3.9 \(\int \frac{(a+b x) (a c-b c x)^3}{x^5} \, dx\)

Optimal. Leaf size=50 \[ \frac{2 a^3 b c^3}{3 x^3}-\frac{a^4 c^3}{4 x^4}-\frac{2 a b^3 c^3}{x}-b^4 c^3 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0208749, antiderivative size = 50, 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} \[ \frac{2 a^3 b c^3}{3 x^3}-\frac{a^4 c^3}{4 x^4}-\frac{2 a b^3 c^3}{x}-b^4 c^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^5,x]

[Out]

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

Mathematica [A]  time = 0.007259, size = 42, normalized size = 0.84 \[ c^3 \left (\frac{2 a^3 b}{3 x^3}-\frac{a^4}{4 x^4}-\frac{2 a b^3}{x}-b^4 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^5,x]

[Out]

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

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Maple [A]  time = 0.006, size = 47, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}{c}^{3}}{4\,{x}^{4}}}+{\frac{2\,{a}^{3}b{c}^{3}}{3\,{x}^{3}}}-2\,{\frac{a{b}^{3}{c}^{3}}{x}}-{b}^{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^5,x)

[Out]

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

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Maxima [A]  time = 0.986706, size = 63, normalized size = 1.26 \begin{align*} -b^{4} c^{3} \log \left (x\right ) - \frac{24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{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^5,x, algorithm="maxima")

[Out]

-b^4*c^3*log(x) - 1/12*(24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4

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Fricas [A]  time = 1.95638, size = 111, normalized size = 2.22 \begin{align*} -\frac{12 \, b^{4} c^{3} x^{4} \log \left (x\right ) + 24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{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^5,x, algorithm="fricas")

[Out]

-1/12*(12*b^4*c^3*x^4*log(x) + 24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4

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Sympy [A]  time = 0.422533, size = 49, normalized size = 0.98 \begin{align*} - b^{4} c^{3} \log{\left (x \right )} - \frac{3 a^{4} c^{3} - 8 a^{3} b c^{3} x + 24 a b^{3} c^{3} x^{3}}{12 x^{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**5,x)

[Out]

-b**4*c**3*log(x) - (3*a**4*c**3 - 8*a**3*b*c**3*x + 24*a*b**3*c**3*x**3)/(12*x**4)

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Giac [A]  time = 1.18806, size = 65, normalized size = 1.3 \begin{align*} -b^{4} c^{3} \log \left ({\left | x \right |}\right ) - \frac{24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{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^5,x, algorithm="giac")

[Out]

-b^4*c^3*log(abs(x)) - 1/12*(24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4