3.41 \(\int \frac{(a+b x) (a c-b c x)^5}{x^{10}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{5 a^4 b^2 c^5}{7 x^7}+\frac{a^2 b^4 c^5}{x^5}+\frac{a^5 b c^5}{2 x^8}-\frac{a^6 c^5}{9 x^9}-\frac{a b^5 c^5}{x^4}+\frac{b^6 c^5}{3 x^3} \]

[Out]

-(a^6*c^5)/(9*x^9) + (a^5*b*c^5)/(2*x^8) - (5*a^4*b^2*c^5)/(7*x^7) + (a^2*b^4*c^5)/x^5 - (a*b^5*c^5)/x^4 + (b^
6*c^5)/(3*x^3)

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Rubi [A]  time = 0.0357538, antiderivative size = 82, 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} \[ -\frac{5 a^4 b^2 c^5}{7 x^7}+\frac{a^2 b^4 c^5}{x^5}+\frac{a^5 b c^5}{2 x^8}-\frac{a^6 c^5}{9 x^9}-\frac{a b^5 c^5}{x^4}+\frac{b^6 c^5}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^10,x]

[Out]

-(a^6*c^5)/(9*x^9) + (a^5*b*c^5)/(2*x^8) - (5*a^4*b^2*c^5)/(7*x^7) + (a^2*b^4*c^5)/x^5 - (a*b^5*c^5)/x^4 + (b^
6*c^5)/(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)^5}{x^{10}} \, dx &=\int \left (\frac{a^6 c^5}{x^{10}}-\frac{4 a^5 b c^5}{x^9}+\frac{5 a^4 b^2 c^5}{x^8}-\frac{5 a^2 b^4 c^5}{x^6}+\frac{4 a b^5 c^5}{x^5}-\frac{b^6 c^5}{x^4}\right ) \, dx\\ &=-\frac{a^6 c^5}{9 x^9}+\frac{a^5 b c^5}{2 x^8}-\frac{5 a^4 b^2 c^5}{7 x^7}+\frac{a^2 b^4 c^5}{x^5}-\frac{a b^5 c^5}{x^4}+\frac{b^6 c^5}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0066495, size = 68, normalized size = 0.83 \[ c^5 \left (-\frac{5 a^4 b^2}{7 x^7}+\frac{a^2 b^4}{x^5}+\frac{a^5 b}{2 x^8}-\frac{a^6}{9 x^9}-\frac{a b^5}{x^4}+\frac{b^6}{3 x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^10,x]

[Out]

c^5*(-a^6/(9*x^9) + (a^5*b)/(2*x^8) - (5*a^4*b^2)/(7*x^7) + (a^2*b^4)/x^5 - (a*b^5)/x^4 + b^6/(3*x^3))

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Maple [A]  time = 0.005, size = 61, normalized size = 0.7 \begin{align*}{c}^{5} \left ({\frac{{b}^{6}}{3\,{x}^{3}}}+{\frac{{a}^{2}{b}^{4}}{{x}^{5}}}-{\frac{a{b}^{5}}{{x}^{4}}}+{\frac{{a}^{5}b}{2\,{x}^{8}}}-{\frac{5\,{a}^{4}{b}^{2}}{7\,{x}^{7}}}-{\frac{{a}^{6}}{9\,{x}^{9}}} \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^10,x)

[Out]

c^5*(1/3*b^6/x^3+a^2*b^4/x^5-a*b^5/x^4+1/2*a^5*b/x^8-5/7*a^4*b^2/x^7-1/9*a^6/x^9)

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Maxima [A]  time = 1.16106, size = 101, normalized size = 1.23 \begin{align*} \frac{42 \, b^{6} c^{5} x^{6} - 126 \, a b^{5} c^{5} x^{5} + 126 \, a^{2} b^{4} c^{5} x^{4} - 90 \, a^{4} b^{2} c^{5} x^{2} + 63 \, a^{5} b c^{5} x - 14 \, a^{6} c^{5}}{126 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^10,x, algorithm="maxima")

[Out]

1/126*(42*b^6*c^5*x^6 - 126*a*b^5*c^5*x^5 + 126*a^2*b^4*c^5*x^4 - 90*a^4*b^2*c^5*x^2 + 63*a^5*b*c^5*x - 14*a^6
*c^5)/x^9

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Fricas [A]  time = 1.60913, size = 163, normalized size = 1.99 \begin{align*} \frac{42 \, b^{6} c^{5} x^{6} - 126 \, a b^{5} c^{5} x^{5} + 126 \, a^{2} b^{4} c^{5} x^{4} - 90 \, a^{4} b^{2} c^{5} x^{2} + 63 \, a^{5} b c^{5} x - 14 \, a^{6} c^{5}}{126 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^10,x, algorithm="fricas")

[Out]

1/126*(42*b^6*c^5*x^6 - 126*a*b^5*c^5*x^5 + 126*a^2*b^4*c^5*x^4 - 90*a^4*b^2*c^5*x^2 + 63*a^5*b*c^5*x - 14*a^6
*c^5)/x^9

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Sympy [A]  time = 0.721534, size = 80, normalized size = 0.98 \begin{align*} \frac{- 14 a^{6} c^{5} + 63 a^{5} b c^{5} x - 90 a^{4} b^{2} c^{5} x^{2} + 126 a^{2} b^{4} c^{5} x^{4} - 126 a b^{5} c^{5} x^{5} + 42 b^{6} c^{5} x^{6}}{126 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**10,x)

[Out]

(-14*a**6*c**5 + 63*a**5*b*c**5*x - 90*a**4*b**2*c**5*x**2 + 126*a**2*b**4*c**5*x**4 - 126*a*b**5*c**5*x**5 +
42*b**6*c**5*x**6)/(126*x**9)

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Giac [A]  time = 1.2117, size = 101, normalized size = 1.23 \begin{align*} \frac{42 \, b^{6} c^{5} x^{6} - 126 \, a b^{5} c^{5} x^{5} + 126 \, a^{2} b^{4} c^{5} x^{4} - 90 \, a^{4} b^{2} c^{5} x^{2} + 63 \, a^{5} b c^{5} x - 14 \, a^{6} c^{5}}{126 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^10,x, algorithm="giac")

[Out]

1/126*(42*b^6*c^5*x^6 - 126*a*b^5*c^5*x^5 + 126*a^2*b^4*c^5*x^4 - 90*a^4*b^2*c^5*x^2 + 63*a^5*b*c^5*x - 14*a^6
*c^5)/x^9