3.45 \(\int \frac{(a+b x) (a c-b c x)^6}{x^{10}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}-\frac{c^6 (a-b x)^7}{9 x^9}-\frac{11 b c^6 (a-b x)^7}{72 a x^8} \]

[Out]

-(c^6*(a - b*x)^7)/(9*x^9) - (11*b*c^6*(a - b*x)^7)/(72*a*x^8) - (11*b^2*c^6*(a
- b*x)^7)/(504*a^2*x^7)

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Rubi [A]  time = 0.0732524, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}-\frac{c^6 (a-b x)^7}{9 x^9}-\frac{11 b c^6 (a-b x)^7}{72 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

-(c^6*(a - b*x)^7)/(9*x^9) - (11*b*c^6*(a - b*x)^7)/(72*a*x^8) - (11*b^2*c^6*(a
- b*x)^7)/(504*a^2*x^7)

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Rubi in Sympy [A]  time = 16.9989, size = 60, normalized size = 0.92 \[ - \frac{c^{6} \left (a - b x\right )^{7}}{9 x^{9}} - \frac{11 b c^{6} \left (a - b x\right )^{7}}{72 a x^{8}} - \frac{11 b^{2} c^{6} \left (a - b x\right )^{7}}{504 a^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**6/x**10,x)

[Out]

-c**6*(a - b*x)**7/(9*x**9) - 11*b*c**6*(a - b*x)**7/(72*a*x**8) - 11*b**2*c**6*
(a - b*x)**7/(504*a**2*x**7)

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Mathematica [A]  time = 0.012375, size = 116, normalized size = 1.78 \[ -\frac{a^7 c^6}{9 x^9}+\frac{5 a^6 b c^6}{8 x^8}-\frac{9 a^5 b^2 c^6}{7 x^7}+\frac{5 a^4 b^3 c^6}{6 x^6}+\frac{a^3 b^4 c^6}{x^5}-\frac{9 a^2 b^5 c^6}{4 x^4}+\frac{5 a b^6 c^6}{3 x^3}-\frac{b^7 c^6}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

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

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Maple [A]  time = 0.007, size = 83, normalized size = 1.3 \[{c}^{6} \left ({\frac{5\,{a}^{6}b}{8\,{x}^{8}}}-{\frac{9\,{a}^{5}{b}^{2}}{7\,{x}^{7}}}-{\frac{{a}^{7}}{9\,{x}^{9}}}-{\frac{{b}^{7}}{2\,{x}^{2}}}+{\frac{{a}^{3}{b}^{4}}{{x}^{5}}}+{\frac{5\,a{b}^{6}}{3\,{x}^{3}}}-{\frac{9\,{a}^{2}{b}^{5}}{4\,{x}^{4}}}+{\frac{5\,{a}^{4}{b}^{3}}{6\,{x}^{6}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^6/x^10,x)

[Out]

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

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Maxima [A]  time = 1.36789, size = 139, normalized size = 2.14 \[ -\frac{252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^10,x, algorithm="maxima")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4
*c^6*x^4 - 420*a^4*b^3*c^6*x^3 + 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*
c^6)/x^9

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Fricas [A]  time = 0.20362, size = 139, normalized size = 2.14 \[ -\frac{252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^10,x, algorithm="fricas")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4
*c^6*x^4 - 420*a^4*b^3*c^6*x^3 + 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*
c^6)/x^9

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Sympy [A]  time = 3.94577, size = 112, normalized size = 1.72 \[ - \frac{56 a^{7} c^{6} - 315 a^{6} b c^{6} x + 648 a^{5} b^{2} c^{6} x^{2} - 420 a^{4} b^{3} c^{6} x^{3} - 504 a^{3} b^{4} c^{6} x^{4} + 1134 a^{2} b^{5} c^{6} x^{5} - 840 a b^{6} c^{6} x^{6} + 252 b^{7} c^{6} x^{7}}{504 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**6/x**10,x)

[Out]

-(56*a**7*c**6 - 315*a**6*b*c**6*x + 648*a**5*b**2*c**6*x**2 - 420*a**4*b**3*c**
6*x**3 - 504*a**3*b**4*c**6*x**4 + 1134*a**2*b**5*c**6*x**5 - 840*a*b**6*c**6*x*
*6 + 252*b**7*c**6*x**7)/(504*x**9)

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GIAC/XCAS [A]  time = 0.25027, size = 139, normalized size = 2.14 \[ -\frac{252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^10,x, algorithm="giac")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4
*c^6*x^4 - 420*a^4*b^3*c^6*x^3 + 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*
c^6)/x^9