3.43 \(\int \frac{(a+b x) (a c-b c x)^6}{x^8} \, dx\)

Optimal. Leaf size=113 \[ -\frac{a^7 c^6}{7 x^7}+\frac{5 a^6 b c^6}{6 x^6}-\frac{9 a^5 b^2 c^6}{5 x^5}+\frac{5 a^4 b^3 c^6}{4 x^4}+\frac{5 a^3 b^4 c^6}{3 x^3}-\frac{9 a^2 b^5 c^6}{2 x^2}+\frac{5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.133524, antiderivative size = 113, 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 \[ -\frac{a^7 c^6}{7 x^7}+\frac{5 a^6 b c^6}{6 x^6}-\frac{9 a^5 b^2 c^6}{5 x^5}+\frac{5 a^4 b^3 c^6}{4 x^4}+\frac{5 a^3 b^4 c^6}{3 x^3}-\frac{9 a^2 b^5 c^6}{2 x^2}+\frac{5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a*c - b*c*x)^6)/x^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 42.72, size = 116, normalized size = 1.03 \[ - \frac{a^{7} c^{6}}{7 x^{7}} + \frac{5 a^{6} b c^{6}}{6 x^{6}} - \frac{9 a^{5} b^{2} c^{6}}{5 x^{5}} + \frac{5 a^{4} b^{3} c^{6}}{4 x^{4}} + \frac{5 a^{3} b^{4} c^{6}}{3 x^{3}} - \frac{9 a^{2} b^{5} c^{6}}{2 x^{2}} + \frac{5 a b^{6} c^{6}}{x} + b^{7} c^{6} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(-b*c*x+a*c)**6/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.0179226, size = 113, normalized size = 1. \[ -\frac{a^7 c^6}{7 x^7}+\frac{5 a^6 b c^6}{6 x^6}-\frac{9 a^5 b^2 c^6}{5 x^5}+\frac{5 a^4 b^3 c^6}{4 x^4}+\frac{5 a^3 b^4 c^6}{3 x^3}-\frac{9 a^2 b^5 c^6}{2 x^2}+\frac{5 a b^6 c^6}{x}+b^7 c^6 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^8,x]

[Out]

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

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Maple [A]  time = 0.009, size = 102, normalized size = 0.9 \[ -{\frac{{a}^{7}{c}^{6}}{7\,{x}^{7}}}+{\frac{5\,{a}^{6}b{c}^{6}}{6\,{x}^{6}}}-{\frac{9\,{a}^{5}{b}^{2}{c}^{6}}{5\,{x}^{5}}}+{\frac{5\,{a}^{4}{b}^{3}{c}^{6}}{4\,{x}^{4}}}+{\frac{5\,{a}^{3}{b}^{4}{c}^{6}}{3\,{x}^{3}}}-{\frac{9\,{a}^{2}{b}^{5}{c}^{6}}{2\,{x}^{2}}}+5\,{\frac{a{b}^{6}{c}^{6}}{x}}+{b}^{7}{c}^{6}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(-b*c*x+a*c)^6/x^8,x)

[Out]

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

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Maxima [A]  time = 1.37068, size = 138, normalized size = 1.22 \[ b^{7} c^{6} \log \left (x\right ) + \frac{2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^8,x, algorithm="maxima")

[Out]

b^7*c^6*log(x) + 1/420*(2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*a^3*b^4*
c^6*x^4 + 525*a^4*b^3*c^6*x^3 - 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x - 60*a^7*c
^6)/x^7

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Fricas [A]  time = 0.212408, size = 142, normalized size = 1.26 \[ \frac{420 \, b^{7} c^{6} x^{7} \log \left (x\right ) + 2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^8,x, algorithm="fricas")

[Out]

1/420*(420*b^7*c^6*x^7*log(x) + 2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*
a^3*b^4*c^6*x^4 + 525*a^4*b^3*c^6*x^3 - 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x -
60*a^7*c^6)/x^7

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Sympy [A]  time = 2.93247, size = 109, normalized size = 0.96 \[ b^{7} c^{6} \log{\left (x \right )} + \frac{- 60 a^{7} c^{6} + 350 a^{6} b c^{6} x - 756 a^{5} b^{2} c^{6} x^{2} + 525 a^{4} b^{3} c^{6} x^{3} + 700 a^{3} b^{4} c^{6} x^{4} - 1890 a^{2} b^{5} c^{6} x^{5} + 2100 a b^{6} c^{6} x^{6}}{420 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(-b*c*x+a*c)**6/x**8,x)

[Out]

b**7*c**6*log(x) + (-60*a**7*c**6 + 350*a**6*b*c**6*x - 756*a**5*b**2*c**6*x**2
+ 525*a**4*b**3*c**6*x**3 + 700*a**3*b**4*c**6*x**4 - 1890*a**2*b**5*c**6*x**5 +
2100*a*b**6*c**6*x**6)/(420*x**7)

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GIAC/XCAS [A]  time = 0.268075, size = 139, normalized size = 1.23 \[ b^{7} c^{6}{\rm ln}\left ({\left | x \right |}\right ) + \frac{2100 \, a b^{6} c^{6} x^{6} - 1890 \, a^{2} b^{5} c^{6} x^{5} + 700 \, a^{3} b^{4} c^{6} x^{4} + 525 \, a^{4} b^{3} c^{6} x^{3} - 756 \, a^{5} b^{2} c^{6} x^{2} + 350 \, a^{6} b c^{6} x - 60 \, a^{7} c^{6}}{420 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^6*(b*x + a)/x^8,x, algorithm="giac")

[Out]

b^7*c^6*ln(abs(x)) + 1/420*(2100*a*b^6*c^6*x^6 - 1890*a^2*b^5*c^6*x^5 + 700*a^3*
b^4*c^6*x^4 + 525*a^4*b^3*c^6*x^3 - 756*a^5*b^2*c^6*x^2 + 350*a^6*b*c^6*x - 60*a
^7*c^6)/x^7