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

Optimal. Leaf size=145 \[ -\frac {a^2 A}{x}+a^2 B \log (x)+\frac {1}{5} x^5 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac {1}{3} x^3 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a x (a C+2 A b)+\frac {1}{4} B x^4 \left (2 a c+b^2\right )+a b B x^2+\frac {1}{7} c x^7 (A c+2 b C)+\frac {1}{3} b B c x^6+\frac {1}{8} B c^2 x^8+\frac {1}{9} c^2 C x^9 \]

[Out]

-a^2*A/x+a*(2*A*b+C*a)*x+a*b*B*x^2+1/3*(A*(2*a*c+b^2)+2*a*b*C)*x^3+1/4*B*(2*a*c+b^2)*x^4+1/5*(2*A*b*c+(2*a*c+b
^2)*C)*x^5+1/3*b*B*c*x^6+1/7*c*(A*c+2*C*b)*x^7+1/8*B*c^2*x^8+1/9*c^2*C*x^9+a^2*B*ln(x)

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Rubi [A]  time = 0.12, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1628} \[ -\frac {a^2 A}{x}+a^2 B \log (x)+\frac {1}{5} x^5 \left (C \left (2 a c+b^2\right )+2 A b c\right )+\frac {1}{3} x^3 \left (A \left (2 a c+b^2\right )+2 a b C\right )+a x (a C+2 A b)+\frac {1}{4} B x^4 \left (2 a c+b^2\right )+a b B x^2+\frac {1}{7} c x^7 (A c+2 b C)+\frac {1}{3} b B c x^6+\frac {1}{8} B c^2 x^8+\frac {1}{9} c^2 C x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*A)/x) + a*(2*A*b + a*C)*x + a*b*B*x^2 + ((A*(b^2 + 2*a*c) + 2*a*b*C)*x^3)/3 + (B*(b^2 + 2*a*c)*x^4)/4 +
((2*A*b*c + (b^2 + 2*a*c)*C)*x^5)/5 + (b*B*c*x^6)/3 + (c*(A*c + 2*b*C)*x^7)/7 + (B*c^2*x^8)/8 + (c^2*C*x^9)/9
+ a^2*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 )^2}{x^2} \, dx &=\int \left (a (2 A b+a C)+\frac {a^2 A}{x^2}+\frac {a^2 B}{x}+2 a b B x+\left (A \left (b^2+2 a c\right )+2 a b C\right ) x^2+B \left (b^2+2 a c\right ) x^3+\left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^4+2 b B c x^5+c (A c+2 b C) x^6+B c^2 x^7+c^2 C x^8\right ) \, dx\\ &=-\frac {a^2 A}{x}+a (2 A b+a C) x+a b B x^2+\frac {1}{3} \left (A \left (b^2+2 a c\right )+2 a b C\right ) x^3+\frac {1}{4} B \left (b^2+2 a c\right ) x^4+\frac {1}{5} \left (2 A b c+\left (b^2+2 a c\right ) C\right ) x^5+\frac {1}{3} b B c x^6+\frac {1}{7} c (A c+2 b C) x^7+\frac {1}{8} B c^2 x^8+\frac {1}{9} c^2 C x^9+a^2 B \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 145, normalized size = 1.00 \[ -\frac {a^2 A}{x}+a^2 B \log (x)+\frac {1}{5} x^5 \left (2 a c C+2 A b c+b^2 C\right )+\frac {1}{3} x^3 \left (2 a A c+2 a b C+A b^2\right )+a x (a C+2 A b)+\frac {1}{4} B x^4 \left (2 a c+b^2\right )+a b B x^2+\frac {1}{7} c x^7 (A c+2 b C)+\frac {1}{3} b B c x^6+\frac {1}{8} B c^2 x^8+\frac {1}{9} c^2 C x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*A)/x) + a*(2*A*b + a*C)*x + a*b*B*x^2 + ((A*b^2 + 2*a*A*c + 2*a*b*C)*x^3)/3 + (B*(b^2 + 2*a*c)*x^4)/4 +
((2*A*b*c + b^2*C + 2*a*c*C)*x^5)/5 + (b*B*c*x^6)/3 + (c*(A*c + 2*b*C)*x^7)/7 + (B*c^2*x^8)/8 + (c^2*C*x^9)/9
+ a^2*B*Log[x]

