3.6 \(\int \frac{(a+b x) (a c-b c x)^3}{x^2} \, dx\)

Optimal. Leaf size=47 \[ -2 a^3 b c^3 \log (x)-\frac{a^4 c^3}{x}+a b^3 c^3 x^2-\frac{1}{3} b^4 c^3 x^3 \]

[Out]

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

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Rubi [A]  time = 0.0206908, antiderivative size = 47, 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} \[ -2 a^3 b c^3 \log (x)-\frac{a^4 c^3}{x}+a b^3 c^3 x^2-\frac{1}{3} b^4 c^3 x^3 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0069652, size = 39, normalized size = 0.83 \[ c^3 \left (-2 a^3 b \log (x)-\frac{a^4}{x}+a b^3 x^2-\frac{b^4 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^2,x]

[Out]

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

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Maple [A]  time = 0.004, size = 46, normalized size = 1. \begin{align*} -{\frac{{a}^{4}{c}^{3}}{x}}+a{b}^{3}{c}^{3}{x}^{2}-{\frac{{b}^{4}{c}^{3}{x}^{3}}{3}}-2\,{a}^{3}b{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^2,x)

[Out]

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

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Maxima [A]  time = 1.04786, size = 61, normalized size = 1.3 \begin{align*} -\frac{1}{3} \, b^{4} c^{3} x^{3} + a b^{3} c^{3} x^{2} - 2 \, a^{3} b c^{3} \log \left (x\right ) - \frac{a^{4} c^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92426, size = 101, normalized size = 2.15 \begin{align*} -\frac{b^{4} c^{3} x^{4} - 3 \, a b^{3} c^{3} x^{3} + 6 \, a^{3} b c^{3} x \log \left (x\right ) + 3 \, a^{4} c^{3}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.312491, size = 44, normalized size = 0.94 \begin{align*} - \frac{a^{4} c^{3}}{x} - 2 a^{3} b c^{3} \log{\left (x \right )} + a b^{3} c^{3} x^{2} - \frac{b^{4} c^{3} x^{3}}{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**2,x)

[Out]

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

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Giac [A]  time = 1.20103, size = 62, normalized size = 1.32 \begin{align*} -\frac{1}{3} \, b^{4} c^{3} x^{3} + a b^{3} c^{3} x^{2} - 2 \, a^{3} b c^{3} \log \left ({\left | x \right |}\right ) - \frac{a^{4} c^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^2,x, algorithm="giac")

[Out]

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