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

Optimal. Leaf size=76 \[ a^5 c^4 \log (x)-3 a^4 b c^4 x+a^3 b^2 c^4 x^2+\frac{2}{3} a^2 b^3 c^4 x^3-\frac{3}{4} a b^4 c^4 x^4+\frac{1}{5} b^5 c^4 x^5 \]

[Out]

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

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Rubi [A]  time = 0.0760721, antiderivative size = 76, 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 \[ a^5 c^4 \log (x)-3 a^4 b c^4 x+a^3 b^2 c^4 x^2+\frac{2}{3} a^2 b^3 c^4 x^3-\frac{3}{4} a b^4 c^4 x^4+\frac{1}{5} b^5 c^4 x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0308659, size = 70, normalized size = 0.92 \[ \frac{1}{60} c^4 \left (60 a^5 \log (-b c x)+113 a^5-180 a^4 b x+60 a^3 b^2 x^2+40 a^2 b^3 x^3-45 a b^4 x^4+12 b^5 x^5\right ) \]

Antiderivative was successfully verified.

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

[Out]

(c^4*(113*a^5 - 180*a^4*b*x + 60*a^3*b^2*x^2 + 40*a^2*b^3*x^3 - 45*a*b^4*x^4 + 1
2*b^5*x^5 + 60*a^5*Log[-(b*c*x)]))/60

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Maple [A]  time = 0.004, size = 71, normalized size = 0.9 \[ -3\,{a}^{4}b{c}^{4}x+{a}^{3}{b}^{2}{c}^{4}{x}^{2}+{\frac{2\,{a}^{2}{b}^{3}{c}^{4}{x}^{3}}{3}}-{\frac{3\,a{b}^{4}{c}^{4}{x}^{4}}{4}}+{\frac{{b}^{5}{c}^{4}{x}^{5}}{5}}+{a}^{5}{c}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^4/x,x)

[Out]

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

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Maxima [A]  time = 1.33748, size = 95, normalized size = 1.25 \[ \frac{1}{5} \, b^{5} c^{4} x^{5} - \frac{3}{4} \, a b^{4} c^{4} x^{4} + \frac{2}{3} \, a^{2} b^{3} c^{4} x^{3} + a^{3} b^{2} c^{4} x^{2} - 3 \, a^{4} b c^{4} x + a^{5} c^{4} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203911, size = 95, normalized size = 1.25 \[ \frac{1}{5} \, b^{5} c^{4} x^{5} - \frac{3}{4} \, a b^{4} c^{4} x^{4} + \frac{2}{3} \, a^{2} b^{3} c^{4} x^{3} + a^{3} b^{2} c^{4} x^{2} - 3 \, a^{4} b c^{4} x + a^{5} c^{4} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.637898, size = 78, normalized size = 1.03 \[ a^{5} c^{4} \log{\left (x \right )} - 3 a^{4} b c^{4} x + a^{3} b^{2} c^{4} x^{2} + \frac{2 a^{2} b^{3} c^{4} x^{3}}{3} - \frac{3 a b^{4} c^{4} x^{4}}{4} + \frac{b^{5} c^{4} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.240822, size = 96, normalized size = 1.26 \[ \frac{1}{5} \, b^{5} c^{4} x^{5} - \frac{3}{4} \, a b^{4} c^{4} x^{4} + \frac{2}{3} \, a^{2} b^{3} c^{4} x^{3} + a^{3} b^{2} c^{4} x^{2} - 3 \, a^{4} b c^{4} x + a^{5} c^{4}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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