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fricas [A]  time = 0.65, size = 145, normalized size = 1.00 \[ \frac {280 \, C c^{2} x^{10} + 315 \, B c^{2} x^{9} + 840 \, B b c x^{7} + 360 \, {\left (2 \, C b c + A c^{2}\right )} x^{8} + 504 \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{6} + 2520 \, B a b x^{3} + 630 \, {\left (B b^{2} + 2 \, B a c\right )} x^{5} + 840 \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 2520 \, B a^{2} x \log \relax (x) - 2520 \, A a^{2} + 2520 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{2520 \, x} \]

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)^2/x^2,x, algorithm="fricas")

[Out]

1/2520*(280*C*c^2*x^10 + 315*B*c^2*x^9 + 840*B*b*c*x^7 + 360*(2*C*b*c + A*c^2)*x^8 + 504*(C*b^2 + 2*(C*a + A*b
)*c)*x^6 + 2520*B*a*b*x^3 + 630*(B*b^2 + 2*B*a*c)*x^5 + 840*(2*C*a*b + A*b^2 + 2*A*a*c)*x^4 + 2520*B*a^2*x*log
(x) - 2520*A*a^2 + 2520*(C*a^2 + 2*A*a*b)*x^2)/x

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giac [A]  time = 0.28, size = 147, normalized size = 1.01 \[ \frac {1}{9} \, C c^{2} x^{9} + \frac {1}{8} \, B c^{2} x^{8} + \frac {2}{7} \, C b c x^{7} + \frac {1}{7} \, A c^{2} x^{7} + \frac {1}{3} \, B b c x^{6} + \frac {1}{5} \, C b^{2} x^{5} + \frac {2}{5} \, C a c x^{5} + \frac {2}{5} \, A b c x^{5} + \frac {1}{4} \, B b^{2} x^{4} + \frac {1}{2} \, B a c x^{4} + \frac {2}{3} \, C a b x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {2}{3} \, A a c x^{3} + B a b x^{2} + C a^{2} x + 2 \, A a b x + B a^{2} \log \left ({\left | x \right |}\right ) - \frac {A a^{2}}{x} \]

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)^2/x^2,x, algorithm="giac")

[Out]

1/9*C*c^2*x^9 + 1/8*B*c^2*x^8 + 2/7*C*b*c*x^7 + 1/7*A*c^2*x^7 + 1/3*B*b*c*x^6 + 1/5*C*b^2*x^5 + 2/5*C*a*c*x^5
+ 2/5*A*b*c*x^5 + 1/4*B*b^2*x^4 + 1/2*B*a*c*x^4 + 2/3*C*a*b*x^3 + 1/3*A*b^2*x^3 + 2/3*A*a*c*x^3 + B*a*b*x^2 +
C*a^2*x + 2*A*a*b*x + B*a^2*log(abs(x)) - A*a^2/x

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maple [A]  time = 0.01, size = 147, normalized size = 1.01 \[ \frac {C \,c^{2} x^{9}}{9}+\frac {B \,c^{2} x^{8}}{8}+\frac {A \,c^{2} x^{7}}{7}+\frac {2 C b c \,x^{7}}{7}+\frac {B b c \,x^{6}}{3}+\frac {2 A b c \,x^{5}}{5}+\frac {2 C a c \,x^{5}}{5}+\frac {C \,b^{2} x^{5}}{5}+\frac {B a c \,x^{4}}{2}+\frac {B \,b^{2} x^{4}}{4}+\frac {2 A a c \,x^{3}}{3}+\frac {A \,b^{2} x^{3}}{3}+\frac {2 C a b \,x^{3}}{3}+B a b \,x^{2}+2 A a b x +B \,a^{2} \ln \relax (x )+C \,a^{2} x -\frac {A \,a^{2}}{x} \]

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)^2/x^2,x)

[Out]

1/9*c^2*C*x^9+1/8*B*c^2*x^8+1/7*A*x^7*c^2+2/7*C*x^7*b*c+1/3*b*B*c*x^6+2/5*A*x^5*b*c+2/5*C*x^5*a*c+1/5*C*x^5*b^
2+1/2*B*x^4*a*c+1/4*B*x^4*b^2+2/3*A*x^3*a*c+1/3*A*x^3*b^2+2/3*C*x^3*a*b+a*b*B*x^2+2*A*a*b*x+C*a^2*x+a^2*B*ln(x
)-a^2*A/x

