### 3.5 $$\int \frac{(A+B x+C x^2) (a+b x^2+c x^4)}{x^2} \, dx$$

Optimal. Leaf size=63 $x (a C+A b)-\frac{a A}{x}+a B \log (x)+\frac{1}{3} x^3 (A c+b C)+\frac{1}{2} b B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5$

[Out]

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

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Rubi [A]  time = 0.0505412, 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} $x (a C+A b)-\frac{a A}{x}+a B \log (x)+\frac{1}{3} x^3 (A c+b C)+\frac{1}{2} b B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0230996, size = 63, normalized size = 1. $x (a C+A b)-\frac{a A}{x}+a B \log (x)+\frac{1}{3} x^3 (A c+b C)+\frac{1}{2} b B x^2+\frac{1}{4} B c x^4+\frac{1}{5} c C x^5$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 57, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{5}}{5}}+{\frac{Bc{x}^{4}}{4}}+{\frac{A{x}^{3}c}{3}}+{\frac{C{x}^{3}b}{3}}+{\frac{bB{x}^{2}}{2}}+Abx+aCx-{\frac{Aa}{x}}+aB\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^2,x)

[Out]

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

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Maxima [A]  time = 0.965118, size = 74, normalized size = 1.17 \begin{align*} \frac{1}{5} \, C c x^{5} + \frac{1}{4} \, B c x^{4} + \frac{1}{2} \, B b x^{2} + \frac{1}{3} \,{\left (C b + A c\right )} x^{3} + B a \log \left (x\right ) +{\left (C a + A b\right )} x - \frac{A a}{x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.28934, size = 157, normalized size = 2.49 \begin{align*} \frac{12 \, C c x^{6} + 15 \, B c x^{5} + 30 \, B b x^{3} + 20 \,{\left (C b + A c\right )} x^{4} + 60 \, B a x \log \left (x\right ) + 60 \,{\left (C a + A b\right )} x^{2} - 60 \, A a}{60 \, x} \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^2,x, algorithm="fricas")

[Out]

1/60*(12*C*c*x^6 + 15*B*c*x^5 + 30*B*b*x^3 + 20*(C*b + A*c)*x^4 + 60*B*a*x*log(x) + 60*(C*a + A*b)*x^2 - 60*A*
a)/x

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Sympy [A]  time = 0.49202, size = 58, normalized size = 0.92 \begin{align*} - \frac{A a}{x} + B a \log{\left (x \right )} + \frac{B b x^{2}}{2} + \frac{B c x^{4}}{4} + \frac{C c x^{5}}{5} + x^{3} \left (\frac{A c}{3} + \frac{C b}{3}\right ) + x \left (A b + C a\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**2,x)

[Out]

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

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Giac [A]  time = 1.09517, size = 77, normalized size = 1.22 \begin{align*} \frac{1}{5} \, C c x^{5} + \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, C b x^{3} + \frac{1}{3} \, A c x^{3} + \frac{1}{2} \, B b x^{2} + C a x + A b x + B a \log \left ({\left | x \right |}\right ) - \frac{A a}{x} \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^2,x, algorithm="giac")

[Out]

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