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

Optimal. Leaf size=162 \[ a^3 A \log (x)+\frac{1}{2} a^2 x^2 (a B+3 A b)+\frac{3}{10} c x^{10} \left (a B c+A b c+b^2 B\right )+\frac{3}{4} a x^4 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{8} x^8 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{6} x^6 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{12} c^2 x^{12} (A c+3 b B)+\frac{1}{14} B c^3 x^{14} \]

[Out]

(a^2*(3*A*b + a*B)*x^2)/2 + (3*a*(a*b*B + A*(b^2 + a*c))*x^4)/4 + ((3*a*B*(b^2 +
a*c) + A*(b^3 + 6*a*b*c))*x^6)/6 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)
*x^8)/8 + (3*c*(b^2*B + A*b*c + a*B*c)*x^10)/10 + (c^2*(3*b*B + A*c)*x^12)/12 +
(B*c^3*x^14)/14 + a^3*A*Log[x]

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Rubi [A]  time = 0.439926, 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 \[ a^3 A \log (x)+\frac{1}{2} a^2 x^2 (a B+3 A b)+\frac{3}{10} c x^{10} \left (a B c+A b c+b^2 B\right )+\frac{3}{4} a x^4 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{8} x^8 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{6} x^6 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{12} c^2 x^{12} (A c+3 b B)+\frac{1}{14} B c^3 x^{14} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(3*A*b + a*B)*x^2)/2 + (3*a*(a*b*B + A*(b^2 + a*c))*x^4)/4 + ((3*a*B*(b^2 +
a*c) + A*(b^3 + 6*a*b*c))*x^6)/6 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)
*x^8)/8 + (3*c*(b^2*B + A*b*c + a*B*c)*x^10)/10 + (c^2*(3*b*B + A*c)*x^12)/12 +
(B*c^3*x^14)/14 + a^3*A*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{A a^{3} \log{\left (x^{2} \right )}}{2} + \frac{B c^{3} x^{14}}{14} + \frac{3 a \left (A a c + A b^{2} + B a b\right ) \int ^{x^{2}} x\, dx}{2} + \frac{c^{2} x^{12} \left (A c + 3 B b\right )}{12} + \frac{3 c x^{10} \left (A b c + B a c + B b^{2}\right )}{10} + x^{8} \left (\frac{3 A a c^{2}}{8} + \frac{3 A b^{2} c}{8} + \frac{3 B a b c}{4} + \frac{B b^{3}}{8}\right ) + x^{6} \left (A a b c + \frac{A b^{3}}{6} + \frac{B a^{2} c}{2} + \frac{B a b^{2}}{2}\right ) + \left (\frac{3 A b}{2} + \frac{B a}{2}\right ) \int ^{x^{2}} a^{2}\, dx \]

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,x)

[Out]

A*a**3*log(x**2)/2 + B*c**3*x**14/14 + 3*a*(A*a*c + A*b**2 + B*a*b)*Integral(x,
(x, x**2))/2 + c**2*x**12*(A*c + 3*B*b)/12 + 3*c*x**10*(A*b*c + B*a*c + B*b**2)/
10 + x**8*(3*A*a*c**2/8 + 3*A*b**2*c/8 + 3*B*a*b*c/4 + B*b**3/8) + x**6*(A*a*b*c
+ A*b**3/6 + B*a**2*c/2 + B*a*b**2/2) + (3*A*b/2 + B*a/2)*Integral(a**2, (x, x*
*2))

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Mathematica [A]  time = 0.123168, size = 162, normalized size = 1. \[ a^3 A \log (x)+\frac{1}{2} a^2 x^2 (a B+3 A b)+\frac{3}{10} c x^{10} \left (a B c+A b c+b^2 B\right )+\frac{3}{4} a x^4 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{8} x^8 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{6} x^6 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{12} c^2 x^{12} (A c+3 b B)+\frac{1}{14} B c^3 x^{14} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(3*A*b + a*B)*x^2)/2 + (3*a*(a*b*B + A*(b^2 + a*c))*x^4)/4 + ((3*a*B*(b^2 +
a*c) + A*(b^3 + 6*a*b*c))*x^6)/6 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)
*x^8)/8 + (3*c*(b^2*B + A*b*c + a*B*c)*x^10)/10 + (c^2*(3*b*B + A*c)*x^12)/12 +
(B*c^3*x^14)/14 + a^3*A*Log[x]

