3.101 \(\int \frac{\left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3}{x^3} \, dx\)

Optimal. Leaf size=162 \[ -\frac{a^3 A}{2 x^2}+a^2 \log (x) (a B+3 A b)+\frac{3}{8} c x^8 \left (a B c+A b c+b^2 B\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{10} c^2 x^{10} (A c+3 b B)+\frac{1}{12} B c^3 x^{12} \]

[Out]

-(a^3*A)/(2*x^2) + (3*a*(a*b*B + A*(b^2 + a*c))*x^2)/2 + ((3*a*B*(b^2 + a*c) + A
*(b^3 + 6*a*b*c))*x^4)/4 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^6)/6 +
(3*c*(b^2*B + A*b*c + a*B*c)*x^8)/8 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^12
)/12 + a^2*(3*A*b + a*B)*Log[x]

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Rubi [A]  time = 0.507832, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{a^3 A}{2 x^2}+a^2 \log (x) (a B+3 A b)+\frac{3}{8} c x^8 \left (a B c+A b c+b^2 B\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{10} c^2 x^{10} (A c+3 b B)+\frac{1}{12} B c^3 x^{12} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^3,x]

[Out]

-(a^3*A)/(2*x^2) + (3*a*(a*b*B + A*(b^2 + a*c))*x^2)/2 + ((3*a*B*(b^2 + a*c) + A
*(b^3 + 6*a*b*c))*x^4)/4 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^6)/6 +
(3*c*(b^2*B + A*b*c + a*B*c)*x^8)/8 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^12
)/12 + a^2*(3*A*b + a*B)*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{3}}{2 x^{2}} + \frac{B c^{3} x^{12}}{12} + \frac{a^{2} \left (3 A b + B a\right ) \log{\left (x^{2} \right )}}{2} + \frac{3 a x^{2} \left (A a c + A b^{2} + B a b\right )}{2} + \frac{c^{2} x^{10} \left (A c + 3 B b\right )}{10} + \frac{3 c x^{8} \left (A b c + B a c + B b^{2}\right )}{8} + x^{6} \left (\frac{A a c^{2}}{2} + \frac{A b^{2} c}{2} + B a b c + \frac{B b^{3}}{6}\right ) + \left (3 A a b c + \frac{A b^{3}}{2} + \frac{3 B a^{2} c}{2} + \frac{3 B a b^{2}}{2}\right ) \int ^{x^{2}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2+a)**3/x**3,x)

[Out]

-A*a**3/(2*x**2) + B*c**3*x**12/12 + a**2*(3*A*b + B*a)*log(x**2)/2 + 3*a*x**2*(
A*a*c + A*b**2 + B*a*b)/2 + c**2*x**10*(A*c + 3*B*b)/10 + 3*c*x**8*(A*b*c + B*a*
c + B*b**2)/8 + x**6*(A*a*c**2/2 + A*b**2*c/2 + B*a*b*c + B*b**3/6) + (3*A*a*b*c
+ A*b**3/2 + 3*B*a**2*c/2 + 3*B*a*b**2/2)*Integral(x, (x, x**2))

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Mathematica [A]  time = 0.178136, size = 162, normalized size = 1. \[ -\frac{a^3 A}{2 x^2}+a^2 \log (x) (a B+3 A b)+\frac{3}{8} c x^8 \left (a B c+A b c+b^2 B\right )+\frac{3}{2} a x^2 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{10} c^2 x^{10} (A c+3 b B)+\frac{1}{12} B c^3 x^{12} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(a + b*x^2 + c*x^4)^3)/x^3,x]

[Out]

-(a^3*A)/(2*x^2) + (3*a*(a*b*B + A*(b^2 + a*c))*x^2)/2 + ((3*a*B*(b^2 + a*c) + A
*(b^3 + 6*a*b*c))*x^4)/4 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^6)/6 +
(3*c*(b^2*B + A*b*c + a*B*c)*x^8)/8 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^12
)/12 + a^2*(3*A*b + a*B)*Log[x]

