### 3.6 $$\int \frac{(A+B x+C x^2) (a+b x^2+c x^4)}{x^3} \, dx$$

Optimal. Leaf size=63 $\log (x) (a C+A b)-\frac{a A}{2 x^2}-\frac{a B}{x}+\frac{1}{2} x^2 (A c+b C)+b B x+\frac{1}{3} B c x^3+\frac{1}{4} c C x^4$

[Out]

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

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Rubi [A]  time = 0.048346, 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} $\log (x) (a C+A b)-\frac{a A}{2 x^2}-\frac{a B}{x}+\frac{1}{2} x^2 (A c+b C)+b B x+\frac{1}{3} B c x^3+\frac{1}{4} c C x^4$

Antiderivative was successfully veriﬁed.

[In]

Int[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0398518, size = 58, normalized size = 0.92 $\log (x) (a C+A b)-\frac{a (A+2 B x)}{2 x^2}+\frac{1}{12} x \left (c x \left (6 A+4 B x+3 C x^2\right )+6 b (2 B+C x)\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 58, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{4}}{4}}+{\frac{Bc{x}^{3}}{3}}+{\frac{A{x}^{2}c}{2}}+{\frac{C{x}^{2}b}{2}}+bBx-{\frac{Ba}{x}}-{\frac{Aa}{2\,{x}^{2}}}+A\ln \left ( x \right ) b+C\ln \left ( x \right ) a \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^3,x)

[Out]

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

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Maxima [A]  time = 0.945481, size = 74, normalized size = 1.17 \begin{align*} \frac{1}{4} \, C c x^{4} + \frac{1}{3} \, B c x^{3} + B b x + \frac{1}{2} \,{\left (C b + A c\right )} x^{2} +{\left (C a + A b\right )} \log \left (x\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.21064, size = 154, normalized size = 2.44 \begin{align*} \frac{3 \, C c x^{6} + 4 \, B c x^{5} + 12 \, B b x^{3} + 6 \,{\left (C b + A c\right )} x^{4} + 12 \,{\left (C a + A b\right )} x^{2} \log \left (x\right ) - 12 \, B a x - 6 \, A a}{12 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.670233, size = 60, normalized size = 0.95 \begin{align*} B b x + \frac{B c x^{3}}{3} + \frac{C c x^{4}}{4} + x^{2} \left (\frac{A c}{2} + \frac{C b}{2}\right ) + \left (A b + C a\right ) \log{\left (x \right )} - \frac{A a + 2 B a x}{2 x^{2}} \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**3,x)

[Out]

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

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Giac [A]  time = 1.08334, size = 78, normalized size = 1.24 \begin{align*} \frac{1}{4} \, C c x^{4} + \frac{1}{3} \, B c x^{3} + \frac{1}{2} \, C b x^{2} + \frac{1}{2} \, A c x^{2} + B b x +{\left (C a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \, B a x + A a}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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