### 3.10 $$\int \frac{(A+B x+C x^2) (a+b x^2+c x^4)}{x^7} \, dx$$

Optimal. Leaf size=68 $-\frac{a C+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{a B}{5 x^5}-\frac{A c+b C}{2 x^2}-\frac{b B}{3 x^3}-\frac{B c}{x}+c C \log (x)$

[Out]

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

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Rubi [A]  time = 0.0482934, antiderivative size = 68, 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} $-\frac{a C+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{a B}{5 x^5}-\frac{A c+b C}{2 x^2}-\frac{b B}{3 x^3}-\frac{B c}{x}+c C \log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^7,x]

[Out]

-(a*A)/(6*x^6) - (a*B)/(5*x^5) - (A*b + a*C)/(4*x^4) - (b*B)/(3*x^3) - (A*c + b*C)/(2*x^2) - (B*c)/x + c*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^7} \, dx &=\int \left (\frac{a A}{x^7}+\frac{a B}{x^6}+\frac{A b+a C}{x^5}+\frac{b B}{x^4}+\frac{A c+b C}{x^3}+\frac{B c}{x^2}+\frac{c C}{x}\right ) \, dx\\ &=-\frac{a A}{6 x^6}-\frac{a B}{5 x^5}-\frac{A b+a C}{4 x^4}-\frac{b B}{3 x^3}-\frac{A c+b C}{2 x^2}-\frac{B c}{x}+c C \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0475365, size = 68, normalized size = 1. $c C \log (x)-\frac{a (10 A+3 x (4 B+5 C x))+5 x^2 \left (3 A \left (b+2 c x^2\right )+2 x \left (2 b B+3 b C x+6 B c x^2\right )\right )}{60 x^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^7,x]

[Out]

-(a*(10*A + 3*x*(4*B + 5*C*x)) + 5*x^2*(3*A*(b + 2*c*x^2) + 2*x*(2*b*B + 3*b*C*x + 6*B*c*x^2)))/(60*x^6) + c*C
*Log[x]

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Maple [A]  time = 0.006, size = 63, normalized size = 0.9 \begin{align*} -{\frac{Bc}{x}}-{\frac{Ac}{2\,{x}^{2}}}-{\frac{bC}{2\,{x}^{2}}}-{\frac{Ba}{5\,{x}^{5}}}-{\frac{Ab}{4\,{x}^{4}}}-{\frac{aC}{4\,{x}^{4}}}-{\frac{Aa}{6\,{x}^{6}}}-{\frac{Bb}{3\,{x}^{3}}}+cC\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^7,x)

[Out]

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

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Maxima [A]  time = 0.967971, size = 80, normalized size = 1.18 \begin{align*} C c \log \left (x\right ) - \frac{60 \, B c x^{5} + 20 \, B b x^{3} + 30 \,{\left (C b + A c\right )} x^{4} + 12 \, B a x + 15 \,{\left (C a + A b\right )} x^{2} + 10 \, A a}{60 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

C*c*log(x) - 1/60*(60*B*c*x^5 + 20*B*b*x^3 + 30*(C*b + A*c)*x^4 + 12*B*a*x + 15*(C*a + A*b)*x^2 + 10*A*a)/x^6

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Fricas [A]  time = 1.27163, size = 159, normalized size = 2.34 \begin{align*} \frac{60 \, C c x^{6} \log \left (x\right ) - 60 \, B c x^{5} - 20 \, B b x^{3} - 30 \,{\left (C b + A c\right )} x^{4} - 12 \, B a x - 15 \,{\left (C a + A b\right )} x^{2} - 10 \, A a}{60 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

1/60*(60*C*c*x^6*log(x) - 60*B*c*x^5 - 20*B*b*x^3 - 30*(C*b + A*c)*x^4 - 12*B*a*x - 15*(C*a + A*b)*x^2 - 10*A*
a)/x^6

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Sympy [A]  time = 16.138, size = 66, normalized size = 0.97 \begin{align*} C c \log{\left (x \right )} - \frac{10 A a + 12 B a x + 20 B b x^{3} + 60 B c x^{5} + x^{4} \left (30 A c + 30 C b\right ) + x^{2} \left (15 A b + 15 C a\right )}{60 x^{6}} \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**7,x)

[Out]

C*c*log(x) - (10*A*a + 12*B*a*x + 20*B*b*x**3 + 60*B*c*x**5 + x**4*(30*A*c + 30*C*b) + x**2*(15*A*b + 15*C*a))
/(60*x**6)

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Giac [A]  time = 1.09615, size = 81, normalized size = 1.19 \begin{align*} C c \log \left ({\left | x \right |}\right ) - \frac{60 \, B c x^{5} + 20 \, B b x^{3} + 30 \,{\left (C b + A c\right )} x^{4} + 12 \, B a x + 15 \,{\left (C a + A b\right )} x^{2} + 10 \, A a}{60 \, x^{6}} \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^7,x, algorithm="giac")

[Out]

C*c*log(abs(x)) - 1/60*(60*B*c*x^5 + 20*B*b*x^3 + 30*(C*b + A*c)*x^4 + 12*B*a*x + 15*(C*a + A*b)*x^2 + 10*A*a)
/x^6