3.12 \(\int \frac{(a+b x) (a c-b c x)^3}{x^8} \, dx\)

Optimal. Leaf size=55 \[ \frac{a^3 b c^3}{3 x^6}-\frac{a^4 c^3}{7 x^7}-\frac{a b^3 c^3}{2 x^4}+\frac{b^4 c^3}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0199046, antiderivative size = 55, 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} \[ \frac{a^3 b c^3}{3 x^6}-\frac{a^4 c^3}{7 x^7}-\frac{a b^3 c^3}{2 x^4}+\frac{b^4 c^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^8,x]

[Out]

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

Mathematica [A]  time = 0.0063669, size = 47, normalized size = 0.85 \[ c^3 \left (\frac{a^3 b}{3 x^6}-\frac{a^4}{7 x^7}-\frac{a b^3}{2 x^4}+\frac{b^4}{3 x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^8,x]

[Out]

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

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Maple [A]  time = 0.006, size = 40, normalized size = 0.7 \begin{align*}{c}^{3} \left ({\frac{{b}^{4}}{3\,{x}^{3}}}-{\frac{a{b}^{3}}{2\,{x}^{4}}}+{\frac{{a}^{3}b}{3\,{x}^{6}}}-{\frac{{a}^{4}}{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)^3/x^8,x)

[Out]

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

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Maxima [A]  time = 1.04941, size = 63, normalized size = 1.15 \begin{align*} \frac{14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^8,x, algorithm="maxima")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7

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Fricas [A]  time = 1.93638, size = 101, normalized size = 1.84 \begin{align*} \frac{14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^8,x, algorithm="fricas")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7

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Sympy [A]  time = 0.490534, size = 49, normalized size = 0.89 \begin{align*} \frac{- 6 a^{4} c^{3} + 14 a^{3} b c^{3} x - 21 a b^{3} c^{3} x^{3} + 14 b^{4} c^{3} x^{4}}{42 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**8,x)

[Out]

(-6*a**4*c**3 + 14*a**3*b*c**3*x - 21*a*b**3*c**3*x**3 + 14*b**4*c**3*x**4)/(42*x**7)

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Giac [A]  time = 1.19342, size = 63, normalized size = 1.15 \begin{align*} \frac{14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^8,x, algorithm="giac")

[Out]

1/42*(14*b^4*c^3*x^4 - 21*a*b^3*c^3*x^3 + 14*a^3*b*c^3*x - 6*a^4*c^3)/x^7