3.87 \(\int \frac{(a+b x)^5}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0507455, 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 \[ -\frac{a^5}{3 x^3}-\frac{5 a^4 b}{2 x^2}-\frac{10 a^3 b^2}{x}+10 a^2 b^3 \log (x)+5 a b^4 x+\frac{b^5 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/x^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.00615775, size = 60, normalized size = 1. \[ -\frac{a^5}{3 x^3}-\frac{5 a^4 b}{2 x^2}-\frac{10 a^3 b^2}{x}+10 a^2 b^3 \log (x)+5 a b^4 x+\frac{b^5 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/x^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/x^4,x)

[Out]

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

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Maxima [A]  time = 1.34088, size = 74, normalized size = 1.23 \[ \frac{1}{2} \, b^{5} x^{2} + 5 \, a b^{4} x + 10 \, a^{2} b^{3} \log \left (x\right ) - \frac{60 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x + 2 \, a^{5}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^4,x, algorithm="maxima")

[Out]

1/2*b^5*x^2 + 5*a*b^4*x + 10*a^2*b^3*log(x) - 1/6*(60*a^3*b^2*x^2 + 15*a^4*b*x +
2*a^5)/x^3

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Fricas [A]  time = 0.206678, size = 80, normalized size = 1.33 \[ \frac{3 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} \log \left (x\right ) - 60 \, a^{3} b^{2} x^{2} - 15 \, a^{4} b x - 2 \, a^{5}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.58521, size = 58, normalized size = 0.97 \[ 10 a^{2} b^{3} \log{\left (x \right )} + 5 a b^{4} x + \frac{b^{5} x^{2}}{2} - \frac{2 a^{5} + 15 a^{4} b x + 60 a^{3} b^{2} x^{2}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215715, size = 76, normalized size = 1.27 \[ \frac{1}{2} \, b^{5} x^{2} + 5 \, a b^{4} x + 10 \, a^{2} b^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{60 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x + 2 \, a^{5}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/x^4,x, algorithm="giac")

[Out]

1/2*b^5*x^2 + 5*a*b^4*x + 10*a^2*b^3*ln(abs(x)) - 1/6*(60*a^3*b^2*x^2 + 15*a^4*b
*x + 2*a^5)/x^3