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

Optimal. Leaf size=149 \[ -\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}+\frac {1}{3} x^3 \left (C \left (2 a c+b^2\right )+2 A b c\right )+x \left (A \left (2 a c+b^2\right )+2 a b C\right )-\frac {a (a C+2 A b)}{x}+\frac {1}{2} B x^2 \left (2 a c+b^2\right )+2 a b B \log (x)+\frac {1}{5} c x^5 (A c+2 b C)+\frac {1}{2} b B c x^4+\frac {1}{6} B c^2 x^6+\frac {1}{7} c^2 C x^7 \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 149, 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} \[ -\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}+\frac {1}{3} x^3 \left (C \left (2 a c+b^2\right )+2 A b c\right )+x \left (A \left (2 a c+b^2\right )+2 a b C\right )-\frac {a (a C+2 A b)}{x}+\frac {1}{2} B x^2 \left (2 a c+b^2\right )+2 a b B \log (x)+\frac {1}{5} c x^5 (A c+2 b C)+\frac {1}{2} b B c x^4+\frac {1}{6} B c^2 x^6+\frac {1}{7} c^2 C x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.08, size = 151, normalized size = 1.01 \[ \frac {a^2 (-C)-2 a A b}{x}-\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}+\frac {1}{3} x^3 \left (2 a c C+2 A b c+b^2 C\right )+x \left (2 a A c+2 a b C+A b^2\right )+\frac {1}{2} B x^2 \left (2 a c+b^2\right )+2 a b B \log (x)+\frac {1}{5} c x^5 (A c+2 b C)+\frac {1}{2} b B c x^4+\frac {1}{6} B c^2 x^6+\frac {1}{7} c^2 C x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(a^2*A)/x^3 - (a^2*B)/(2*x^2) + (-2*a*A*b - a^2*C)/x + (A*b^2 + 2*a*A*c + 2*a*b*C)*x + (B*(b^2 + 2*a*c)*x
^2)/2 + ((2*A*b*c + b^2*C + 2*a*c*C)*x^3)/3 + (b*B*c*x^4)/2 + (c*(A*c + 2*b*C)*x^5)/5 + (B*c^2*x^6)/6 + (c^2*C
*x^7)/7 + 2*a*b*B*Log[x]

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fricas [A]  time = 0.75, size = 145, normalized size = 0.97 \[ \frac {30 \, C c^{2} x^{10} + 35 \, B c^{2} x^{9} + 105 \, B b c x^{7} + 42 \, {\left (2 \, C b c + A c^{2}\right )} x^{8} + 70 \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{6} + 420 \, B a b x^{3} \log \relax (x) + 105 \, {\left (B b^{2} + 2 \, B a c\right )} x^{5} + 210 \, {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 105 \, B a^{2} x - 70 \, A a^{2} - 210 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{210 \, x^{3}} \]

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^4,x, algorithm="fricas")

[Out]

1/210*(30*C*c^2*x^10 + 35*B*c^2*x^9 + 105*B*b*c*x^7 + 42*(2*C*b*c + A*c^2)*x^8 + 70*(C*b^2 + 2*(C*a + A*b)*c)*
x^6 + 420*B*a*b*x^3*log(x) + 105*(B*b^2 + 2*B*a*c)*x^5 + 210*(2*C*a*b + A*b^2 + 2*A*a*c)*x^4 - 105*B*a^2*x - 7
0*A*a^2 - 210*(C*a^2 + 2*A*a*b)*x^2)/x^3

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giac [A]  time = 0.28, size = 146, normalized size = 0.98 \[ \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{6} \, B c^{2} x^{6} + \frac {2}{5} \, C b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{2} \, B b c x^{4} + \frac {1}{3} \, C b^{2} x^{3} + \frac {2}{3} \, C a c x^{3} + \frac {2}{3} \, A b c x^{3} + \frac {1}{2} \, B b^{2} x^{2} + B a c x^{2} + 2 \, C a b x + A b^{2} x + 2 \, A a c x + 2 \, B a b \log \left ({\left | x \right |}\right ) - \frac {3 \, B a^{2} x + 2 \, A a^{2} + 6 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \]

