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

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

[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 = 0.0309691, 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, Rules used = {75} \[ a^3 b^2 c^4 x^2+\frac{2}{3} a^2 b^3 c^4 x^3-3 a^4 b c^4 x+a^5 c^4 \log (x)-\frac{3}{4} a b^4 c^4 x^4+\frac{1}{5} b^5 c^4 x^5 \]

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]

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x) (a c-b c x)^4}{x} \, dx &=\int \left (-3 a^4 b c^4+\frac{a^5 c^4}{x}+2 a^3 b^2 c^4 x+2 a^2 b^3 c^4 x^2-3 a b^4 c^4 x^3+b^5 c^4 x^4\right ) \, dx\\ &=-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+a^5 c^4 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0196865, size = 70, normalized size = 0.92 \[ \frac{1}{60} c^4 \left (60 a^3 b^2 x^2+40 a^2 b^3 x^3+60 a^5 \log (-b c x)-180 a^4 b x+113 a^5-45 a b^4 x^4+12 b^5 x^5\right ) \]

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 + 12*b^5*x^5 + 60*a^5*Log[-(b*c*x
)]))/60

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Maple [A]  time = 0.002, size = 71, normalized size = 0.9 \begin{align*} -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 ) \end{align*}

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 = 1.06798, size = 95, normalized size = 1.25 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/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 = 1.9674, size = 149, normalized size = 1.96 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/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 = 0.312422, size = 78, normalized size = 1.03 \begin{align*} 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} \end{align*}

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 [A]  time = 1.1905, size = 96, normalized size = 1.26 \begin{align*} \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 ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/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*log(abs(
x))