalized size = 1.2 \[ \frac{1}{3} \, b^{5} x^{3} + \frac{5}{2} \, a b^{4} x^{2} + 10 \, a^{2} b^{3} x + 10 \, a^{3} b^{2} \log \left (x\right ) - \frac{10 \, a^{4} b x + a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.231677, size = 80, normalized size = 1.33 \[ \frac{2 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} \log \left (x\right ) - 30 \, a^{4} b x - 3 \, a^{5}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*x^5 + 15*a*b^4*x^4 + 60*a^2*b^3*x^3 + 60*a^3*b^2*x^2*log(x) - 30*a^4*
b*x - 3*a^5)/x^2

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Sympy [A]  time = 1.40683, size = 58, normalized size = 0.97 \[ 10 a^{3} b^{2} \log{\left (x \right )} + 10 a^{2} b^{3} x + \frac{5 a b^{4} x^{2}}{2} + \frac{b^{5} x^{3}}{3} - \frac{a^{5} + 10 a^{4} b x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**3,x)

[Out]

10*a**3*b**2*log(x) + 10*a**2*b**3*x + 5*a*b**4*x**2/2 + b**5*x**3/3 - (a**5 + 1
0*a**4*b*x)/(2*x**2)

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GIAC/XCAS [A]  time = 0.206866, size = 73, normalized size = 1.22 \[ \frac{1}{3} \, b^{5} x^{3} + \frac{5}{2} \, a b^{4} x^{2} + 10 \, a^{2} b^{3} x + 10 \, a^{3} b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{10 \, a^{4} b x + a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^3,x, algorithm="giac")

[Out]

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