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^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 146, normalized size = 0.98 \[ \frac {C \,c^{2} x^{7}}{7}+\frac {B \,c^{2} x^{6}}{6}+\frac {A \,c^{2} x^{5}}{5}+\frac {2 C b c \,x^{5}}{5}+\frac {B b c \,x^{4}}{2}+\frac {2 A b c \,x^{3}}{3}+\frac {2 C a c \,x^{3}}{3}+\frac {C \,b^{2} x^{3}}{3}+B a c \,x^{2}+\frac {B \,b^{2} x^{2}}{2}+2 A a c x +A \,b^{2} x +2 B a b \ln \relax (x )+2 C a b x -\frac {2 A a b}{x}-\frac {C \,a^{2}}{x}-\frac {B \,a^{2}}{2 x^{2}}-\frac {A \,a^{2}}{3 x^{3}} \]

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^4,x)

[Out]

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

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maxima [A]  time = 0.68, size = 140, normalized size = 0.94 \[ \frac {1}{7} \, C c^{2} x^{7} + \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{2} \, B b c x^{4} + \frac {1}{5} \, {\left (2 \, C b c + A c^{2}\right )} x^{5} + \frac {1}{3} \, {\left (C b^{2} + 2 \, {\left (C a + A b\right )} c\right )} x^{3} + 2 \, B a b \log \relax (x) + \frac {1}{2} \, {\left (B b^{2} + 2 \, B a c\right )} x^{2} + {\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x - \frac {3 \, B a^{2} x + 2 \, A a^{2} + 6 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \]

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^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 137, normalized size = 0.92 \[ x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,C\,b\,c}{5}\right )-\frac {x^2\,\left (C\,a^2+2\,A\,b\,a\right )+\frac {A\,a^2}{3}+\frac {B\,a^2\,x}{2}}{x^3}+x\,\left (A\,b^2+2\,C\,a\,b+2\,A\,a\,c\right )+x^3\,\left (\frac {C\,b^2}{3}+\frac {2\,A\,c\,b}{3}+\frac {2\,C\,a\,c}{3}\right )+\frac {B\,c^2\,x^6}{6}+\frac {C\,c^2\,x^7}{7}+\frac {B\,x^2\,\left (b^2+2\,a\,c\right )}{2}+\frac {B\,b\,c\,x^4}{2}+2\,B\,a\,b\,\ln \relax (x) \]

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^4,x)

[Out]

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

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sympy [A]  time = 0.72, size = 160, normalized size = 1.07 \[ 2 B a b \log {\relax (x )} + \frac {B b c x^{4}}{2} + \frac {B c^{2} x^{6}}{6} + \frac {C c^{2} x^{7}}{7} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 C b c}{5}\right ) + x^{3} \left (\frac {2 A b c}{3} + \frac {2 C a c}{3} + \frac {C b^{2}}{3}\right ) + x^{2} \left (B a c + \frac {B b^{2}}{2}\right ) + x \left (2 A a c + A b^{2} + 2 C a b\right ) + \frac {- 2 A a^{2} - 3 B a^{2} x + x^{2} \left (- 12 A a b - 6 C a^{2}\right )}{6 x^{3}} \]

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**4,x)

[Out]

2*B*a*b*log(x) + B*b*c*x**4/2 + B*c**2*x**6/6 + C*c**2*x**7/7 + x**5*(A*c**2/5 + 2*C*b*c/5) + x**3*(2*A*b*c/3
+ 2*C*a*c/3 + C*b**2/3) + x**2*(B*a*c + B*b**2/2) + x*(2*A*a*c + A*b**2 + 2*C*a*b) + (-2*A*a**2 - 3*B*a**2*x +
x**2*(-12*A*a*b - 6*C*a**2))/(6*x**3)

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