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Maple [A]  time = 0.004, size = 191, normalized size = 1.2 \[{\frac{B{c}^{3}{x}^{14}}{14}}+{\frac{A{x}^{12}{c}^{3}}{12}}+{\frac{B{x}^{12}b{c}^{2}}{4}}+{\frac{3\,A{x}^{10}b{c}^{2}}{10}}+{\frac{3\,B{x}^{10}a{c}^{2}}{10}}+{\frac{3\,B{x}^{10}{b}^{2}c}{10}}+{\frac{3\,A{x}^{8}a{c}^{2}}{8}}+{\frac{3\,A{x}^{8}{b}^{2}c}{8}}+{\frac{3\,B{x}^{8}abc}{4}}+{\frac{B{x}^{8}{b}^{3}}{8}}+A{x}^{6}abc+{\frac{A{x}^{6}{b}^{3}}{6}}+{\frac{B{x}^{6}{a}^{2}c}{2}}+{\frac{B{x}^{6}a{b}^{2}}{2}}+{\frac{3\,A{x}^{4}{a}^{2}c}{4}}+{\frac{3\,A{x}^{4}a{b}^{2}}{4}}+{\frac{3\,B{x}^{4}{a}^{2}b}{4}}+{\frac{3\,A{x}^{2}{a}^{2}b}{2}}+{\frac{B{x}^{2}{a}^{3}}{2}}+{a}^{3}A\ln \left ( x \right ) \]

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,x)

[Out]

1/14*B*c^3*x^14+1/12*A*x^12*c^3+1/4*B*x^12*b*c^2+3/10*A*x^10*b*c^2+3/10*B*x^10*a
*c^2+3/10*B*x^10*b^2*c+3/8*A*x^8*a*c^2+3/8*A*x^8*b^2*c+3/4*B*x^8*a*b*c+1/8*B*x^8
*b^3+A*x^6*a*b*c+1/6*A*x^6*b^3+1/2*B*x^6*a^2*c+1/2*B*x^6*a*b^2+3/4*A*x^4*a^2*c+3
/4*A*x^4*a*b^2+3/4*B*x^4*a^2*b+3/2*A*x^2*a^2*b+1/2*B*x^2*a^3+a^3*A*ln(x)

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Maxima [A]  time = 0.703261, size = 225, normalized size = 1.39 \[ \frac{1}{14} \, B c^{3} x^{14} + \frac{1}{12} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{12} + \frac{3}{10} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{10} + \frac{1}{8} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + \frac{1}{6} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + \frac{3}{4} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} + \frac{1}{2} \, A a^{3} \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]

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,x, algorithm="maxima")

[Out]

1/14*B*c^3*x^14 + 1/12*(3*B*b*c^2 + A*c^3)*x^12 + 3/10*(B*b^2*c + (B*a + A*b)*c^
2)*x^10 + 1/8*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^8 + 1/6*(3*B*a*b^2 +
A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^6 + 3/4*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^4 + 1/
2*A*a^3*log(x^2) + 1/2*(B*a^3 + 3*A*a^2*b)*x^2

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Fricas [A]  time = 0.255229, size = 221, normalized size = 1.36 \[ \frac{1}{14} \, B c^{3} x^{14} + \frac{1}{12} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{12} + \frac{3}{10} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{10} + \frac{1}{8} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{8} + \frac{1}{6} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + \frac{3}{4} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{2} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]

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,x, algorithm="fricas")

[Out]

1/14*B*c^3*x^14 + 1/12*(3*B*b*c^2 + A*c^3)*x^12 + 3/10*(B*b^2*c + (B*a + A*b)*c^
2)*x^10 + 1/8*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^8 + 1/6*(3*B*a*b^2 +
A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^6 + 3/4*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^4 + A*
a^3*log(x) + 1/2*(B*a^3 + 3*A*a^2*b)*x^2

