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

Optimal. Leaf size=63 \[ -\frac{a C+A b}{x}-\frac{a A}{3 x^3}-\frac{a B}{2 x^2}+x (A c+b C)+b B \log (x)+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.10621, 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 \[ -\frac{a C+A b}{x}-\frac{a A}{3 x^3}-\frac{a B}{2 x^2}+x (A c+b C)+b B \log (x)+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

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**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0877233, size = 60, normalized size = 0.95 \[ -\frac{a (2 A+3 x (B+2 C x))}{6 x^3}-\frac{A b}{x}+A c x+b B \log (x)+b C x+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3 \]

Antiderivative was successfully verified.

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

[Out]

-((A*b)/x) + A*c*x + b*C*x + (B*c*x^2)/2 + (c*C*x^3)/3 - (a*(2*A + 3*x*(B + 2*C*
x)))/(6*x^3) + b*B*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 57, normalized size = 0.9 \[{\frac{cC{x}^{3}}{3}}+{\frac{Bc{x}^{2}}{2}}+Acx+Cxb-{\frac{Aa}{3\,{x}^{3}}}+bB\ln \left ( x \right ) -{\frac{Ab}{x}}-{\frac{aC}{x}}-{\frac{Ba}{2\,{x}^{2}}} \]

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^4,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.703787, size = 76, normalized size = 1.21 \[ \frac{1}{3} \, C c x^{3} + \frac{1}{2} \, B c x^{2} + B b \log \left (x\right ) +{\left (C b + A c\right )} x - \frac{3 \, B a x + 6 \,{\left (C a + A b\right )} x^{2} + 2 \, A a}{6 \, x^{3}} \]

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^4,x, algorithm="maxima")

[Out]

1/3*C*c*x^3 + 1/2*B*c*x^2 + B*b*log(x) + (C*b + A*c)*x - 1/6*(3*B*a*x + 6*(C*a +
A*b)*x^2 + 2*A*a)/x^3

_______________________________________________________________________________________

Fricas [A]  time = 0.246375, size = 84, normalized size = 1.33 \[ \frac{2 \, C c x^{6} + 3 \, B c x^{5} + 6 \, B b x^{3} \log \left (x\right ) + 6 \,{\left (C b + A c\right )} x^{4} - 3 \, B a x - 6 \,{\left (C a + A b\right )} x^{2} - 2 \, A a}{6 \, x^{3}} \]

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^4,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.5344, size = 61, normalized size = 0.97 \[ B b \log{\left (x \right )} + \frac{B c x^{2}}{2} + \frac{C c x^{3}}{3} + x \left (A c + C b\right ) - \frac{2 A a + 3 B a x + x^{2} \left (6 A b + 6 C a\right )}{6 x^{3}} \]

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**4,x)

[Out]

B*b*log(x) + B*c*x**2/2 + C*c*x**3/3 + x*(A*c + C*b) - (2*A*a + 3*B*a*x + x**2*(
6*A*b + 6*C*a))/(6*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27994, size = 76, normalized size = 1.21 \[ \frac{1}{3} \, C c x^{3} + \frac{1}{2} \, B c x^{2} + C b x + A c x + B b{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, B a x + 6 \,{\left (C a + A b\right )} x^{2} + 2 \, A a}{6 \, x^{3}} \]

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^4,x, algorithm="giac")

[Out]

1/3*C*c*x^3 + 1/2*B*c*x^2 + C*b*x + A*c*x + B*b*ln(abs(x)) - 1/6*(3*B*a*x + 6*(C
*a + A*b)*x^2 + 2*A*a)/x^3