### 3.8 $$\int \frac{(A+B x+C x^2) (a+b x^2+c x^4)}{x^5} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0510479, 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} $-\frac{a C+A b}{2 x^2}-\frac{a A}{4 x^4}-\frac{a B}{3 x^3}+\log (x) (A c+b C)-\frac{b B}{x}+B c x+\frac{1}{2} c C x^2$

Antiderivative was successfully veriﬁed.

[In]

Int[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^5,x]

[Out]

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

Mathematica [A]  time = 0.0296124, size = 62, normalized size = 0.98 $-\frac{a \left (3 A+4 B x+6 C x^2\right )}{12 x^4}+\frac{-A b-2 b B x+c x^3 (2 B+C x)}{2 x^2}+\log (x) (A c+b C)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^5,x]

[Out]

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

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Maple [A]  time = 0.007, size = 58, normalized size = 0.9 \begin{align*}{\frac{cC{x}^{2}}{2}}+Bcx-{\frac{Bb}{x}}-{\frac{Aa}{4\,{x}^{4}}}-{\frac{Ab}{2\,{x}^{2}}}-{\frac{aC}{2\,{x}^{2}}}-{\frac{Ba}{3\,{x}^{3}}}+A\ln \left ( x \right ) c+C\ln \left ( x \right ) b \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^5,x)

[Out]

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

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Maxima [A]  time = 0.969201, size = 76, normalized size = 1.21 \begin{align*} \frac{1}{2} \, C c x^{2} + B c x +{\left (C b + A c\right )} \log \left (x\right ) - \frac{12 \, B b x^{3} + 4 \, B a x + 6 \,{\left (C a + A b\right )} x^{2} + 3 \, A a}{12 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.29137, size = 154, normalized size = 2.44 \begin{align*} \frac{6 \, C c x^{6} + 12 \, B c x^{5} + 12 \,{\left (C b + A c\right )} x^{4} \log \left (x\right ) - 12 \, B b x^{3} - 4 \, B a x - 6 \,{\left (C a + A b\right )} x^{2} - 3 \, A a}{12 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.98161, size = 61, normalized size = 0.97 \begin{align*} B c x + \frac{C c x^{2}}{2} + \left (A c + C b\right ) \log{\left (x \right )} - \frac{3 A a + 4 B a x + 12 B b x^{3} + x^{2} \left (6 A b + 6 C a\right )}{12 x^{4}} \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**5,x)

[Out]

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

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Giac [A]  time = 1.09652, size = 77, normalized size = 1.22 \begin{align*} \frac{1}{2} \, C c x^{2} + B c x +{\left (C b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac{12 \, B b x^{3} + 4 \, B a x + 6 \,{\left (C a + A b\right )} x^{2} + 3 \, A a}{12 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

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