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Sympy [A]  time = 1.73224, size = 199, normalized size = 1.23 \[ A a^{3} \log{\left (x \right )} + \frac{B c^{3} x^{14}}{14} + x^{12} \left (\frac{A c^{3}}{12} + \frac{B b c^{2}}{4}\right ) + x^{10} \left (\frac{3 A b c^{2}}{10} + \frac{3 B a c^{2}}{10} + \frac{3 B b^{2} c}{10}\right ) + x^{8} \left (\frac{3 A a c^{2}}{8} + \frac{3 A b^{2} c}{8} + \frac{3 B a b c}{4} + \frac{B b^{3}}{8}\right ) + x^{6} \left (A a b c + \frac{A b^{3}}{6} + \frac{B a^{2} c}{2} + \frac{B a b^{2}}{2}\right ) + x^{4} \left (\frac{3 A a^{2} c}{4} + \frac{3 A a b^{2}}{4} + \frac{3 B a^{2} b}{4}\right ) + x^{2} \left (\frac{3 A a^{2} b}{2} + \frac{B a^{3}}{2}\right ) \]

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,x)

[Out]

A*a**3*log(x) + B*c**3*x**14/14 + x**12*(A*c**3/12 + B*b*c**2/4) + x**10*(3*A*b*
c**2/10 + 3*B*a*c**2/10 + 3*B*b**2*c/10) + x**8*(3*A*a*c**2/8 + 3*A*b**2*c/8 + 3
*B*a*b*c/4 + B*b**3/8) + x**6*(A*a*b*c + A*b**3/6 + B*a**2*c/2 + B*a*b**2/2) + x
**4*(3*A*a**2*c/4 + 3*A*a*b**2/4 + 3*B*a**2*b/4) + x**2*(3*A*a**2*b/2 + B*a**3/2
)

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GIAC/XCAS [A]  time = 0.264959, size = 261, normalized size = 1.61 \[ \frac{1}{14} \, B c^{3} x^{14} + \frac{1}{4} \, B b c^{2} x^{12} + \frac{1}{12} \, A c^{3} x^{12} + \frac{3}{10} \, B b^{2} c x^{10} + \frac{3}{10} \, B a c^{2} x^{10} + \frac{3}{10} \, A b c^{2} x^{10} + \frac{1}{8} \, B b^{3} x^{8} + \frac{3}{4} \, B a b c x^{8} + \frac{3}{8} \, A b^{2} c x^{8} + \frac{3}{8} \, A a c^{2} x^{8} + \frac{1}{2} \, B a b^{2} x^{6} + \frac{1}{6} \, A b^{3} x^{6} + \frac{1}{2} \, B a^{2} c x^{6} + A a b c x^{6} + \frac{3}{4} \, B a^{2} b x^{4} + \frac{3}{4} \, A a b^{2} x^{4} + \frac{3}{4} \, A a^{2} c x^{4} + \frac{1}{2} \, B a^{3} x^{2} + \frac{3}{2} \, A a^{2} b x^{2} + \frac{1}{2} \, A a^{3}{\rm ln}\left (x^{2}\right ) \]

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,x, algorithm="giac")

[Out]

1/14*B*c^3*x^14 + 1/4*B*b*c^2*x^12 + 1/12*A*c^3*x^12 + 3/10*B*b^2*c*x^10 + 3/10*
B*a*c^2*x^10 + 3/10*A*b*c^2*x^10 + 1/8*B*b^3*x^8 + 3/4*B*a*b*c*x^8 + 3/8*A*b^2*c
*x^8 + 3/8*A*a*c^2*x^8 + 1/2*B*a*b^2*x^6 + 1/6*A*b^3*x^6 + 1/2*B*a^2*c*x^6 + A*a
*b*c*x^6 + 3/4*B*a^2*b*x^4 + 3/4*A*a*b^2*x^4 + 3/4*A*a^2*c*x^4 + 1/2*B*a^3*x^2 +
3/2*A*a^2*b*x^2 + 1/2*A*a^3*ln(x^2)