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Maple [A]  time = 0.01, size = 190, normalized size = 1.2 \[{\frac{B{c}^{3}{x}^{12}}{12}}+{\frac{A{x}^{10}{c}^{3}}{10}}+{\frac{3\,B{x}^{10}b{c}^{2}}{10}}+{\frac{3\,A{x}^{8}b{c}^{2}}{8}}+{\frac{3\,B{x}^{8}a{c}^{2}}{8}}+{\frac{3\,B{x}^{8}{b}^{2}c}{8}}+{\frac{A{x}^{6}a{c}^{2}}{2}}+{\frac{A{x}^{6}{b}^{2}c}{2}}+B{x}^{6}abc+{\frac{B{x}^{6}{b}^{3}}{6}}+{\frac{3\,A{x}^{4}abc}{2}}+{\frac{A{x}^{4}{b}^{3}}{4}}+{\frac{3\,B{x}^{4}{a}^{2}c}{4}}+{\frac{3\,B{x}^{4}a{b}^{2}}{4}}+{\frac{3\,A{x}^{2}{a}^{2}c}{2}}+{\frac{3\,A{x}^{2}a{b}^{2}}{2}}+{\frac{3\,B{x}^{2}{a}^{2}b}{2}}+3\,A\ln \left ( x \right ){a}^{2}b+B\ln \left ( x \right ){a}^{3}-{\frac{A{a}^{3}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2+a)^3/x^3,x)

[Out]

1/12*B*c^3*x^12+1/10*A*x^10*c^3+3/10*B*x^10*b*c^2+3/8*A*x^8*b*c^2+3/8*B*x^8*a*c^
2+3/8*B*x^8*b^2*c+1/2*A*x^6*a*c^2+1/2*A*x^6*b^2*c+B*x^6*a*b*c+1/6*B*x^6*b^3+3/2*
A*x^4*a*b*c+1/4*A*x^4*b^3+3/4*B*x^4*a^2*c+3/4*B*x^4*a*b^2+3/2*A*x^2*a^2*c+3/2*A*
x^2*a*b^2+3/2*B*x^2*a^2*b+3*A*ln(x)*a^2*b+B*ln(x)*a^3-1/2*a^3*A/x^2

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Maxima [A]  time = 0.704833, size = 225, normalized size = 1.39 \[ \frac{1}{12} \, B c^{3} x^{12} + \frac{1}{10} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + \frac{3}{8} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{8} + \frac{1}{6} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{6} + \frac{1}{4} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + \frac{3}{2} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - \frac{A a^{3}}{2 \, x^{2}} + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3*(B*x^2 + A)/x^3,x, algorithm="maxima")

[Out]

1/12*B*c^3*x^12 + 1/10*(3*B*b*c^2 + A*c^3)*x^10 + 3/8*(B*b^2*c + (B*a + A*b)*c^2
)*x^8 + 1/6*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^6 + 1/4*(3*B*a*b^2 + A
*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^4 + 3/2*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 - 1/2*
A*a^3/x^2 + 1/2*(B*a^3 + 3*A*a^2*b)*log(x^2)

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Fricas [A]  time = 0.246103, size = 230, normalized size = 1.42 \[ \frac{10 \, B c^{3} x^{14} + 12 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{12} + 45 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{10} + 20 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + 30 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + 180 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} - 60 \, A a^{3} + 120 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \log \left (x\right )}{120 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3*(B*x^2 + A)/x^3,x, algorithm="fricas")

[Out]

1/120*(10*B*c^3*x^14 + 12*(3*B*b*c^2 + A*c^3)*x^12 + 45*(B*b^2*c + (B*a + A*b)*c
^2)*x^10 + 20*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^8 + 30*(3*B*a*b^2 +
A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^6 + 180*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^4 - 60*
A*a^3 + 120*(B*a^3 + 3*A*a^2*b)*x^2*log(x))/x^2