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maxima [A]  time = 0.73, size = 137, normalized size = 0.94 \[ \frac {1}{9} \, C c^{2} x^{9} + \frac {1}{8} \, B c^{2} x^{8} + \frac {1}{3} \, B b c x^{6} + \frac {1}{7} \, {\left (2 \, C b c + A c^{2}\right )} x^{7} + \frac {1}{5} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{5} + B a b x^{2} + \frac {1}{4} \, {\left (B b^{2} + 2 \, B a c\right )} x^{4} + \frac {1}{3} \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{3} + B a^{2} \log \relax (x) - \frac {A a^{2}}{x} + {\left (C a^{2} + 2 \, A a b\right )} x \]

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)^2/x^2,x, algorithm="maxima")

[Out]

1/9*C*c^2*x^9 + 1/8*B*c^2*x^8 + 1/3*B*b*c*x^6 + 1/7*(2*C*b*c + A*c^2)*x^7 + 1/5*(C*b^2 + 2*(C*a + A*b)*c)*x^5
+ B*a*b*x^2 + 1/4*(B*b^2 + 2*B*a*c)*x^4 + 1/3*(2*C*a*b + A*b^2 + 2*A*a*c)*x^3 + B*a^2*log(x) - A*a^2/x + (C*a^
2 + 2*A*a*b)*x

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mupad [B]  time = 0.80, size = 135, normalized size = 0.93 \[ x^7\,\left (\frac {A\,c^2}{7}+\frac {2\,C\,b\,c}{7}\right )+x^3\,\left (\frac {A\,b^2}{3}+\frac {2\,C\,a\,b}{3}+\frac {2\,A\,a\,c}{3}\right )+x^5\,\left (\frac {C\,b^2}{5}+\frac {2\,A\,c\,b}{5}+\frac {2\,C\,a\,c}{5}\right )+x\,\left (C\,a^2+2\,A\,b\,a\right )-\frac {A\,a^2}{x}+\frac {B\,c^2\,x^8}{8}+\frac {C\,c^2\,x^9}{9}+B\,a^2\,\ln \relax (x)+\frac {B\,x^4\,\left (b^2+2\,a\,c\right )}{4}+B\,a\,b\,x^2+\frac {B\,b\,c\,x^6}{3} \]

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)^2)/x^2,x)

[Out]

x^7*((A*c^2)/7 + (2*C*b*c)/7) + x^3*((A*b^2)/3 + (2*A*a*c)/3 + (2*C*a*b)/3) + x^5*((C*b^2)/5 + (2*A*b*c)/5 + (
2*C*a*c)/5) + x*(C*a^2 + 2*A*a*b) - (A*a^2)/x + (B*c^2*x^8)/8 + (C*c^2*x^9)/9 + B*a^2*log(x) + (B*x^4*(2*a*c +
b^2))/4 + B*a*b*x^2 + (B*b*c*x^6)/3

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sympy [A]  time = 0.32, size = 156, normalized size = 1.08 \[ - \frac {A a^{2}}{x} + B a^{2} \log {\relax (x )} + B a b x^{2} + \frac {B b c x^{6}}{3} + \frac {B c^{2} x^{8}}{8} + \frac {C c^{2} x^{9}}{9} + x^{7} \left (\frac {A c^{2}}{7} + \frac {2 C b c}{7}\right ) + x^{5} \left (\frac {2 A b c}{5} + \frac {2 C a c}{5} + \frac {C b^{2}}{5}\right ) + x^{4} \left (\frac {B a c}{2} + \frac {B b^{2}}{4}\right ) + x^{3} \left (\frac {2 A a c}{3} + \frac {A b^{2}}{3} + \frac {2 C a b}{3}\right ) + x \left (2 A a b + C a^{2}\right ) \]

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)**2/x**2,x)

[Out]

-A*a**2/x + B*a**2*log(x) + B*a*b*x**2 + B*b*c*x**6/3 + B*c**2*x**8/8 + C*c**2*x**9/9 + x**7*(A*c**2/7 + 2*C*b
*c/7) + x**5*(2*A*b*c/5 + 2*C*a*c/5 + C*b**2/5) + x**4*(B*a*c/2 + B*b**2/4) + x**3*(2*A*a*c/3 + A*b**2/3 + 2*C
*a*b/3) + x*(2*A*a*b + C*a**2)

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