3.1.9 \(\int \frac {(A+B x+C x^2) (a+b x^2+c x^4)}{x^6} \, dx\)

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

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Rubi [A]  time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a C+A b}{3 x^3}-\frac {a A}{5 x^5}-\frac {a B}{4 x^4}-\frac {A c+b C}{x}-\frac {b B}{2 x^2}+B c \log (x)+c C x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^6,x]

[Out]

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

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Mathematica [A]  time = 0.05, size = 63, normalized size = 1.00 \begin {gather*} B c \log (x)-\frac {12 a A+5 a x (3 B+4 C x)+20 A x^2 \left (b+3 c x^2\right )+30 b x^3 (B+2 C x)-60 c C x^6}{60 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^6,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^6,x]

[Out]

IntegrateAlgebraic[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^6, x]

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fricas [A]  time = 1.28, size = 62, normalized size = 0.98 \begin {gather*} \frac {60 \, C c x^{6} + 60 \, B c x^{5} \log \relax (x) - 30 \, B b x^{3} - 60 \, {\left (C b + A c\right )} x^{4} - 15 \, B a x - 20 \, {\left (C a + A b\right )} x^{2} - 12 \, A a}{60 \, x^{5}} \end {gather*}

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^6,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.26, size = 57, normalized size = 0.90 \begin {gather*} C c x + B c \log \left ({\left | x \right |}\right ) - \frac {30 \, B b x^{3} + 60 \, {\left (C b + A c\right )} x^{4} + 15 \, B a x + 20 \, {\left (C a + A b\right )} x^{2} + 12 \, A a}{60 \, x^{5}} \end {gather*}

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^6,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 60, normalized size = 0.95 \begin {gather*} B c \ln \relax (x )+C c x -\frac {A c}{x}-\frac {C b}{x}-\frac {B b}{2 x^{2}}-\frac {A b}{3 x^{3}}-\frac {C a}{3 x^{3}}-\frac {B a}{4 x^{4}}-\frac {A a}{5 x^{5}} \end {gather*}

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^6,x)

[Out]

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

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maxima [A]  time = 0.74, size = 56, normalized size = 0.89 \begin {gather*} C c x + B c \log \relax (x) - \frac {30 \, B b x^{3} + 60 \, {\left (C b + A c\right )} x^{4} + 15 \, B a x + 20 \, {\left (C a + A b\right )} x^{2} + 12 \, A a}{60 \, x^{5}} \end {gather*}

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^6,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.78, size = 56, normalized size = 0.89 \begin {gather*} C\,c\,x-\frac {\left (A\,c+C\,b\right )\,x^4+\frac {B\,b\,x^3}{2}+\left (\frac {A\,b}{3}+\frac {C\,a}{3}\right )\,x^2+\frac {B\,a\,x}{4}+\frac {A\,a}{5}}{x^5}+B\,c\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x + C*x^2)*(a + b*x^2 + c*x^4))/x^6,x)

[Out]

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

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sympy [A]  time = 5.70, size = 66, normalized size = 1.05 \begin {gather*} B c \log {\relax (x )} + C c x + \frac {- 12 A a - 15 B a x - 30 B b x^{3} + x^{4} \left (- 60 A c - 60 C b\right ) + x^{2} \left (- 20 A b - 20 C a\right )}{60 x^{5}} \end {gather*}

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**6,x)

[Out]

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

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