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Sympy [A]  time = 2.09926, size = 197, normalized size = 1.22 \[ - \frac{A a^{3}}{2 x^{2}} + \frac{B c^{3} x^{12}}{12} + a^{2} \left (3 A b + B a\right ) \log{\left (x \right )} + x^{10} \left (\frac{A c^{3}}{10} + \frac{3 B b c^{2}}{10}\right ) + x^{8} \left (\frac{3 A b c^{2}}{8} + \frac{3 B a c^{2}}{8} + \frac{3 B b^{2} c}{8}\right ) + x^{6} \left (\frac{A a c^{2}}{2} + \frac{A b^{2} c}{2} + B a b c + \frac{B b^{3}}{6}\right ) + x^{4} \left (\frac{3 A a b c}{2} + \frac{A b^{3}}{4} + \frac{3 B a^{2} c}{4} + \frac{3 B a b^{2}}{4}\right ) + x^{2} \left (\frac{3 A a^{2} c}{2} + \frac{3 A a b^{2}}{2} + \frac{3 B a^{2} b}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2+a)**3/x**3,x)

[Out]

-A*a**3/(2*x**2) + B*c**3*x**12/12 + a**2*(3*A*b + B*a)*log(x) + x**10*(A*c**3/1
0 + 3*B*b*c**2/10) + x**8*(3*A*b*c**2/8 + 3*B*a*c**2/8 + 3*B*b**2*c/8) + x**6*(A
*a*c**2/2 + A*b**2*c/2 + B*a*b*c + B*b**3/6) + x**4*(3*A*a*b*c/2 + A*b**3/4 + 3*
B*a**2*c/4 + 3*B*a*b**2/4) + x**2*(3*A*a**2*c/2 + 3*A*a*b**2/2 + 3*B*a**2*b/2)

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GIAC/XCAS [A]  time = 0.267513, size = 286, normalized size = 1.77 \[ \frac{1}{12} \, B c^{3} x^{12} + \frac{3}{10} \, B b c^{2} x^{10} + \frac{1}{10} \, A c^{3} x^{10} + \frac{3}{8} \, B b^{2} c x^{8} + \frac{3}{8} \, B a c^{2} x^{8} + \frac{3}{8} \, A b c^{2} x^{8} + \frac{1}{6} \, B b^{3} x^{6} + B a b c x^{6} + \frac{1}{2} \, A b^{2} c x^{6} + \frac{1}{2} \, A a c^{2} x^{6} + \frac{3}{4} \, B a b^{2} x^{4} + \frac{1}{4} \, A b^{3} x^{4} + \frac{3}{4} \, B a^{2} c x^{4} + \frac{3}{2} \, A a b c x^{4} + \frac{3}{2} \, B a^{2} b x^{2} + \frac{3}{2} \, A a b^{2} x^{2} + \frac{3}{2} \, A a^{2} c x^{2} + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )}{\rm ln}\left (x^{2}\right ) - \frac{B a^{3} x^{2} + 3 \, A a^{2} b x^{2} + A a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^3*(B*x^2 + A)/x^3,x, algorithm="giac")

[Out]

1/12*B*c^3*x^12 + 3/10*B*b*c^2*x^10 + 1/10*A*c^3*x^10 + 3/8*B*b^2*c*x^8 + 3/8*B*
a*c^2*x^8 + 3/8*A*b*c^2*x^8 + 1/6*B*b^3*x^6 + B*a*b*c*x^6 + 1/2*A*b^2*c*x^6 + 1/
2*A*a*c^2*x^6 + 3/4*B*a*b^2*x^4 + 1/4*A*b^3*x^4 + 3/4*B*a^2*c*x^4 + 3/2*A*a*b*c*
x^4 + 3/2*B*a^2*b*x^2 + 3/2*A*a*b^2*x^2 + 3/2*A*a^2*c*x^2 + 1/2*(B*a^3 + 3*A*a^2
*b)*ln(x^2) - 1/2*(B*a^3*x^2 + 3*A*a^2*b*x^2 + A*a^3)/x^2