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

Optimal. Leaf size=18 \[ -\frac{c^3 (a-b x)^4}{2 x^2} \]

[Out]

-(c^3*(a - b*x)^4)/(2*x^2)

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Rubi [A]  time = 0.0029245, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {74} \[ -\frac{c^3 (a-b x)^4}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^3*(a - b*x)^4)/(2*x^2)

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

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

Mathematica [B]  time = 0.0066017, size = 41, normalized size = 2.28 \[ c^3 \left (\frac{2 a^3 b}{x}-\frac{a^4}{2 x^2}+2 a b^3 x-\frac{b^4 x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.004, size = 38, normalized size = 2.1 \begin{align*}{c}^{3} \left ( -{\frac{{b}^{4}{x}^{2}}{2}}+2\,a{b}^{3}x-{\frac{{a}^{4}}{2\,{x}^{2}}}+2\,{\frac{{a}^{3}b}{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^3,x)

[Out]

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

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Maxima [B]  time = 1.01585, size = 62, normalized size = 3.44 \begin{align*} -\frac{1}{2} \, b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + \frac{4 \, a^{3} b c^{3} x - a^{4} c^{3}}{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)^3/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.82898, size = 92, normalized size = 5.11 \begin{align*} -\frac{b^{4} c^{3} x^{4} - 4 \, a b^{3} c^{3} x^{3} - 4 \, a^{3} b c^{3} x + a^{4} c^{3}}{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)^3/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.31966, size = 46, normalized size = 2.56 \begin{align*} 2 a b^{3} c^{3} x - \frac{b^{4} c^{3} x^{2}}{2} + \frac{- a^{4} c^{3} + 4 a^{3} b c^{3} 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)**3/x**3,x)

[Out]

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

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Giac [B]  time = 1.19523, size = 62, normalized size = 3.44 \begin{align*} -\frac{1}{2} \, b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + \frac{4 \, a^{3} b c^{3} x - a^{4} c^{3}}{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)^3/x^3,x, algorithm="giac")

[Out]

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