3.46 \(\int \frac{(a+b x) (a c-b c x)^6}{x^{11}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7}-\frac{13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac{c^6 (a-b x)^7}{10 x^{10}}-\frac{13 b c^6 (a-b x)^7}{90 a x^9} \]

[Out]

-(c^6*(a - b*x)^7)/(10*x^10) - (13*b*c^6*(a - b*x)^7)/(90*a*x^9) - (13*b^2*c^6*(
a - b*x)^7)/(360*a^2*x^8) - (13*b^3*c^6*(a - b*x)^7)/(2520*a^3*x^7)

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Rubi [A]  time = 0.0972803, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7}-\frac{13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac{c^6 (a-b x)^7}{10 x^{10}}-\frac{13 b c^6 (a-b x)^7}{90 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^6)/x^11,x]

[Out]

-(c^6*(a - b*x)^7)/(10*x^10) - (13*b*c^6*(a - b*x)^7)/(90*a*x^9) - (13*b^2*c^6*(
a - b*x)^7)/(360*a^2*x^8) - (13*b^3*c^6*(a - b*x)^7)/(2520*a^3*x^7)

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Rubi in Sympy [A]  time = 24.7376, size = 83, normalized size = 0.93 \[ - \frac{c^{6} \left (a - b x\right )^{7}}{10 x^{10}} - \frac{13 b c^{6} \left (a - b x\right )^{7}}{90 a x^{9}} - \frac{13 b^{2} c^{6} \left (a - b x\right )^{7}}{360 a^{2} x^{8}} - \frac{13 b^{3} c^{6} \left (a - b x\right )^{7}}{2520 a^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**6/x**11,x)

[Out]

-c**6*(a - b*x)**7/(10*x**10) - 13*b*c**6*(a - b*x)**7/(90*a*x**9) - 13*b**2*c**
6*(a - b*x)**7/(360*a**2*x**8) - 13*b**3*c**6*(a - b*x)**7/(2520*a**3*x**7)

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Mathematica [A]  time = 0.0135778, size = 119, normalized size = 1.34 \[ -\frac{a^7 c^6}{10 x^{10}}+\frac{5 a^6 b c^6}{9 x^9}-\frac{9 a^5 b^2 c^6}{8 x^8}+\frac{5 a^4 b^3 c^6}{7 x^7}+\frac{5 a^3 b^4 c^6}{6 x^6}-\frac{9 a^2 b^5 c^6}{5 x^5}+\frac{5 a b^6 c^6}{4 x^4}-\frac{b^7 c^6}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^11,x]

[Out]

-(a^7*c^6)/(10*x^10) + (5*a^6*b*c^6)/(9*x^9) - (9*a^5*b^2*c^6)/(8*x^8) + (5*a^4*
b^3*c^6)/(7*x^7) + (5*a^3*b^4*c^6)/(6*x^6) - (9*a^2*b^5*c^6)/(5*x^5) + (5*a*b^6*
c^6)/(4*x^4) - (b^7*c^6)/(3*x^3)

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Maple [A]  time = 0.009, size = 84, normalized size = 0.9 \[{c}^{6} \left ( -{\frac{9\,{a}^{5}{b}^{2}}{8\,{x}^{8}}}+{\frac{5\,{a}^{4}{b}^{3}}{7\,{x}^{7}}}+{\frac{5\,{a}^{6}b}{9\,{x}^{9}}}-{\frac{9\,{a}^{2}{b}^{5}}{5\,{x}^{5}}}-{\frac{{a}^{7}}{10\,{x}^{10}}}-{\frac{{b}^{7}}{3\,{x}^{3}}}+{\frac{5\,a{b}^{6}}{4\,{x}^{4}}}+{\frac{5\,{a}^{3}{b}^{4}}{6\,{x}^{6}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^6/x^11,x)

[Out]

c^6*(-9/8*a^5*b^2/x^8+5/7*a^4*b^3/x^7+5/9*a^6*b/x^9-9/5*a^2*b^5/x^5-1/10*a^7/x^1
0-1/3*b^7/x^3+5/4*a*b^6/x^4+5/6*a^3*b^4/x^6)

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Maxima [A]  time = 1.35529, size = 139, normalized size = 1.56 \[ -\frac{840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^11,x, algorithm="maxima")

[Out]

-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 2100*a^3*
b^4*c^6*x^4 - 1800*a^4*b^3*c^6*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6*b*c^6*x + 2
52*a^7*c^6)/x^10

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Fricas [A]  time = 0.200069, size = 139, normalized size = 1.56 \[ -\frac{840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^11,x, algorithm="fricas")

[Out]

-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 2100*a^3*
b^4*c^6*x^4 - 1800*a^4*b^3*c^6*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6*b*c^6*x + 2
52*a^7*c^6)/x^10

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Sympy [A]  time = 3.71471, size = 112, normalized size = 1.26 \[ - \frac{252 a^{7} c^{6} - 1400 a^{6} b c^{6} x + 2835 a^{5} b^{2} c^{6} x^{2} - 1800 a^{4} b^{3} c^{6} x^{3} - 2100 a^{3} b^{4} c^{6} x^{4} + 4536 a^{2} b^{5} c^{6} x^{5} - 3150 a b^{6} c^{6} x^{6} + 840 b^{7} c^{6} x^{7}}{2520 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**6/x**11,x)

[Out]

-(252*a**7*c**6 - 1400*a**6*b*c**6*x + 2835*a**5*b**2*c**6*x**2 - 1800*a**4*b**3
*c**6*x**3 - 2100*a**3*b**4*c**6*x**4 + 4536*a**2*b**5*c**6*x**5 - 3150*a*b**6*c
**6*x**6 + 840*b**7*c**6*x**7)/(2520*x**10)

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GIAC/XCAS [A]  time = 0.237591, size = 139, normalized size = 1.56 \[ -\frac{840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^11,x, algorithm="giac")

[Out]

-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 2100*a^3*
b^4*c^6*x^4 - 1800*a^4*b^3*c^6*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6*b*c^6*x + 2
52*a^7*c^6)/x^10