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

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

[Out]

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

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Rubi [A]  time = 0.0162868, 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} \[ -2 a^3 b c^3 x+a^4 c^3 \log (x)+\frac{2}{3} a b^3 c^3 x^3-\frac{1}{4} b^4 c^3 x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x,x]

[Out]

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

Mathematica [A]  time = 0.014532, size = 48, normalized size = 1.02 \[ c^3 \left (\frac{1}{12} \left (-24 a^3 b x+19 a^4+8 a b^3 x^3-3 b^4 x^4\right )+a^4 \log (-b x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*((19*a^4 - 24*a^3*b*x + 8*a*b^3*x^3 - 3*b^4*x^4)/12 + a^4*Log[-(b*x)])

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Maple [A]  time = 0.002, size = 44, normalized size = 0.9 \begin{align*} -2\,{a}^{3}b{c}^{3}x+{\frac{2\,a{b}^{3}{c}^{3}{x}^{3}}{3}}-{\frac{{b}^{4}{c}^{3}{x}^{4}}{4}}+{a}^{4}{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,x)

[Out]

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

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Maxima [A]  time = 0.991816, size = 58, normalized size = 1.23 \begin{align*} -\frac{1}{4} \, b^{4} c^{3} x^{4} + \frac{2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \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)^3/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.92242, size = 96, normalized size = 2.04 \begin{align*} -\frac{1}{4} \, b^{4} c^{3} x^{4} + \frac{2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \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)^3/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.288638, size = 48, normalized size = 1.02 \begin{align*} a^{4} c^{3} \log{\left (x \right )} - 2 a^{3} b c^{3} x + \frac{2 a b^{3} c^{3} x^{3}}{3} - \frac{b^{4} c^{3} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15326, size = 59, normalized size = 1.26 \begin{align*} -\frac{1}{4} \, b^{4} c^{3} x^{4} + \frac{2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \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)^3/x,x, algorithm="giac")

[Out]

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