3.20 \(\int \frac{\left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^2}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.316564, 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 \[ -\frac{a^2 A}{6 x^6}-\frac{a^2 B}{5 x^5}-\frac{A \left (2 a c+b^2\right )+2 a b C}{2 x^2}+\log (x) \left (C \left (2 a c+b^2\right )+2 A b c\right )-\frac{a (a C+2 A b)}{4 x^4}-\frac{B \left (2 a c+b^2\right )}{x}-\frac{2 a b B}{3 x^3}+\frac{1}{2} c x^2 (A c+2 b C)+2 b B c x+\frac{1}{3} B c^2 x^3+\frac{1}{4} c^2 C x^4 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^7,x]

[Out]

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

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

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**7,x)

[Out]

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

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Mathematica [A]  time = 0.208139, size = 144, normalized size = 0.97 \[ -\frac{a^2 (10 A+3 x (4 B+5 C x))}{60 x^6}+\log (x) \left (C \left (2 a c+b^2\right )+2 A b c\right )-\frac{a \left (3 A \left (b+2 c x^2\right )+2 x \left (2 b B+3 b C x+6 B c x^2\right )\right )}{6 x^4}+\frac{A \left (c^2 x^4-b^2\right )}{2 x^2}-\frac{b^2 B}{x}+b c x (2 B+C x)+\frac{1}{12} c^2 x^3 (4 B+3 C x) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^2)/x^7,x]

[Out]

-((b^2*B)/x) + b*c*x*(2*B + C*x) + (c^2*x^3*(4*B + 3*C*x))/12 + (A*(-b^2 + c^2*x
^4))/(2*x^2) - (a^2*(10*A + 3*x*(4*B + 5*C*x)))/(60*x^6) - (a*(3*A*(b + 2*c*x^2)
+ 2*x*(2*b*B + 3*b*C*x + 6*B*c*x^2)))/(6*x^4) + (2*A*b*c + (b^2 + 2*a*c)*C)*Log
[x]

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Maple [A]  time = 0.012, size = 148, normalized size = 1. \[{\frac{{c}^{2}C{x}^{4}}{4}}+{\frac{B{c}^{2}{x}^{3}}{3}}+{\frac{A{x}^{2}{c}^{2}}{2}}+C{x}^{2}bc+2\,bBcx-{\frac{2\,abB}{3\,{x}^{3}}}-{\frac{A{a}^{2}}{6\,{x}^{6}}}+2\,A\ln \left ( x \right ) bc+2\,C\ln \left ( x \right ) ac+C\ln \left ( x \right ){b}^{2}-2\,{\frac{aBc}{x}}-{\frac{{b}^{2}B}{x}}-{\frac{aAc}{{x}^{2}}}-{\frac{A{b}^{2}}{2\,{x}^{2}}}-{\frac{abC}{{x}^{2}}}-{\frac{B{a}^{2}}{5\,{x}^{5}}}-{\frac{abA}{2\,{x}^{4}}}-{\frac{{a}^{2}C}{4\,{x}^{4}}} \]

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^7,x)

[Out]

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

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Maxima [A]  time = 0.704074, size = 189, normalized size = 1.27 \[ \frac{1}{4} \, C c^{2} x^{4} + \frac{1}{3} \, B c^{2} x^{3} + 2 \, B b c x + \frac{1}{2} \,{\left (2 \, C b c + A c^{2}\right )} x^{2} +{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} \log \left (x\right ) - \frac{40 \, B a b x^{3} + 60 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + 30 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 12 \, B a^{2} x + 10 \, A a^{2} + 15 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \]

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^7,x, algorithm="maxima")

[Out]

1/4*C*c^2*x^4 + 1/3*B*c^2*x^3 + 2*B*b*c*x + 1/2*(2*C*b*c + A*c^2)*x^2 + (C*b^2 +
2*(C*a + A*b)*c)*log(x) - 1/60*(40*B*a*b*x^3 + 60*(B*b^2 + 2*B*a*c)*x^5 + 30*(2
*C*a*b + A*b^2 + 2*A*a*c)*x^4 + 12*B*a^2*x + 10*A*a^2 + 15*(C*a^2 + 2*A*a*b)*x^2
)/x^6

