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

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

[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 = 0.0361433, 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, Rules used = {75} \[ -\frac{2 a^3 b^2 c^4}{x}+2 a^2 b^3 c^4 \log (x)+\frac{3 a^4 b c^4}{2 x^2}-\frac{a^5 c^4}{3 x^3}-3 a b^4 c^4 x+\frac{1}{2} b^5 c^4 x^2 \]

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]

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^4} \, dx &=\int \left (-3 a b^4 c^4+\frac{a^5 c^4}{x^4}-\frac{3 a^4 b c^4}{x^3}+\frac{2 a^3 b^2 c^4}{x^2}+\frac{2 a^2 b^3 c^4}{x}+b^5 c^4 x\right ) \, dx\\ &=-\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}-3 a b^4 c^4 x+\frac{1}{2} b^5 c^4 x^2+2 a^2 b^3 c^4 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0081004, size = 78, normalized size = 1. \[ -\frac{2 a^3 b^2 c^4}{x}+2 a^2 b^3 c^4 \log (x)+\frac{3 a^4 b c^4}{2 x^2}-\frac{a^5 c^4}{3 x^3}-3 a b^4 c^4 x+\frac{1}{2} b^5 c^4 x^2 \]

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 = 0.006, size = 73, normalized size = 0.9 \begin{align*} -{\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 ) \end{align*}

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 = 1.13871, size = 99, normalized size = 1.27 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/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 = 1.94815, size = 163, normalized size = 2.09 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/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 = 0.439783, size = 78, normalized size = 1. \begin{align*} 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}} \end{align*}

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 [A]  time = 1.17911, size = 100, normalized size = 1.28 \begin{align*} \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 ({\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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