3.43 \(\int \frac{(a+b x) (a c-b c x)^5}{x^{12}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{5 a^4 b^2 c^5}{9 x^9}+\frac{5 a^2 b^4 c^5}{7 x^7}+\frac{2 a^5 b c^5}{5 x^{10}}-\frac{a^6 c^5}{11 x^{11}}-\frac{2 a b^5 c^5}{3 x^6}+\frac{b^6 c^5}{5 x^5} \]

[Out]

-(a^6*c^5)/(11*x^11) + (2*a^5*b*c^5)/(5*x^10) - (5*a^4*b^2*c^5)/(9*x^9) + (5*a^2*b^4*c^5)/(7*x^7) - (2*a*b^5*c
^5)/(3*x^6) + (b^6*c^5)/(5*x^5)

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Rubi [A]  time = 0.0345958, 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} \[ -\frac{5 a^4 b^2 c^5}{9 x^9}+\frac{5 a^2 b^4 c^5}{7 x^7}+\frac{2 a^5 b c^5}{5 x^{10}}-\frac{a^6 c^5}{11 x^{11}}-\frac{2 a b^5 c^5}{3 x^6}+\frac{b^6 c^5}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^12,x]

[Out]

-(a^6*c^5)/(11*x^11) + (2*a^5*b*c^5)/(5*x^10) - (5*a^4*b^2*c^5)/(9*x^9) + (5*a^2*b^4*c^5)/(7*x^7) - (2*a*b^5*c
^5)/(3*x^6) + (b^6*c^5)/(5*x^5)

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^{12}} \, dx &=\int \left (\frac{a^6 c^5}{x^{12}}-\frac{4 a^5 b c^5}{x^{11}}+\frac{5 a^4 b^2 c^5}{x^{10}}-\frac{5 a^2 b^4 c^5}{x^8}+\frac{4 a b^5 c^5}{x^7}-\frac{b^6 c^5}{x^6}\right ) \, dx\\ &=-\frac{a^6 c^5}{11 x^{11}}+\frac{2 a^5 b c^5}{5 x^{10}}-\frac{5 a^4 b^2 c^5}{9 x^9}+\frac{5 a^2 b^4 c^5}{7 x^7}-\frac{2 a b^5 c^5}{3 x^6}+\frac{b^6 c^5}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.006681, size = 73, normalized size = 0.84 \[ c^5 \left (-\frac{5 a^4 b^2}{9 x^9}+\frac{5 a^2 b^4}{7 x^7}+\frac{2 a^5 b}{5 x^{10}}-\frac{a^6}{11 x^{11}}-\frac{2 a b^5}{3 x^6}+\frac{b^6}{5 x^5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^12,x]

[Out]

c^5*(-a^6/(11*x^11) + (2*a^5*b)/(5*x^10) - (5*a^4*b^2)/(9*x^9) + (5*a^2*b^4)/(7*x^7) - (2*a*b^5)/(3*x^6) + b^6
/(5*x^5))

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Maple [A]  time = 0.005, size = 62, normalized size = 0.7 \begin{align*}{c}^{5} \left ({\frac{{b}^{6}}{5\,{x}^{5}}}-{\frac{{a}^{6}}{11\,{x}^{11}}}-{\frac{2\,a{b}^{5}}{3\,{x}^{6}}}+{\frac{5\,{a}^{2}{b}^{4}}{7\,{x}^{7}}}-{\frac{5\,{a}^{4}{b}^{2}}{9\,{x}^{9}}}+{\frac{2\,{a}^{5}b}{5\,{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^12,x)

[Out]

c^5*(1/5*b^6/x^5-1/11*a^6/x^11-2/3*a*b^5/x^6+5/7*a^2*b^4/x^7-5/9*a^4*b^2/x^9+2/5*a^5*b/x^10)

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Maxima [A]  time = 1.11149, size = 101, normalized size = 1.16 \begin{align*} \frac{693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^12,x, algorithm="maxima")

[Out]

1/3465*(693*b^6*c^5*x^6 - 2310*a*b^5*c^5*x^5 + 2475*a^2*b^4*c^5*x^4 - 1925*a^4*b^2*c^5*x^2 + 1386*a^5*b*c^5*x
- 315*a^6*c^5)/x^11

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Fricas [A]  time = 1.73175, size = 177, normalized size = 2.03 \begin{align*} \frac{693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^12,x, algorithm="fricas")

[Out]

1/3465*(693*b^6*c^5*x^6 - 2310*a*b^5*c^5*x^5 + 2475*a^2*b^4*c^5*x^4 - 1925*a^4*b^2*c^5*x^2 + 1386*a^5*b*c^5*x
- 315*a^6*c^5)/x^11

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Sympy [A]  time = 0.824168, size = 80, normalized size = 0.92 \begin{align*} \frac{- 315 a^{6} c^{5} + 1386 a^{5} b c^{5} x - 1925 a^{4} b^{2} c^{5} x^{2} + 2475 a^{2} b^{4} c^{5} x^{4} - 2310 a b^{5} c^{5} x^{5} + 693 b^{6} c^{5} x^{6}}{3465 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**12,x)

[Out]

(-315*a**6*c**5 + 1386*a**5*b*c**5*x - 1925*a**4*b**2*c**5*x**2 + 2475*a**2*b**4*c**5*x**4 - 2310*a*b**5*c**5*
x**5 + 693*b**6*c**5*x**6)/(3465*x**11)

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Giac [A]  time = 1.32076, size = 101, normalized size = 1.16 \begin{align*} \frac{693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^12,x, algorithm="giac")

[Out]

1/3465*(693*b^6*c^5*x^6 - 2310*a*b^5*c^5*x^5 + 2475*a^2*b^4*c^5*x^4 - 1925*a^4*b^2*c^5*x^2 + 1386*a^5*b*c^5*x
- 315*a^6*c^5)/x^11