3.19 \(\int \frac{(a+b x) (a c-b c x)^4}{x^2} \, dx\)

Optimal. Leaf size=73 \[ a^2 b^3 c^4 x^2+2 a^3 b^2 c^4 x-3 a^4 b c^4 \log (x)-\frac{a^5 c^4}{x}-a b^4 c^4 x^3+\frac{1}{4} b^5 c^4 x^4 \]

[Out]

-((a^5*c^4)/x) + 2*a^3*b^2*c^4*x + a^2*b^3*c^4*x^2 - a*b^4*c^4*x^3 + (b^5*c^4*x^4)/4 - 3*a^4*b*c^4*Log[x]

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Rubi [A]  time = 0.0349168, antiderivative size = 73, 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} \[ a^2 b^3 c^4 x^2+2 a^3 b^2 c^4 x-3 a^4 b c^4 \log (x)-\frac{a^5 c^4}{x}-a b^4 c^4 x^3+\frac{1}{4} b^5 c^4 x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0084405, size = 73, normalized size = 1. \[ a^2 b^3 c^4 x^2+2 a^3 b^2 c^4 x-3 a^4 b c^4 \log (x)-\frac{a^5 c^4}{x}-a b^4 c^4 x^3+\frac{1}{4} b^5 c^4 x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^2,x]

[Out]

-((a^5*c^4)/x) + 2*a^3*b^2*c^4*x + a^2*b^3*c^4*x^2 - a*b^4*c^4*x^3 + (b^5*c^4*x^4)/4 - 3*a^4*b*c^4*Log[x]

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Maple [A]  time = 0.006, size = 72, normalized size = 1. \begin{align*} -{\frac{{a}^{5}{c}^{4}}{x}}+2\,{a}^{3}{b}^{2}{c}^{4}x+{a}^{2}{b}^{3}{c}^{4}{x}^{2}-a{b}^{4}{c}^{4}{x}^{3}+{\frac{{b}^{5}{c}^{4}{x}^{4}}{4}}-3\,{a}^{4}b{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^2,x)

[Out]

-a^5*c^4/x+2*a^3*b^2*c^4*x+a^2*b^3*c^4*x^2-a*b^4*c^4*x^3+1/4*b^5*c^4*x^4-3*a^4*b*c^4*ln(x)

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Maxima [A]  time = 1.0459, size = 96, normalized size = 1.32 \begin{align*} \frac{1}{4} \, b^{5} c^{4} x^{4} - a b^{4} c^{4} x^{3} + a^{2} b^{3} c^{4} x^{2} + 2 \, a^{3} b^{2} c^{4} x - 3 \, a^{4} b c^{4} \log \left (x\right ) - \frac{a^{5} c^{4}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^2,x, algorithm="maxima")

[Out]

1/4*b^5*c^4*x^4 - a*b^4*c^4*x^3 + a^2*b^3*c^4*x^2 + 2*a^3*b^2*c^4*x - 3*a^4*b*c^4*log(x) - a^5*c^4/x

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Fricas [A]  time = 1.97873, size = 155, normalized size = 2.12 \begin{align*} \frac{b^{5} c^{4} x^{5} - 4 \, a b^{4} c^{4} x^{4} + 4 \, a^{2} b^{3} c^{4} x^{3} + 8 \, a^{3} b^{2} c^{4} x^{2} - 12 \, a^{4} b c^{4} x \log \left (x\right ) - 4 \, a^{5} c^{4}}{4 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^2,x, algorithm="fricas")

[Out]

1/4*(b^5*c^4*x^5 - 4*a*b^4*c^4*x^4 + 4*a^2*b^3*c^4*x^3 + 8*a^3*b^2*c^4*x^2 - 12*a^4*b*c^4*x*log(x) - 4*a^5*c^4
)/x

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Sympy [A]  time = 0.377275, size = 71, normalized size = 0.97 \begin{align*} - \frac{a^{5} c^{4}}{x} - 3 a^{4} b c^{4} \log{\left (x \right )} + 2 a^{3} b^{2} c^{4} x + a^{2} b^{3} c^{4} x^{2} - a b^{4} c^{4} x^{3} + \frac{b^{5} c^{4} 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)**4/x**2,x)

[Out]

-a**5*c**4/x - 3*a**4*b*c**4*log(x) + 2*a**3*b**2*c**4*x + a**2*b**3*c**4*x**2 - a*b**4*c**4*x**3 + b**5*c**4*
x**4/4

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Giac [A]  time = 1.19384, size = 97, normalized size = 1.33 \begin{align*} \frac{1}{4} \, b^{5} c^{4} x^{4} - a b^{4} c^{4} x^{3} + a^{2} b^{3} c^{4} x^{2} + 2 \, a^{3} b^{2} c^{4} x - 3 \, a^{4} b c^{4} \log \left ({\left | x \right |}\right ) - \frac{a^{5} c^{4}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^2,x, algorithm="giac")

[Out]

1/4*b^5*c^4*x^4 - a*b^4*c^4*x^3 + a^2*b^3*c^4*x^2 + 2*a^3*b^2*c^4*x - 3*a^4*b*c^4*log(abs(x)) - a^5*c^4/x