3.19 \(\int \frac{(a+b x) (a c-b c x)^4}{x^3} \, dx\)

Optimal. Leaf size=78 \[ -\frac{a^5 c^4}{2 x^2}+\frac{3 a^4 b c^4}{x}+2 a^3 b^2 c^4 \log (x)+2 a^2 b^3 c^4 x-\frac{3}{2} a b^4 c^4 x^2+\frac{1}{3} b^5 c^4 x^3 \]

[Out]

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

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Rubi [A]  time = 0.0928805, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^5 c^4}{2 x^2}+\frac{3 a^4 b c^4}{x}+2 a^3 b^2 c^4 \log (x)+2 a^2 b^3 c^4 x-\frac{3}{2} a b^4 c^4 x^2+\frac{1}{3} b^5 c^4 x^3 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^4)/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**4/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0128985, size = 78, normalized size = 1. \[ -\frac{a^5 c^4}{2 x^2}+\frac{3 a^4 b c^4}{x}+2 a^3 b^2 c^4 \log (x)+2 a^2 b^3 c^4 x-\frac{3}{2} a b^4 c^4 x^2+\frac{1}{3} b^5 c^4 x^3 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 73, normalized size = 0.9 \[ -{\frac{{a}^{5}{c}^{4}}{2\,{x}^{2}}}+3\,{\frac{{a}^{4}b{c}^{4}}{x}}+2\,{a}^{2}{b}^{3}{c}^{4}x-{\frac{3\,a{b}^{4}{c}^{4}{x}^{2}}{2}}+{\frac{{b}^{5}{c}^{4}{x}^{3}}{3}}+2\,{a}^{3}{b}^{2}{c}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^4/x^3,x)

[Out]

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

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Maxima [A]  time = 1.3403, size = 99, normalized size = 1.27 \[ \frac{1}{3} \, b^{5} c^{4} x^{3} - \frac{3}{2} \, a b^{4} c^{4} x^{2} + 2 \, a^{2} b^{3} c^{4} x + 2 \, a^{3} b^{2} c^{4} \log \left (x\right ) + \frac{6 \, a^{4} b c^{4} x - a^{5} c^{4}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.201447, size = 104, normalized size = 1.33 \[ \frac{2 \, b^{5} c^{4} x^{5} - 9 \, a b^{4} c^{4} x^{4} + 12 \, a^{2} b^{3} c^{4} x^{3} + 12 \, a^{3} b^{2} c^{4} x^{2} \log \left (x\right ) + 18 \, a^{4} b c^{4} x - 3 \, a^{5} c^{4}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*c^4*x^5 - 9*a*b^4*c^4*x^4 + 12*a^2*b^3*c^4*x^3 + 12*a^3*b^2*c^4*x^2*l
og(x) + 18*a^4*b*c^4*x - 3*a^5*c^4)/x^2

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Sympy [A]  time = 0.751934, size = 78, normalized size = 1. \[ 2 a^{3} b^{2} c^{4} \log{\left (x \right )} + 2 a^{2} b^{3} c^{4} x - \frac{3 a b^{4} c^{4} x^{2}}{2} + \frac{b^{5} c^{4} x^{3}}{3} + \frac{- a^{5} c^{4} + 6 a^{4} b c^{4} x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**4/x**3,x)

[Out]

2*a**3*b**2*c**4*log(x) + 2*a**2*b**3*c**4*x - 3*a*b**4*c**4*x**2/2 + b**5*c**4*
x**3/3 + (-a**5*c**4 + 6*a**4*b*c**4*x)/(2*x**2)

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GIAC/XCAS [A]  time = 0.292718, size = 100, normalized size = 1.28 \[ \frac{1}{3} \, b^{5} c^{4} x^{3} - \frac{3}{2} \, a b^{4} c^{4} x^{2} + 2 \, a^{2} b^{3} c^{4} x + 2 \, a^{3} b^{2} c^{4}{\rm ln}\left ({\left | x \right |}\right ) + \frac{6 \, a^{4} b c^{4} x - a^{5} c^{4}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)/x^3,x, algorithm="giac")

[Out]

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