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

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

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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*} 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 \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 63, normalized size = 1.00 \begin {gather*} 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 \end {gather*}

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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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^2} \, 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^2,x]

[Out]

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

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fricas [A]  time = 1.23, size = 62, normalized size = 0.98 \begin {gather*} \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 \relax (x) + 60 \, {\left (C a + A b\right )} x^{2} - 60 \, A a}{60 \, x} \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^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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giac [A]  time = 0.30, size = 57, normalized size = 0.90 \begin {gather*} \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 {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^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

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maple [A]  time = 0.01, size = 57, normalized size = 0.90 \begin {gather*} \frac {C c \,x^{5}}{5}+\frac {B c \,x^{4}}{4}+\frac {A c \,x^{3}}{3}+\frac {C b \,x^{3}}{3}+\frac {B b \,x^{2}}{2}+A b x +B a \ln \relax (x )+C a x -\frac {A a}{x} \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^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*B*ln(x)-a*A/x

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maxima [A]  time = 0.68, size = 55, normalized size = 0.87 \begin {gather*} \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 \relax (x) + {\left (C a + A b\right )} x - \frac {A a}{x} \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^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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mupad [B]  time = 0.04, size = 56, normalized size = 0.89 \begin {gather*} x\,\left (A\,b+C\,a\right )+x^3\,\left (\frac {A\,c}{3}+\frac {C\,b}{3}\right )-\frac {A\,a}{x}+\frac {B\,b\,x^2}{2}+\frac {B\,c\,x^4}{4}+\frac {C\,c\,x^5}{5}+B\,a\,\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^2,x)

[Out]

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

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sympy [A]  time = 0.18, size = 58, normalized size = 0.92 \begin {gather*} - \frac {A a}{x} + B a \log {\relax (x )} + \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 {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**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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