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

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]

-(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]

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Rubi [A]  time = 0.30541, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{a^2 A}{3 x^3}-\frac{a^2 B}{2 x^2}+\frac{1}{3} x^3 \left (C \left (2 a c+b^2\right )+2 A b c\right )+x \left (A \left (2 a c+b^2\right )+2 a b C\right )-\frac{a (a C+2 A b)}{x}+\frac{1}{2} B x^2 \left (2 a c+b^2\right )+2 a b B \log (x)+\frac{1}{5} c x^5 (A c+2 b C)+\frac{1}{2} b B c x^4+\frac{1}{6} B c^2 x^6+\frac{1}{7} c^2 C x^7 \]

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]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{3 x^{3}} - \frac{B a^{2}}{2 x^{2}} + 2 B a b \log{\left (x \right )} + \frac{B b c x^{4}}{2} + \frac{B c^{2} x^{6}}{6} + B \left (2 a c + b^{2}\right ) \int x\, dx + \frac{C c^{2} x^{7}}{7} - \frac{a \left (2 A b + C a\right )}{x} + \frac{c x^{5} \left (A c + 2 C b\right )}{5} + x^{3} \left (\frac{2 A b c}{3} + \frac{2 C a c}{3} + \frac{C b^{2}}{3}\right ) + \frac{\left (A b^{2} + 2 a \left (A c + C b\right )\right ) \int A\, dx}{A} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.174244, 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]

-(a^2*A)/(3*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*Lo
g[x]

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Maple [A]  time = 0.01, size = 146, normalized size = 1. \[{\frac{{c}^{2}C{x}^{7}}{7}}+{\frac{B{c}^{2}{x}^{6}}{6}}+{\frac{A{x}^{5}{c}^{2}}{5}}+{\frac{2\,C{x}^{5}bc}{5}}+{\frac{bBc{x}^{4}}{2}}+{\frac{2\,A{x}^{3}bc}{3}}+{\frac{2\,C{x}^{3}ac}{3}}+{\frac{C{x}^{3}{b}^{2}}{3}}+B{x}^{2}ac+{\frac{B{x}^{2}{b}^{2}}{2}}+2\,Axac+Ax{b}^{2}+2\,Cxab-{\frac{A{a}^{2}}{3\,{x}^{3}}}+2\,abB\ln \left ( x \right ) -2\,{\frac{abA}{x}}-{\frac{{a}^{2}C}{x}}-{\frac{B{a}^{2}}{2\,{x}^{2}}} \]

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

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Maxima [A]  time = 0.689953, size = 189, normalized size = 1.27 \[ \frac{1}{7} \, C c^{2} x^{7} + \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{2} \, B b c x^{4} + \frac{1}{5} \,{\left (2 \, C b c + A c^{2}\right )} x^{5} + \frac{1}{3} \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{3} + 2 \, B a b \log \left (x\right ) + \frac{1}{2} \,{\left (B b^{2} + 2 \, B a c\right )} x^{2} +{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x - \frac{3 \, B a^{2} x + 2 \, A a^{2} + 6 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(C*x^2 + B*x + A)/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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Fricas [A]  time = 0.248056, size = 196, normalized size = 1.32 \[ \frac{30 \, C c^{2} x^{10} + 35 \, B c^{2} x^{9} + 105 \, B b c x^{7} + 42 \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + 70 \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} + 420 \, B a b x^{3} \log \left (x\right ) + 105 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + 210 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 105 \, B a^{2} x - 70 \, A a^{2} - 210 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{210 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(C*x^2 + B*x + A)/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 - 70*A*a^2 - 210*(C*a^2
+ 2*A*a*b)*x^2)/x^3

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Sympy [A]  time = 3.33591, size = 158, normalized size = 1.06 \[ 2 B a b \log{\left (x \right )} + \frac{B b c x^{4}}{2} + \frac{B c^{2} x^{6}}{6} + \frac{C c^{2} x^{7}}{7} + x^{5} \left (\frac{A c^{2}}{5} + \frac{2 C b c}{5}\right ) + x^{3} \left (\frac{2 A b c}{3} + \frac{2 C a c}{3} + \frac{C b^{2}}{3}\right ) + x^{2} \left (B a c + \frac{B b^{2}}{2}\right ) + x \left (2 A a c + A b^{2} + 2 C a b\right ) - \frac{2 A a^{2} + 3 B a^{2} x + x^{2} \left (12 A a b + 6 C a^{2}\right )}{6 x^{3}} \]

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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GIAC/XCAS [A]  time = 0.285649, size = 197, normalized size = 1.32 \[ \frac{1}{7} \, C c^{2} x^{7} + \frac{1}{6} \, B c^{2} x^{6} + \frac{2}{5} \, C b c x^{5} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{2} \, B b c x^{4} + \frac{1}{3} \, C b^{2} x^{3} + \frac{2}{3} \, C a c x^{3} + \frac{2}{3} \, A b c x^{3} + \frac{1}{2} \, B b^{2} x^{2} + B a c x^{2} + 2 \, C a b x + A b^{2} x + 2 \, A a c x + 2 \, B a b{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, B a^{2} x + 2 \, A a^{2} + 6 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^2*(C*x^2 + B*x + A)/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*ln(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