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Fricas [A]  time = 0.250922, size = 196, normalized size = 1.32 \[ \frac{15 \, C c^{2} x^{10} + 20 \, B c^{2} x^{9} + 120 \, B b c x^{7} + 30 \,{\left (2 \, C b c + A c^{2}\right )} x^{8} + 60 \,{\left (C b^{2} + 2 \,{\left (C a + A b\right )} c\right )} x^{6} \log \left (x\right ) - 40 \, B a b x^{3} - 60 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} - 30 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} - 12 \, B a^{2} x - 10 \, A a^{2} - 15 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \]

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^7,x, algorithm="fricas")

[Out]

1/60*(15*C*c^2*x^10 + 20*B*c^2*x^9 + 120*B*b*c*x^7 + 30*(2*C*b*c + A*c^2)*x^8 +
60*(C*b^2 + 2*(C*a + A*b)*c)*x^6*log(x) - 40*B*a*b*x^3 - 60*(B*b^2 + 2*B*a*c)*x^
5 - 30*(2*C*a*b + A*b^2 + 2*A*a*c)*x^4 - 12*B*a^2*x - 10*A*a^2 - 15*(C*a^2 + 2*A
*a*b)*x^2)/x^6

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Sympy [A]  time = 98.9006, size = 153, normalized size = 1.03 \[ 2 B b c x + \frac{B c^{2} x^{3}}{3} + \frac{C c^{2} x^{4}}{4} + x^{2} \left (\frac{A c^{2}}{2} + C b c\right ) + \left (2 A b c + 2 C a c + C b^{2}\right ) \log{\left (x \right )} - \frac{10 A a^{2} + 12 B a^{2} x + 40 B a b x^{3} + x^{5} \left (120 B a c + 60 B b^{2}\right ) + x^{4} \left (60 A a c + 30 A b^{2} + 60 C a b\right ) + x^{2} \left (30 A a b + 15 C a^{2}\right )}{60 x^{6}} \]

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**7,x)

[Out]

2*B*b*c*x + B*c**2*x**3/3 + C*c**2*x**4/4 + x**2*(A*c**2/2 + C*b*c) + (2*A*b*c +
2*C*a*c + C*b**2)*log(x) - (10*A*a**2 + 12*B*a**2*x + 40*B*a*b*x**3 + x**5*(120
*B*a*c + 60*B*b**2) + x**4*(60*A*a*c + 30*A*b**2 + 60*C*a*b) + x**2*(30*A*a*b +
15*C*a**2))/(60*x**6)

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GIAC/XCAS [A]  time = 0.282073, size = 190, normalized size = 1.28 \[ \frac{1}{4} \, C c^{2} x^{4} + \frac{1}{3} \, B c^{2} x^{3} + C b c x^{2} + \frac{1}{2} \, A c^{2} x^{2} + 2 \, B b c x +{\left (C b^{2} + 2 \, C a c + 2 \, A b c\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{40 \, B a b x^{3} + 60 \,{\left (B b^{2} + 2 \, B a c\right )} x^{5} + 30 \,{\left (2 \, C a b + A b^{2} + 2 \, A a c\right )} x^{4} + 12 \, B a^{2} x + 10 \, A a^{2} + 15 \,{\left (C a^{2} + 2 \, A a b\right )} x^{2}}{60 \, x^{6}} \]

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^7,x, algorithm="giac")

[Out]

1/4*C*c^2*x^4 + 1/3*B*c^2*x^3 + C*b*c*x^2 + 1/2*A*c^2*x^2 + 2*B*b*c*x + (C*b^2 +
2*C*a*c + 2*A*b*c)*ln(abs(x)) - 1/60*(40*B*a*b*x^3 + 60*(B*b^2 + 2*B*a*c)*x^5 +
30*(2*C*a*b + A*b^2 + 2*A*a*c)*x^4 + 12*B*a^2*x + 10*A*a^2 + 15*(C*a^2 + 2*A*a*
b)*x^2)/x^6