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

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

[Out]

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

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Rubi [A]  time = 0.097031, 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 \[ -\frac{a^5 c^4}{3 x^3}+\frac{3 a^4 b c^4}{2 x^2}-\frac{2 a^3 b^2 c^4}{x}+2 a^2 b^3 c^4 \log (x)-3 a b^4 c^4 x+\frac{1}{2} b^5 c^4 x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0123702, size = 78, normalized size = 1. \[ -\frac{a^5 c^4}{3 x^3}+\frac{3 a^4 b c^4}{2 x^2}-\frac{2 a^3 b^2 c^4}{x}+2 a^2 b^3 c^4 \log (x)-3 a b^4 c^4 x+\frac{1}{2} b^5 c^4 x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 73, normalized size = 0.9 \[ -{\frac{{a}^{5}{c}^{4}}{3\,{x}^{3}}}+{\frac{3\,{a}^{4}b{c}^{4}}{2\,{x}^{2}}}-2\,{\frac{{a}^{3}{b}^{2}{c}^{4}}{x}}-3\,a{b}^{4}{c}^{4}x+{\frac{{b}^{5}{c}^{4}{x}^{2}}{2}}+2\,{a}^{2}{b}^{3}{c}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.32848, size = 99, normalized size = 1.27 \[ \frac{1}{2} \, b^{5} c^{4} x^{2} - 3 \, a b^{4} c^{4} x + 2 \, a^{2} b^{3} c^{4} \log \left (x\right ) - \frac{12 \, a^{3} b^{2} c^{4} x^{2} - 9 \, a^{4} b c^{4} x + 2 \, a^{5} c^{4}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203335, size = 104, normalized size = 1.33 \[ \frac{3 \, b^{5} c^{4} x^{5} - 18 \, a b^{4} c^{4} x^{4} + 12 \, a^{2} b^{3} c^{4} x^{3} \log \left (x\right ) - 12 \, a^{3} b^{2} c^{4} x^{2} + 9 \, a^{4} b c^{4} x - 2 \, a^{5} c^{4}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.86951, size = 78, normalized size = 1. \[ 2 a^{2} b^{3} c^{4} \log{\left (x \right )} - 3 a b^{4} c^{4} x + \frac{b^{5} c^{4} x^{2}}{2} - \frac{2 a^{5} c^{4} - 9 a^{4} b c^{4} x + 12 a^{3} b^{2} c^{4} x^{2}}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*b**3*c**4*log(x) - 3*a*b**4*c**4*x + b**5*c**4*x**2/2 - (2*a**5*c**4 - 9*
a**4*b*c**4*x + 12*a**3*b**2*c**4*x**2)/(6*x**3)

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GIAC/XCAS [A]  time = 0.239015, size = 100, normalized size = 1.28 \[ \frac{1}{2} \, b^{5} c^{4} x^{2} - 3 \, a b^{4} c^{4} x + 2 \, a^{2} b^{3} c^{4}{\rm ln}\left ({\left | x \right |}\right ) - \frac{12 \, a^{3} b^{2} c^{4} x^{2} - 9 \, a^{4} b c^{4} x + 2 \, a^{5} c^{4}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^5*c^4*x^2 - 3*a*b^4*c^4*x + 2*a^2*b^3*c^4*ln(abs(x)) - 1/6*(12*a^3*b^2*c^4
*x^2 - 9*a^4*b*c^4*x + 2*a^5*c^4)/x^3