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

Optimal. Leaf size=78 \[ 2 a^2 b^3 c^4 x+2 a^3 b^2 c^4 \log (x)+\frac{3 a^4 b c^4}{x}-\frac{a^5 c^4}{2 x^2}-\frac{3}{2} a b^4 c^4 x^2+\frac{1}{3} b^5 c^4 x^3 \]

[Out]

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

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Rubi [A]  time = 0.0335186, antiderivative size = 78, 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} \[ 2 a^2 b^3 c^4 x+2 a^3 b^2 c^4 \log (x)+\frac{3 a^4 b c^4}{x}-\frac{a^5 c^4}{2 x^2}-\frac{3}{2} a b^4 c^4 x^2+\frac{1}{3} b^5 c^4 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0083213, size = 78, normalized size = 1. \[ 2 a^2 b^3 c^4 x+2 a^3 b^2 c^4 \log (x)+\frac{3 a^4 b c^4}{x}-\frac{a^5 c^4}{2 x^2}-\frac{3}{2} a b^4 c^4 x^2+\frac{1}{3} b^5 c^4 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 73, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}{c}^{4}}{2\,{x}^{2}}}+3\,{\frac{{a}^{4}b{c}^{4}}{x}}+2\,{a}^{2}{b}^{3}{c}^{4}x-{\frac{3\,a{b}^{4}{c}^{4}{x}^{2}}{2}}+{\frac{{b}^{5}{c}^{4}{x}^{3}}{3}}+2\,{a}^{3}{b}^{2}{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^3,x)

[Out]

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

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Maxima [A]  time = 1.09586, size = 99, normalized size = 1.27 \begin{align*} \frac{1}{3} \, b^{5} c^{4} x^{3} - \frac{3}{2} \, a b^{4} c^{4} x^{2} + 2 \, a^{2} b^{3} c^{4} x + 2 \, a^{3} b^{2} c^{4} \log \left (x\right ) + \frac{6 \, a^{4} b c^{4} x - a^{5} c^{4}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^5*c^4*x^3 - 3/2*a*b^4*c^4*x^2 + 2*a^2*b^3*c^4*x + 2*a^3*b^2*c^4*log(x) + 1/2*(6*a^4*b*c^4*x - a^5*c^4)/x
^2

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Fricas [A]  time = 1.95981, size = 163, normalized size = 2.09 \begin{align*} \frac{2 \, b^{5} c^{4} x^{5} - 9 \, a b^{4} c^{4} x^{4} + 12 \, a^{2} b^{3} c^{4} x^{3} + 12 \, a^{3} b^{2} c^{4} x^{2} \log \left (x\right ) + 18 \, a^{4} b c^{4} x - 3 \, a^{5} c^{4}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^5*c^4*x^5 - 9*a*b^4*c^4*x^4 + 12*a^2*b^3*c^4*x^3 + 12*a^3*b^2*c^4*x^2*log(x) + 18*a^4*b*c^4*x - 3*a^5
*c^4)/x^2

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Sympy [A]  time = 0.394437, size = 78, normalized size = 1. \begin{align*} 2 a^{3} b^{2} c^{4} \log{\left (x \right )} + 2 a^{2} b^{3} c^{4} x - \frac{3 a b^{4} c^{4} x^{2}}{2} + \frac{b^{5} c^{4} x^{3}}{3} + \frac{- a^{5} c^{4} + 6 a^{4} b c^{4} x}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*b**2*c**4*log(x) + 2*a**2*b**3*c**4*x - 3*a*b**4*c**4*x**2/2 + b**5*c**4*x**3/3 + (-a**5*c**4 + 6*a**4*
b*c**4*x)/(2*x**2)

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Giac [A]  time = 1.19352, size = 100, normalized size = 1.28 \begin{align*} \frac{1}{3} \, b^{5} c^{4} x^{3} - \frac{3}{2} \, a b^{4} c^{4} x^{2} + 2 \, a^{2} b^{3} c^{4} x + 2 \, a^{3} b^{2} c^{4} \log \left ({\left | x \right |}\right ) + \frac{6 \, a^{4} b c^{4} x - a^{5} c^{4}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^5*c^4*x^3 - 3/2*a*b^4*c^4*x^2 + 2*a^2*b^3*c^4*x + 2*a^3*b^2*c^4*log(abs(x)) + 1/2*(6*a^4*b*c^4*x - a^5*c
^4)/x^2