3.38 \(\int \frac{(a+b x) (a c-b c x)^5}{x^8} \, dx\)

Optimal. Leaf size=41 \[ -\frac{c^5 (a-b x)^6}{7 x^7}-\frac{4 b c^5 (a-b x)^6}{21 a x^6} \]

[Out]

-(c^5*(a - b*x)^6)/(7*x^7) - (4*b*c^5*(a - b*x)^6)/(21*a*x^6)

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Rubi [A]  time = 0.0483606, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{c^5 (a-b x)^6}{7 x^7}-\frac{4 b c^5 (a-b x)^6}{21 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

-(c^5*(a - b*x)^6)/(7*x^7) - (4*b*c^5*(a - b*x)^6)/(21*a*x^6)

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Rubi in Sympy [A]  time = 11.6774, size = 36, normalized size = 0.88 \[ - \frac{c^{5} \left (a - b x\right )^{6}}{7 x^{7}} - \frac{4 b c^{5} \left (a - b x\right )^{6}}{21 a x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**5/x**8,x)

[Out]

-c**5*(a - b*x)**6/(7*x**7) - 4*b*c**5*(a - b*x)**6/(21*a*x**6)

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Mathematica [A]  time = 0.0114612, size = 66, normalized size = 1.61 \[ c^5 \left (-\frac{a^6}{7 x^7}+\frac{2 a^5 b}{3 x^6}-\frac{a^4 b^2}{x^5}+\frac{5 a^2 b^4}{3 x^3}-\frac{2 a b^5}{x^2}+\frac{b^6}{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

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

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Maple [A]  time = 0.008, size = 61, normalized size = 1.5 \[{c}^{5} \left ( -{\frac{{a}^{6}}{7\,{x}^{7}}}-2\,{\frac{a{b}^{5}}{{x}^{2}}}-{\frac{{a}^{4}{b}^{2}}{{x}^{5}}}+{\frac{{b}^{6}}{x}}+{\frac{5\,{a}^{2}{b}^{4}}{3\,{x}^{3}}}+{\frac{2\,{a}^{5}b}{3\,{x}^{6}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^5/x^8,x)

[Out]

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

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Maxima [A]  time = 1.35168, size = 101, normalized size = 2.46 \[ \frac{21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^8,x, algorithm="maxima")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^
2 + 14*a^5*b*c^5*x - 3*a^6*c^5)/x^7

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Fricas [A]  time = 0.200362, size = 101, normalized size = 2.46 \[ \frac{21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^8,x, algorithm="fricas")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^
2 + 14*a^5*b*c^5*x - 3*a^6*c^5)/x^7

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Sympy [A]  time = 2.56319, size = 80, normalized size = 1.95 \[ \frac{- 3 a^{6} c^{5} + 14 a^{5} b c^{5} x - 21 a^{4} b^{2} c^{5} x^{2} + 35 a^{2} b^{4} c^{5} x^{4} - 42 a b^{5} c^{5} x^{5} + 21 b^{6} c^{5} x^{6}}{21 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**5/x**8,x)

[Out]

(-3*a**6*c**5 + 14*a**5*b*c**5*x - 21*a**4*b**2*c**5*x**2 + 35*a**2*b**4*c**5*x*
*4 - 42*a*b**5*c**5*x**5 + 21*b**6*c**5*x**6)/(21*x**7)

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GIAC/XCAS [A]  time = 0.25478, size = 101, normalized size = 2.46 \[ \frac{21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*c*x - a*c)^5*(b*x + a)/x^8,x, algorithm="giac")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^
2 + 14*a^5*b*c^5*x - 3*a^6*c^5)/x^7