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

Optimal. Leaf size=45 \[ \frac{a^3 b c^3}{x^2}-\frac{a^4 c^3}{3 x^3}+2 a b^3 c^3 \log (x)-b^4 c^3 x \]

[Out]

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

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Rubi [A]  time = 0.0196018, antiderivative size = 45, 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} \[ \frac{a^3 b c^3}{x^2}-\frac{a^4 c^3}{3 x^3}+2 a b^3 c^3 \log (x)-b^4 c^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0071044, size = 37, normalized size = 0.82 \[ c^3 \left (\frac{a^3 b}{x^2}-\frac{a^4}{3 x^3}+2 a b^3 \log (x)-b^4 x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 44, normalized size = 1. \begin{align*} -{\frac{{a}^{4}{c}^{3}}{3\,{x}^{3}}}+{\frac{{a}^{3}b{c}^{3}}{{x}^{2}}}-{b}^{4}{c}^{3}x+2\,a{b}^{3}{c}^{3}\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)^3/x^4,x)

[Out]

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

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Maxima [A]  time = 1.07924, size = 61, normalized size = 1.36 \begin{align*} -b^{4} c^{3} x + 2 \, a b^{3} c^{3} \log \left (x\right ) + \frac{3 \, a^{3} b c^{3} x - a^{4} c^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.95818, size = 104, normalized size = 2.31 \begin{align*} -\frac{3 \, b^{4} c^{3} x^{4} - 6 \, a b^{3} c^{3} x^{3} \log \left (x\right ) - 3 \, a^{3} b c^{3} x + a^{4} c^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.380268, size = 44, normalized size = 0.98 \begin{align*} 2 a b^{3} c^{3} \log{\left (x \right )} - b^{4} c^{3} x + \frac{- a^{4} c^{3} + 3 a^{3} b c^{3} x}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16029, size = 62, normalized size = 1.38 \begin{align*} -b^{4} c^{3} x + 2 \, a b^{3} c^{3} \log \left ({\left | x \right |}\right ) + \frac{3 \, a^{3} b c^{3} x - a^{4} c^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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