### 3.4 $$\int \frac{(A+B x+C x^2) (a+b x^2+c x^4)}{x} \, dx$$

Optimal. Leaf size=65 $\frac{1}{2} x^2 (a C+A b)+a A \log (x)+a B x+\frac{1}{4} x^4 (A c+b C)+\frac{1}{3} b B x^3+\frac{1}{5} B c x^5+\frac{1}{6} c C x^6$

[Out]

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

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Rubi [A]  time = 0.0400262, antiderivative size = 65, 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} $\frac{1}{2} x^2 (a C+A b)+a A \log (x)+a B x+\frac{1}{4} x^4 (A c+b C)+\frac{1}{3} b B x^3+\frac{1}{5} B c x^5+\frac{1}{6} c C x^6$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a*B*x + ((A*b + a*C)*x^2)/2 + (b*B*x^3)/3 + ((A*c + b*C)*x^4)/4 + (B*c*x^5)/5 + (c*C*x^6)/6 + a*A*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} \, dx &=\int \left (a B+\frac{a A}{x}+(A b+a C) x+b B x^2+(A c+b C) x^3+B c x^4+c C x^5\right ) \, dx\\ &=a B x+\frac{1}{2} (A b+a C) x^2+\frac{1}{3} b B x^3+\frac{1}{4} (A c+b C) x^4+\frac{1}{5} B c x^5+\frac{1}{6} c C x^6+a A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.015423, size = 65, normalized size = 1. $\frac{1}{2} x^2 (a C+A b)+a A \log (x)+a B x+\frac{1}{4} x^4 (A c+b C)+\frac{1}{3} b B x^3+\frac{1}{5} B c x^5+\frac{1}{6} c C x^6$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 60, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{6}}{6}}+{\frac{Bc{x}^{5}}{5}}+{\frac{A{x}^{4}c}{4}}+{\frac{C{x}^{4}b}{4}}+{\frac{bB{x}^{3}}{3}}+{\frac{A{x}^{2}b}{2}}+{\frac{C{x}^{2}a}{2}}+aBx+aA\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,x)

[Out]

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

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Maxima [A]  time = 0.940837, size = 74, normalized size = 1.14 \begin{align*} \frac{1}{6} \, C c x^{6} + \frac{1}{5} \, B c x^{5} + \frac{1}{3} \, B b x^{3} + \frac{1}{4} \,{\left (C b + A c\right )} x^{4} + B a x + \frac{1}{2} \,{\left (C a + A b\right )} x^{2} + A a \log \left (x\right ) \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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.27108, size = 143, normalized size = 2.2 \begin{align*} \frac{1}{6} \, C c x^{6} + \frac{1}{5} \, B c x^{5} + \frac{1}{3} \, B b x^{3} + \frac{1}{4} \,{\left (C b + A c\right )} x^{4} + B a x + \frac{1}{2} \,{\left (C a + A b\right )} x^{2} + A a \log \left (x\right ) \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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.449987, size = 63, normalized size = 0.97 \begin{align*} A a \log{\left (x \right )} + B a x + \frac{B b x^{3}}{3} + \frac{B c x^{5}}{5} + \frac{C c x^{6}}{6} + x^{4} \left (\frac{A c}{4} + \frac{C b}{4}\right ) + x^{2} \left (\frac{A b}{2} + \frac{C a}{2}\right ) \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,x)

[Out]

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

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Giac [A]  time = 1.09484, size = 81, normalized size = 1.25 \begin{align*} \frac{1}{6} \, C c x^{6} + \frac{1}{5} \, B c x^{5} + \frac{1}{4} \, C b x^{4} + \frac{1}{4} \, A c x^{4} + \frac{1}{3} \, B b x^{3} + \frac{1}{2} \, C a x^{2} + \frac{1}{2} \, A b x^{2} + B a x + A a \log \left ({\left | x \right |}\right ) \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,x, algorithm="giac")

[Out]

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