3.10 \(\int \frac{(a+b x) (a c-b c x)^3}{x^6} \, dx\)

Optimal. Leaf size=50 \[ \frac{a^3 b c^3}{2 x^4}-\frac{a^4 c^3}{5 x^5}-\frac{a b^3 c^3}{x^2}+\frac{b^4 c^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0196034, antiderivative size = 50, 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} \[ \frac{a^3 b c^3}{2 x^4}-\frac{a^4 c^3}{5 x^5}-\frac{a b^3 c^3}{x^2}+\frac{b^4 c^3}{x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0063977, size = 42, normalized size = 0.84 \[ c^3 \left (\frac{a^3 b}{2 x^4}-\frac{a^4}{5 x^5}-\frac{a b^3}{x^2}+\frac{b^4}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^6,x]

[Out]

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

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Maple [A]  time = 0.004, size = 39, normalized size = 0.8 \begin{align*}{c}^{3} \left ( -{\frac{{a}^{4}}{5\,{x}^{5}}}+{\frac{{a}^{3}b}{2\,{x}^{4}}}-{\frac{a{b}^{3}}{{x}^{2}}}+{\frac{{b}^{4}}{x}} \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^6,x)

[Out]

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

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Maxima [A]  time = 1.03014, size = 63, normalized size = 1.26 \begin{align*} \frac{10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.94414, size = 100, normalized size = 2. \begin{align*} \frac{10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.488611, size = 49, normalized size = 0.98 \begin{align*} \frac{- 2 a^{4} c^{3} + 5 a^{3} b c^{3} x - 10 a b^{3} c^{3} x^{3} + 10 b^{4} c^{3} x^{4}}{10 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**6,x)

[Out]

(-2*a**4*c**3 + 5*a**3*b*c**3*x - 10*a*b**3*c**3*x**3 + 10*b**4*c**3*x**4)/(10*x**5)

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Giac [A]  time = 1.19147, size = 63, normalized size = 1.26 \begin{align*} \frac{10 \, b^{4} c^{3} x^{4} - 10 \, a b^{3} c^{3} x^{3} + 5 \, a^{3} b c^{3} x - 2 \, a^{4} c^{3}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^6,x, algorithm="giac")

[Out]

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