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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.01, size = 73, normalized size = 0.84 \[ c^5 \left (-\frac {a^6}{11 x^{11}}+\frac {2 a^5 b}{5 x^{10}}-\frac {5 a^4 b^2}{9 x^9}+\frac {5 a^2 b^4}{7 x^7}-\frac {2 a b^5}{3 x^6}+\frac {b^6}{5 x^5}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^5*(-1/11*a^6/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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fricas [A]  time = 0.85, size = 75, normalized size = 0.86 \[ \frac {693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \]

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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giac [A]  time = 1.05, size = 75, normalized size = 0.86 \[ \frac {693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \]

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

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maple [A]  time = 0.01, size = 62, normalized size = 0.71 \[ \left (\frac {b^{6}}{5 x^{5}}-\frac {2 a \,b^{5}}{3 x^{6}}+\frac {5 a^{2} b^{4}}{7 x^{7}}-\frac {5 a^{4} b^{2}}{9 x^{9}}+\frac {2 a^{5} b}{5 x^{10}}-\frac {a^{6}}{11 x^{11}}\right ) c^{5} \]

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

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maxima [A]  time = 1.04, size = 75, normalized size = 0.86 \[ \frac {693 \, b^{6} c^{5} x^{6} - 2310 \, a b^{5} c^{5} x^{5} + 2475 \, a^{2} b^{4} c^{5} x^{4} - 1925 \, a^{4} b^{2} c^{5} x^{2} + 1386 \, a^{5} b c^{5} x - 315 \, a^{6} c^{5}}{3465 \, x^{11}} \]

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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mupad [B]  time = 0.30, size = 75, normalized size = 0.86 \[ -\frac {\frac {a^6\,c^5}{11}-\frac {2\,a^5\,b\,c^5\,x}{5}+\frac {5\,a^4\,b^2\,c^5\,x^2}{9}-\frac {5\,a^2\,b^4\,c^5\,x^4}{7}+\frac {2\,a\,b^5\,c^5\,x^5}{3}-\frac {b^6\,c^5\,x^6}{5}}{x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.58, size = 82, normalized size = 0.94 \[ - \frac {315 a^{6} c^{5} - 1386 a^{5} b c^{5} x + 1925 a^{4} b^{2} c^{5} x^{2} - 2475 a^{2} b^{4} c^{5} x^{4} + 2310 a b^{5} c^{5} x^{5} - 693 b^{6} c^{5} x^{6}}{3465 x^{11}} \]

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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