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

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]

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Rubi [A]  time = 0.0509742, 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} $-\frac{a C+A b}{x}-\frac{a A}{3 x^3}-\frac{a B}{2 x^2}+x (A c+b C)+b B \log (x)+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3$

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]

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^4} \, dx &=\int \left (A c \left (1+\frac{b C}{A c}\right )+\frac{a A}{x^4}+\frac{a B}{x^3}+\frac{A b+a C}{x^2}+\frac{b B}{x}+B c x+c C x^2\right ) \, dx\\ &=-\frac{a A}{3 x^3}-\frac{a B}{2 x^2}-\frac{A b+a C}{x}+(A c+b C) x+\frac{1}{2} B c x^2+\frac{1}{3} c C x^3+b B \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0466659, 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]

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Maple [A]  time = 0.006, size = 57, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{3}}{3}}+{\frac{Bc{x}^{2}}{2}}+Acx+bCx-{\frac{Ab}{x}}-{\frac{aC}{x}}-{\frac{Ba}{2\,{x}^{2}}}-{\frac{Aa}{3\,{x}^{3}}}+bB\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^4,x)

[Out]

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

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Maxima [A]  time = 0.959429, size = 76, normalized size = 1.21 \begin{align*} \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}} \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^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

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Fricas [A]  time = 1.26255, size = 149, normalized size = 2.37 \begin{align*} \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}} \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^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

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Sympy [A]  time = 1.01312, size = 61, normalized size = 0.97 \begin{align*} 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}} \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**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)

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Giac [A]  time = 1.10642, size = 76, normalized size = 1.21 \begin{align*} \frac{1}{3} \, C c x^{3} + \frac{1}{2} \, B c x^{2} + C b x + A c x + B b \log \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}} \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^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*log(abs(x)) - 1/6*(3*B*a*x + 6*(C*a + A*b)*x^2 + 2*A*a)/x^3