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

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]

________________________________________________________________________________________

Rubi [A]  time = 0.0516926, 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, Rules used = {1628} $-\frac{a C+A b}{3 x^3}-\frac{a A}{5 x^5}-\frac{a B}{4 x^4}-\frac{A c+b C}{x}-\frac{b B}{2 x^2}+B c \log (x)+c C x$

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]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \frac{\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^6} \, dx &=\int \left (c C+\frac{a A}{x^6}+\frac{a B}{x^5}+\frac{A b+a C}{x^4}+\frac{b B}{x^3}+\frac{A c+b C}{x^2}+\frac{B c}{x}\right ) \, dx\\ &=-\frac{a A}{5 x^5}-\frac{a B}{4 x^4}-\frac{A b+a C}{3 x^3}-\frac{b B}{2 x^2}-\frac{A c+b C}{x}+c C x+B c \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0584929, 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*Lo
g[x]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 60, normalized size = 1. \begin{align*} cCx-{\frac{Ac}{x}}-{\frac{bC}{x}}-{\frac{Bb}{2\,{x}^{2}}}-{\frac{Aa}{5\,{x}^{5}}}-{\frac{Ba}{4\,{x}^{4}}}-{\frac{Ab}{3\,{x}^{3}}}-{\frac{aC}{3\,{x}^{3}}}+Bc\ln \left ( x \right ) \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 0.952459, size = 76, normalized size = 1.21 \begin{align*} 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}} \end{align*}

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, 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

________________________________________________________________________________________

Fricas [A]  time = 1.20798, size = 159, normalized size = 2.52 \begin{align*} \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}} \end{align*}

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, 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

________________________________________________________________________________________

Sympy [A]  time = 8.34503, size = 63, normalized size = 1. \begin{align*} 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}} \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.09146, size = 77, normalized size = 1.22 \begin{align*} C c x + B c \log \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}} \end{align*}

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, algorithm="giac")

[Out]

C*c*x + B*c*log(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