3.11 \(\int \frac{(a+b x) (a c-b c x)^3}{x^7} \, dx\)

Optimal. Leaf size=55 \[ \frac{2 a^3 b c^3}{5 x^5}-\frac{a^4 c^3}{6 x^6}-\frac{2 a b^3 c^3}{3 x^3}+\frac{b^4 c^3}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0211831, 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} \[ \frac{2 a^3 b c^3}{5 x^5}-\frac{a^4 c^3}{6 x^6}-\frac{2 a b^3 c^3}{3 x^3}+\frac{b^4 c^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

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

Mathematica [A]  time = 0.0059323, size = 47, normalized size = 0.85 \[ c^3 \left (\frac{2 a^3 b}{5 x^5}-\frac{a^4}{6 x^6}-\frac{2 a b^3}{3 x^3}+\frac{b^4}{2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^7,x]

[Out]

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

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Maple [A]  time = 0.006, size = 40, normalized size = 0.7 \begin{align*}{c}^{3} \left ( -{\frac{2\,a{b}^{3}}{3\,{x}^{3}}}+{\frac{2\,{a}^{3}b}{5\,{x}^{5}}}+{\frac{{b}^{4}}{2\,{x}^{2}}}-{\frac{{a}^{4}}{6\,{x}^{6}}} \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^7,x)

[Out]

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

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Maxima [A]  time = 1.0501, size = 63, normalized size = 1.15 \begin{align*} \frac{15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="maxima")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

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Fricas [A]  time = 1.91766, size = 101, normalized size = 1.84 \begin{align*} \frac{15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="fricas")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6

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Sympy [A]  time = 0.46074, size = 49, normalized size = 0.89 \begin{align*} \frac{- 5 a^{4} c^{3} + 12 a^{3} b c^{3} x - 20 a b^{3} c^{3} x^{3} + 15 b^{4} c^{3} x^{4}}{30 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**7,x)

[Out]

(-5*a**4*c**3 + 12*a**3*b*c**3*x - 20*a*b**3*c**3*x**3 + 15*b**4*c**3*x**4)/(30*x**6)

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Giac [A]  time = 1.20704, size = 63, normalized size = 1.15 \begin{align*} \frac{15 \, b^{4} c^{3} x^{4} - 20 \, a b^{3} c^{3} x^{3} + 12 \, a^{3} b c^{3} x - 5 \, a^{4} c^{3}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^7,x, algorithm="giac")

[Out]

1/30*(15*b^4*c^3*x^4 - 20*a*b^3*c^3*x^3 + 12*a^3*b*c^3*x - 5*a^4*c^3)/x^6