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

Optimal result

Integrand size = 26, antiderivative size = 65 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x} \, dx=a B x+\frac {1}{2} (A b+a C) x^2+\frac {1}{3} b B x^3+\frac {1}{4} (A c+b C) x^4+\frac {1}{5} B c x^5+\frac {1}{6} c C x^6+a A \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1642} \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x} \, dx=\frac {1}{2} x^2 (a C+A b)+a A \log (x)+a B x+\frac {1}{4} x^4 (A c+b C)+\frac {1}{3} b B x^3+\frac {1}{5} B c x^5+\frac {1}{6} c C x^6 \]

[In]

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

[Out]

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

Rule 1642

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*} \text {integral}& = \int \left (a B+\frac {a A}{x}+(A b+a C) x+b B x^2+(A c+b C) x^3+B c x^4+c C x^5\right ) \, dx \\ & = a B x+\frac {1}{2} (A b+a C) x^2+\frac {1}{3} b B x^3+\frac {1}{4} (A c+b C) x^4+\frac {1}{5} B c x^5+\frac {1}{6} c C x^6+a A \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x} \, dx=a B x+\frac {1}{2} (A b+a C) x^2+\frac {1}{3} b B x^3+\frac {1}{4} (A c+b C) x^4+\frac {1}{5} B c x^5+\frac {1}{6} c C x^6+a A \log (x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89

method	result	size
norman	\(\left (\frac {A b}{2}+\frac {C a}{2}\right ) x^{2}+\left (\frac {A c}{4}+\frac {C b}{4}\right ) x^{4}+B a x +\frac {B b \,x^{3}}{3}+\frac {B c \,x^{5}}{5}+\frac {c C \,x^{6}}{6}+a A \ln \left (x \right )\)	\(58\)
default	\(\frac {c C \,x^{6}}{6}+\frac {B c \,x^{5}}{5}+\frac {A c \,x^{4}}{4}+\frac {C b \,x^{4}}{4}+\frac {B b \,x^{3}}{3}+\frac {A b \,x^{2}}{2}+\frac {C a \,x^{2}}{2}+B a x +a A \ln \left (x \right )\)	\(60\)
risch	\(\frac {c C \,x^{6}}{6}+\frac {B c \,x^{5}}{5}+\frac {A c \,x^{4}}{4}+\frac {C b \,x^{4}}{4}+\frac {B b \,x^{3}}{3}+\frac {A b \,x^{2}}{2}+\frac {C a \,x^{2}}{2}+B a x +a A \ln \left (x \right )\)	\(60\)
parallelrisch	\(\frac {c C \,x^{6}}{6}+\frac {B c \,x^{5}}{5}+\frac {A c \,x^{4}}{4}+\frac {C b \,x^{4}}{4}+\frac {B b \,x^{3}}{3}+\frac {A b \,x^{2}}{2}+\frac {C a \,x^{2}}{2}+B a x +a A \ln \left (x \right )\)	\(60\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x} \, dx=\frac {1}{6} \, C c x^{6} + \frac {1}{5} \, B c x^{5} + \frac {1}{3} \, B b x^{3} + \frac {1}{4} \, {\left (C b + A c\right )} x^{4} + B a x + \frac {1}{2} \, {\left (C a + A b\right )} x^{2} + A a \log \left (x\right ) \]

[In]

integrate((C*x^2+B*x+A)*(c*x^4+b*x^2+a)/x,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x} \, dx=A a \log {\left (x \right )} + B a x + \frac {B b x^{3}}{3} + \frac {B c x^{5}}{5} + \frac {C c x^{6}}{6} + x^{4} \left (\frac {A c}{4} + \frac {C b}{4}\right ) + x^{2} \left (\frac {A b}{2} + \frac {C a}{2}\right ) \]

[In]

integrate((C*x**2+B*x+A)*(c*x**4+b*x**2+a)/x,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x} \, dx=\frac {1}{6} \, C c x^{6} + \frac {1}{5} \, B c x^{5} + \frac {1}{3} \, B b x^{3} + \frac {1}{4} \, {\left (C b + A c\right )} x^{4} + B a x + \frac {1}{2} \, {\left (C a + A b\right )} x^{2} + A a \log \left (x\right ) \]

[In]

integrate((C*x^2+B*x+A)*(c*x^4+b*x^2+a)/x,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x} \, dx=\frac {1}{6} \, C c x^{6} + \frac {1}{5} \, B c x^{5} + \frac {1}{4} \, C b x^{4} + \frac {1}{4} \, A c x^{4} + \frac {1}{3} \, B b x^{3} + \frac {1}{2} \, C a x^{2} + \frac {1}{2} \, A b x^{2} + B a x + A a \log \left ({\left | x \right |}\right ) \]

[In]

integrate((C*x^2+B*x+A)*(c*x^4+b*x^2+a)/x,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )}{x} \, dx=x^2\,\left (\frac {A\,b}{2}+\frac {C\,a}{2}\right )+x^4\,\left (\frac {A\,c}{4}+\frac {C\,b}{4}\right )+B\,a\,x+\frac {B\,b\,x^3}{3}+\frac {B\,c\,x^5}{5}+\frac {C\,c\,x^6}{6}+A\,a\,\ln \left (x\right ) \]

[In]

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

[Out]

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