3.6 \(\int \frac {(A+B x+C x^2) (a+b x^2+c x^4)}{x^3} \, dx\)

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

[Out]

-1/2*a*A/x^2-a*B/x+b*B*x+1/2*(A*c+C*b)*x^2+1/3*B*c*x^3+1/4*c*C*x^4+(A*b+C*a)*ln(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} \[ \log (x) (a C+A b)-\frac {a A}{2 x^2}-\frac {a B}{x}+\frac {1}{2} x^2 (A c+b C)+b B x+\frac {1}{3} B c x^3+\frac {1}{4} c C x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.04, size = 58, normalized size = 0.92 \[ \log (x) (a C+A b)-\frac {a (A+2 B x)}{2 x^2}+\frac {1}{12} x \left (c x \left (6 A+4 B x+3 C x^2\right )+6 b (2 B+C x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 62, normalized size = 0.98 \[ \frac {3 \, C c x^{6} + 4 \, B c x^{5} + 12 \, B b x^{3} + 6 \, {\left (C b + A c\right )} x^{4} + 12 \, {\left (C a + A b\right )} x^{2} \log \relax (x) - 12 \, B a x - 6 \, A a}{12 \, x^{2}} \]

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^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.29, size = 58, normalized size = 0.92 \[ \frac {1}{4} \, C c x^{4} + \frac {1}{3} \, B c x^{3} + \frac {1}{2} \, C b x^{2} + \frac {1}{2} \, A c x^{2} + B b x + {\left (C a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, B a x + A a}{2 \, x^{2}} \]

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^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 58, normalized size = 0.92 \[ \frac {C c \,x^{4}}{4}+\frac {B c \,x^{3}}{3}+\frac {A c \,x^{2}}{2}+\frac {C b \,x^{2}}{2}+A b \ln \relax (x )+B b x +C a \ln \relax (x )-\frac {B a}{x}-\frac {A a}{2 x^{2}} \]

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^3,x)

[Out]

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

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maxima [A]  time = 0.75, size = 55, normalized size = 0.87 \[ \frac {1}{4} \, C c x^{4} + \frac {1}{3} \, B c x^{3} + B b x + \frac {1}{2} \, {\left (C b + A c\right )} x^{2} + {\left (C a + A b\right )} \log \relax (x) - \frac {2 \, B a x + A a}{2 \, x^{2}} \]

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^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 56, normalized size = 0.89 \[ x^2\,\left (\frac {A\,c}{2}+\frac {C\,b}{2}\right )-\frac {\frac {A\,a}{2}+B\,a\,x}{x^2}+\ln \relax (x)\,\left (A\,b+C\,a\right )+B\,b\,x+\frac {B\,c\,x^3}{3}+\frac {C\,c\,x^4}{4} \]

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^3,x)

[Out]

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

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sympy [A]  time = 0.29, size = 61, normalized size = 0.97 \[ B b x + \frac {B c x^{3}}{3} + \frac {C c x^{4}}{4} + x^{2} \left (\frac {A c}{2} + \frac {C b}{2}\right ) + \left (A b + C a\right ) \log {\relax (x )} + \frac {- A a - 2 B a x}{2 x^{2}} \]

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**3,x)

[Out]

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

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