3.26 \(\int \frac{(a+b x) (a c-b c x)^4}{x^9} \, dx\)

Optimal. Leaf size=87 \[ -\frac{a^3 b^2 c^4}{3 x^6}-\frac{2 a^2 b^3 c^4}{5 x^5}+\frac{3 a^4 b c^4}{7 x^7}-\frac{a^5 c^4}{8 x^8}+\frac{3 a b^4 c^4}{4 x^4}-\frac{b^5 c^4}{3 x^3} \]

[Out]

-(a^5*c^4)/(8*x^8) + (3*a^4*b*c^4)/(7*x^7) - (a^3*b^2*c^4)/(3*x^6) - (2*a^2*b^3*c^4)/(5*x^5) + (3*a*b^4*c^4)/(
4*x^4) - (b^5*c^4)/(3*x^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0353926, antiderivative size = 87, 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} \[ -\frac{a^3 b^2 c^4}{3 x^6}-\frac{2 a^2 b^3 c^4}{5 x^5}+\frac{3 a^4 b c^4}{7 x^7}-\frac{a^5 c^4}{8 x^8}+\frac{3 a b^4 c^4}{4 x^4}-\frac{b^5 c^4}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^9,x]

[Out]

-(a^5*c^4)/(8*x^8) + (3*a^4*b*c^4)/(7*x^7) - (a^3*b^2*c^4)/(3*x^6) - (2*a^2*b^3*c^4)/(5*x^5) + (3*a*b^4*c^4)/(
4*x^4) - (b^5*c^4)/(3*x^3)

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^9} \, dx &=\int \left (\frac{a^5 c^4}{x^9}-\frac{3 a^4 b c^4}{x^8}+\frac{2 a^3 b^2 c^4}{x^7}+\frac{2 a^2 b^3 c^4}{x^6}-\frac{3 a b^4 c^4}{x^5}+\frac{b^5 c^4}{x^4}\right ) \, dx\\ &=-\frac{a^5 c^4}{8 x^8}+\frac{3 a^4 b c^4}{7 x^7}-\frac{a^3 b^2 c^4}{3 x^6}-\frac{2 a^2 b^3 c^4}{5 x^5}+\frac{3 a b^4 c^4}{4 x^4}-\frac{b^5 c^4}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0081894, size = 87, normalized size = 1. \[ -\frac{a^3 b^2 c^4}{3 x^6}-\frac{2 a^2 b^3 c^4}{5 x^5}+\frac{3 a^4 b c^4}{7 x^7}-\frac{a^5 c^4}{8 x^8}+\frac{3 a b^4 c^4}{4 x^4}-\frac{b^5 c^4}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^9,x]

[Out]

-(a^5*c^4)/(8*x^8) + (3*a^4*b*c^4)/(7*x^7) - (a^3*b^2*c^4)/(3*x^6) - (2*a^2*b^3*c^4)/(5*x^5) + (3*a*b^4*c^4)/(
4*x^4) - (b^5*c^4)/(3*x^3)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 62, normalized size = 0.7 \begin{align*}{c}^{4} \left ( -{\frac{{b}^{5}}{3\,{x}^{3}}}-{\frac{2\,{a}^{2}{b}^{3}}{5\,{x}^{5}}}+{\frac{3\,a{b}^{4}}{4\,{x}^{4}}}-{\frac{{a}^{5}}{8\,{x}^{8}}}-{\frac{{a}^{3}{b}^{2}}{3\,{x}^{6}}}+{\frac{3\,{a}^{4}b}{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^9,x)

[Out]

c^4*(-1/3*b^5/x^3-2/5*a^2*b^3/x^5+3/4*a*b^4/x^4-1/8*a^5/x^8-1/3*a^3*b^2/x^6+3/7*a^4*b/x^7)

________________________________________________________________________________________

Maxima [A]  time = 1.02743, size = 101, normalized size = 1.16 \begin{align*} -\frac{280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="maxima")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8

________________________________________________________________________________________

Fricas [A]  time = 1.94555, size = 170, normalized size = 1.95 \begin{align*} -\frac{280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="fricas")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8

________________________________________________________________________________________

Sympy [A]  time = 0.668314, size = 82, normalized size = 0.94 \begin{align*} - \frac{105 a^{5} c^{4} - 360 a^{4} b c^{4} x + 280 a^{3} b^{2} c^{4} x^{2} + 336 a^{2} b^{3} c^{4} x^{3} - 630 a b^{4} c^{4} x^{4} + 280 b^{5} c^{4} x^{5}}{840 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**9,x)

[Out]

-(105*a**5*c**4 - 360*a**4*b*c**4*x + 280*a**3*b**2*c**4*x**2 + 336*a**2*b**3*c**4*x**3 - 630*a*b**4*c**4*x**4
 + 280*b**5*c**4*x**5)/(840*x**8)

________________________________________________________________________________________

Giac [A]  time = 1.30532, size = 101, normalized size = 1.16 \begin{align*} -\frac{280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="giac")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8