3.9 \(\int \frac{\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^6} \, dx\)

Optimal. Leaf size=63 \[ -\frac{a C+A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{a B}{4 x^4}-\frac{A c+b C}{x}-\frac{b B}{2 x^2}+B c \log (x)+c C x \]

[Out]

-(a*A)/(5*x^5) - (a*B)/(4*x^4) - (A*b + a*C)/(3*x^3) - (b*B)/(2*x^2) - (A*c + b*
C)/x + c*C*x + B*c*Log[x]

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Rubi [A]  time = 0.102326, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ -\frac{a C+A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{a B}{4 x^4}-\frac{A c+b C}{x}-\frac{b B}{2 x^2}+B c \log (x)+c C x \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^6,x]

[Out]

-(a*A)/(5*x^5) - (a*B)/(4*x^4) - (A*b + a*C)/(3*x^3) - (b*B)/(2*x^2) - (A*c + b*
C)/x + c*C*x + B*c*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a}{5 x^{5}} - \frac{B a}{4 x^{4}} - \frac{B b}{2 x^{2}} + B c \log{\left (x \right )} + c \int C\, dx - \frac{A c + C b}{x} - \frac{\frac{A b}{3} + \frac{C a}{3}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((C*x**2+B*x+A)*(c*x**4+b*x**2+a)/x**6,x)

[Out]

-A*a/(5*x**5) - B*a/(4*x**4) - B*b/(2*x**2) + B*c*log(x) + c*Integral(C, x) - (A
*c + C*b)/x - (A*b/3 + C*a/3)/x**3

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Mathematica [A]  time = 0.135777, size = 63, normalized size = 1. \[ B c \log (x)-\frac{12 a A+5 a x (3 B+4 C x)+20 A x^2 \left (b+3 c x^2\right )+30 b x^3 (B+2 C x)-60 c C x^6}{60 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^6,x]

[Out]

-(12*a*A - 60*c*C*x^6 + 30*b*x^3*(B + 2*C*x) + 5*a*x*(3*B + 4*C*x) + 20*A*x^2*(b
+ 3*c*x^2))/(60*x^5) + B*c*Log[x]

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Maple [A]  time = 0.009, size = 60, normalized size = 1. \[ cCx-{\frac{Ab}{3\,{x}^{3}}}-{\frac{aC}{3\,{x}^{3}}}+Bc\ln \left ( x \right ) -{\frac{Ac}{x}}-{\frac{bC}{x}}-{\frac{bB}{2\,{x}^{2}}}-{\frac{Aa}{5\,{x}^{5}}}-{\frac{Ba}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)*(c*x^4+b*x^2+a)/x^6,x)

[Out]

c*C*x-1/3/x^3*A*b-1/3/x^3*a*C+B*c*ln(x)-1/x*A*c-1/x*b*C-1/2*b*B/x^2-1/5*a*A/x^5-
1/4*a*B/x^4

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Maxima [A]  time = 0.698668, size = 76, normalized size = 1.21 \[ C c x + B c \log \left (x\right ) - \frac{30 \, B b x^{3} + 60 \,{\left (C b + A c\right )} x^{4} + 15 \, B a x + 20 \,{\left (C a + A b\right )} x^{2} + 12 \, A a}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*(C*x^2 + B*x + A)/x^6,x, algorithm="maxima")

[Out]

C*c*x + B*c*log(x) - 1/60*(30*B*b*x^3 + 60*(C*b + A*c)*x^4 + 15*B*a*x + 20*(C*a
+ A*b)*x^2 + 12*A*a)/x^5

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Fricas [A]  time = 0.251467, size = 84, normalized size = 1.33 \[ \frac{60 \, C c x^{6} + 60 \, B c x^{5} \log \left (x\right ) - 30 \, B b x^{3} - 60 \,{\left (C b + A c\right )} x^{4} - 15 \, B a x - 20 \,{\left (C a + A b\right )} x^{2} - 12 \, A a}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*(C*x^2 + B*x + A)/x^6,x, algorithm="fricas")

[Out]

1/60*(60*C*c*x^6 + 60*B*c*x^5*log(x) - 30*B*b*x^3 - 60*(C*b + A*c)*x^4 - 15*B*a*
x - 20*(C*a + A*b)*x^2 - 12*A*a)/x^5

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Sympy [A]  time = 21.7377, size = 63, normalized size = 1. \[ B c \log{\left (x \right )} + C c x - \frac{12 A a + 15 B a x + 30 B b x^{3} + x^{4} \left (60 A c + 60 C b\right ) + x^{2} \left (20 A b + 20 C a\right )}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x**2+B*x+A)*(c*x**4+b*x**2+a)/x**6,x)

[Out]

B*c*log(x) + C*c*x - (12*A*a + 15*B*a*x + 30*B*b*x**3 + x**4*(60*A*c + 60*C*b) +
x**2*(20*A*b + 20*C*a))/(60*x**5)

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GIAC/XCAS [A]  time = 0.281511, size = 77, normalized size = 1.22 \[ C c x + B c{\rm ln}\left ({\left | x \right |}\right ) - \frac{30 \, B b x^{3} + 60 \,{\left (C b + A c\right )} x^{4} + 15 \, B a x + 20 \,{\left (C a + A b\right )} x^{2} + 12 \, A a}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*(C*x^2 + B*x + A)/x^6,x, algorithm="giac")

[Out]

C*c*x + B*c*ln(abs(x)) - 1/60*(30*B*b*x^3 + 60*(C*b + A*c)*x^4 + 15*B*a*x + 20*(
C*a + A*b)*x^2 + 12*A*a)/x^5