3.42 \(\int \frac{(a+b x) (a c-b c x)^5}{x^{11}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{5 a^4 b^2 c^5}{8 x^8}+\frac{5 a^2 b^4 c^5}{6 x^6}+\frac{4 a^5 b c^5}{9 x^9}-\frac{a^6 c^5}{10 x^{10}}-\frac{4 a b^5 c^5}{5 x^5}+\frac{b^6 c^5}{4 x^4} \]

[Out]

-(a^6*c^5)/(10*x^10) + (4*a^5*b*c^5)/(9*x^9) - (5*a^4*b^2*c^5)/(8*x^8) + (5*a^2*b^4*c^5)/(6*x^6) - (4*a*b^5*c^
5)/(5*x^5) + (b^6*c^5)/(4*x^4)

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Rubi [A]  time = 0.0334295, 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} \[ -\frac{5 a^4 b^2 c^5}{8 x^8}+\frac{5 a^2 b^4 c^5}{6 x^6}+\frac{4 a^5 b c^5}{9 x^9}-\frac{a^6 c^5}{10 x^{10}}-\frac{4 a b^5 c^5}{5 x^5}+\frac{b^6 c^5}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^11,x]

[Out]

-(a^6*c^5)/(10*x^10) + (4*a^5*b*c^5)/(9*x^9) - (5*a^4*b^2*c^5)/(8*x^8) + (5*a^2*b^4*c^5)/(6*x^6) - (4*a*b^5*c^
5)/(5*x^5) + (b^6*c^5)/(4*x^4)

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)^5}{x^{11}} \, dx &=\int \left (\frac{a^6 c^5}{x^{11}}-\frac{4 a^5 b c^5}{x^{10}}+\frac{5 a^4 b^2 c^5}{x^9}-\frac{5 a^2 b^4 c^5}{x^7}+\frac{4 a b^5 c^5}{x^6}-\frac{b^6 c^5}{x^5}\right ) \, dx\\ &=-\frac{a^6 c^5}{10 x^{10}}+\frac{4 a^5 b c^5}{9 x^9}-\frac{5 a^4 b^2 c^5}{8 x^8}+\frac{5 a^2 b^4 c^5}{6 x^6}-\frac{4 a b^5 c^5}{5 x^5}+\frac{b^6 c^5}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0067463, size = 73, normalized size = 0.84 \[ c^5 \left (-\frac{5 a^4 b^2}{8 x^8}+\frac{5 a^2 b^4}{6 x^6}+\frac{4 a^5 b}{9 x^9}-\frac{a^6}{10 x^{10}}-\frac{4 a b^5}{5 x^5}+\frac{b^6}{4 x^4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^11,x]

[Out]

c^5*(-a^6/(10*x^10) + (4*a^5*b)/(9*x^9) - (5*a^4*b^2)/(8*x^8) + (5*a^2*b^4)/(6*x^6) - (4*a*b^5)/(5*x^5) + b^6/
(4*x^4))

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Maple [A]  time = 0.005, size = 62, normalized size = 0.7 \begin{align*}{c}^{5} \left ( -{\frac{4\,a{b}^{5}}{5\,{x}^{5}}}+{\frac{{b}^{6}}{4\,{x}^{4}}}-{\frac{5\,{a}^{4}{b}^{2}}{8\,{x}^{8}}}+{\frac{5\,{a}^{2}{b}^{4}}{6\,{x}^{6}}}+{\frac{4\,{a}^{5}b}{9\,{x}^{9}}}-{\frac{{a}^{6}}{10\,{x}^{10}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^11,x)

[Out]

c^5*(-4/5*a*b^5/x^5+1/4*b^6/x^4-5/8*a^4*b^2/x^8+5/6*a^2*b^4/x^6+4/9*a^5*b/x^9-1/10*a^6/x^10)

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Maxima [A]  time = 1.07391, size = 101, normalized size = 1.16 \begin{align*} \frac{90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^11,x, algorithm="maxima")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^5*x^2 + 160*a^5*b*c^5*x - 36*a
^6*c^5)/x^10

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Fricas [A]  time = 1.53863, size = 167, normalized size = 1.92 \begin{align*} \frac{90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^11,x, algorithm="fricas")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^5*x^2 + 160*a^5*b*c^5*x - 36*a
^6*c^5)/x^10

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Sympy [A]  time = 0.774193, size = 80, normalized size = 0.92 \begin{align*} \frac{- 36 a^{6} c^{5} + 160 a^{5} b c^{5} x - 225 a^{4} b^{2} c^{5} x^{2} + 300 a^{2} b^{4} c^{5} x^{4} - 288 a b^{5} c^{5} x^{5} + 90 b^{6} c^{5} x^{6}}{360 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**11,x)

[Out]

(-36*a**6*c**5 + 160*a**5*b*c**5*x - 225*a**4*b**2*c**5*x**2 + 300*a**2*b**4*c**5*x**4 - 288*a*b**5*c**5*x**5
+ 90*b**6*c**5*x**6)/(360*x**10)

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Giac [A]  time = 1.20982, size = 101, normalized size = 1.16 \begin{align*} \frac{90 \, b^{6} c^{5} x^{6} - 288 \, a b^{5} c^{5} x^{5} + 300 \, a^{2} b^{4} c^{5} x^{4} - 225 \, a^{4} b^{2} c^{5} x^{2} + 160 \, a^{5} b c^{5} x - 36 \, a^{6} c^{5}}{360 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^11,x, algorithm="giac")

[Out]

1/360*(90*b^6*c^5*x^6 - 288*a*b^5*c^5*x^5 + 300*a^2*b^4*c^5*x^4 - 225*a^4*b^2*c^5*x^2 + 160*a^5*b*c^5*x - 36*a
^6*c^5)/x^10