3.25 \(\int \frac{(a+b x) (a c-b c x)^4}{x^8} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 a^3 b^2 c^4}{5 x^5}-\frac{a^2 b^3 c^4}{2 x^4}+\frac{a^4 b c^4}{2 x^6}-\frac{a^5 c^4}{7 x^7}+\frac{a b^4 c^4}{x^3}-\frac{b^5 c^4}{2 x^2} \]

[Out]

-(a^5*c^4)/(7*x^7) + (a^4*b*c^4)/(2*x^6) - (2*a^3*b^2*c^4)/(5*x^5) - (a^2*b^3*c^4)/(2*x^4) + (a*b^4*c^4)/x^3 -
(b^5*c^4)/(2*x^2)

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Rubi [A]  time = 0.0328864, antiderivative size = 84, 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} \[ -\frac{2 a^3 b^2 c^4}{5 x^5}-\frac{a^2 b^3 c^4}{2 x^4}+\frac{a^4 b c^4}{2 x^6}-\frac{a^5 c^4}{7 x^7}+\frac{a b^4 c^4}{x^3}-\frac{b^5 c^4}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

-(a^5*c^4)/(7*x^7) + (a^4*b*c^4)/(2*x^6) - (2*a^3*b^2*c^4)/(5*x^5) - (a^2*b^3*c^4)/(2*x^4) + (a*b^4*c^4)/x^3 -
(b^5*c^4)/(2*x^2)

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^8} \, dx &=\int \left (\frac{a^5 c^4}{x^8}-\frac{3 a^4 b c^4}{x^7}+\frac{2 a^3 b^2 c^4}{x^6}+\frac{2 a^2 b^3 c^4}{x^5}-\frac{3 a b^4 c^4}{x^4}+\frac{b^5 c^4}{x^3}\right ) \, dx\\ &=-\frac{a^5 c^4}{7 x^7}+\frac{a^4 b c^4}{2 x^6}-\frac{2 a^3 b^2 c^4}{5 x^5}-\frac{a^2 b^3 c^4}{2 x^4}+\frac{a b^4 c^4}{x^3}-\frac{b^5 c^4}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0075268, size = 84, normalized size = 1. \[ -\frac{2 a^3 b^2 c^4}{5 x^5}-\frac{a^2 b^3 c^4}{2 x^4}+\frac{a^4 b c^4}{2 x^6}-\frac{a^5 c^4}{7 x^7}+\frac{a b^4 c^4}{x^3}-\frac{b^5 c^4}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

-(a^5*c^4)/(7*x^7) + (a^4*b*c^4)/(2*x^6) - (2*a^3*b^2*c^4)/(5*x^5) - (a^2*b^3*c^4)/(2*x^4) + (a*b^4*c^4)/x^3 -
(b^5*c^4)/(2*x^2)

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Maple [A]  time = 0.005, size = 61, normalized size = 0.7 \begin{align*}{c}^{4} \left ({\frac{a{b}^{4}}{{x}^{3}}}-{\frac{2\,{a}^{3}{b}^{2}}{5\,{x}^{5}}}-{\frac{{a}^{2}{b}^{3}}{2\,{x}^{4}}}-{\frac{{b}^{5}}{2\,{x}^{2}}}+{\frac{{a}^{4}b}{2\,{x}^{6}}}-{\frac{{a}^{5}}{7\,{x}^{7}}} \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^8,x)

[Out]

c^4*(a*b^4/x^3-2/5*a^3*b^2/x^5-1/2*a^2*b^3/x^4-1/2*b^5/x^2+1/2*a^4*b/x^6-1/7*a^5/x^7)

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Maxima [A]  time = 1.08575, size = 101, normalized size = 1.2 \begin{align*} -\frac{35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="maxima")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7

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Fricas [A]  time = 1.94491, size = 161, normalized size = 1.92 \begin{align*} -\frac{35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="fricas")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7

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Sympy [A]  time = 0.6672, size = 82, normalized size = 0.98 \begin{align*} - \frac{10 a^{5} c^{4} - 35 a^{4} b c^{4} x + 28 a^{3} b^{2} c^{4} x^{2} + 35 a^{2} b^{3} c^{4} x^{3} - 70 a b^{4} c^{4} x^{4} + 35 b^{5} c^{4} x^{5}}{70 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**8,x)

[Out]

-(10*a**5*c**4 - 35*a**4*b*c**4*x + 28*a**3*b**2*c**4*x**2 + 35*a**2*b**3*c**4*x**3 - 70*a*b**4*c**4*x**4 + 35
*b**5*c**4*x**5)/(70*x**7)

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Giac [A]  time = 1.19863, size = 101, normalized size = 1.2 \begin{align*} -\frac{35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="giac")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7