3.97 \(\int x \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx\)

Optimal. Leaf size=166 \[ \frac{1}{2} a^3 A x^2+\frac{1}{4} a^2 x^4 (a B+3 A b)+\frac{1}{4} c x^{12} \left (a B c+A b c+b^2 B\right )+\frac{1}{2} a x^6 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{10} x^{10} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{8} x^8 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{14} c^2 x^{14} (A c+3 b B)+\frac{1}{16} B c^3 x^{16} \]

[Out]

(a^3*A*x^2)/2 + (a^2*(3*A*b + a*B)*x^4)/4 + (a*(a*b*B + A*(b^2 + a*c))*x^6)/2 +
((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^8)/8 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*
c + 3*a*A*c^2)*x^10)/10 + (c*(b^2*B + A*b*c + a*B*c)*x^12)/4 + (c^2*(3*b*B + A*c
)*x^14)/14 + (B*c^3*x^16)/16

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Rubi [A]  time = 0.649906, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{1}{2} a^3 A x^2+\frac{1}{4} a^2 x^4 (a B+3 A b)+\frac{1}{4} c x^{12} \left (a B c+A b c+b^2 B\right )+\frac{1}{2} a x^6 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{10} x^{10} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{8} x^8 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{14} c^2 x^{14} (A c+3 b B)+\frac{1}{16} B c^3 x^{16} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*A*x^2)/2 + (a^2*(3*A*b + a*B)*x^4)/4 + (a*(a*b*B + A*(b^2 + a*c))*x^6)/2 +
((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^8)/8 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*
c + 3*a*A*c^2)*x^10)/10 + (c*(b^2*B + A*b*c + a*B*c)*x^12)/4 + (c^2*(3*b*B + A*c
)*x^14)/14 + (B*c^3*x^16)/16

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c^{3} x^{16}}{16} + \frac{a^{3} \int ^{x^{2}} A\, dx}{2} + \frac{a^{2} \left (3 A b + B a\right ) \int ^{x^{2}} x\, dx}{2} + \frac{a x^{6} \left (A a c + A b^{2} + B a b\right )}{2} + \frac{c^{2} x^{14} \left (A c + 3 B b\right )}{14} + \frac{c x^{12} \left (A b c + B a c + B b^{2}\right )}{4} + x^{10} \left (\frac{3 A a c^{2}}{10} + \frac{3 A b^{2} c}{10} + \frac{3 B a b c}{5} + \frac{B b^{3}}{10}\right ) + x^{8} \left (\frac{3 A a b c}{4} + \frac{A b^{3}}{8} + \frac{3 B a^{2} c}{8} + \frac{3 B a b^{2}}{8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c**3*x**16/16 + a**3*Integral(A, (x, x**2))/2 + a**2*(3*A*b + B*a)*Integral(x,
(x, x**2))/2 + a*x**6*(A*a*c + A*b**2 + B*a*b)/2 + c**2*x**14*(A*c + 3*B*b)/14
+ c*x**12*(A*b*c + B*a*c + B*b**2)/4 + x**10*(3*A*a*c**2/10 + 3*A*b**2*c/10 + 3*
B*a*b*c/5 + B*b**3/10) + x**8*(3*A*a*b*c/4 + A*b**3/8 + 3*B*a**2*c/8 + 3*B*a*b**
2/8)

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Mathematica [A]  time = 0.135843, size = 154, normalized size = 0.93 \[ \frac{1}{560} x^2 \left (280 a^3 A+140 a^2 x^2 (a B+3 A b)+140 c x^{10} \left (a B c+A b c+b^2 B\right )+280 a x^4 \left (A \left (a c+b^2\right )+a b B\right )+56 x^8 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+70 x^6 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+40 c^2 x^{12} (A c+3 b B)+35 B c^3 x^{14}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(280*a^3*A + 140*a^2*(3*A*b + a*B)*x^2 + 280*a*(a*b*B + A*(b^2 + a*c))*x^4
+ 70*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^6 + 56*(b^3*B + 3*A*b^2*c + 6*a*b
*B*c + 3*a*A*c^2)*x^8 + 140*c*(b^2*B + A*b*c + a*B*c)*x^10 + 40*c^2*(3*b*B + A*c
)*x^12 + 35*B*c^3*x^14))/560

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Maple [A]  time = 0.001, size = 226, normalized size = 1.4 \[{\frac{B{c}^{3}{x}^{16}}{16}}+{\frac{ \left ( A{c}^{3}+3\,B{c}^{2}b \right ){x}^{14}}{14}}+{\frac{ \left ( 3\,A{c}^{2}b+B \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{12}}{12}}+{\frac{ \left ( A \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{10}}{10}}+{\frac{ \left ( A \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( A \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +3\,B{a}^{2}b \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){x}^{4}}{4}}+{\frac{{a}^{3}A{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*B*c^3*x^16+1/14*(A*c^3+3*B*b*c^2)*x^14+1/12*(3*A*c^2*b+B*(a*c^2+2*b^2*c+c*(
2*a*c+b^2)))*x^12+1/10*(A*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+B*(4*a*b*c+b*(2*a*c+b^2)
))*x^10+1/8*(A*(4*a*b*c+b*(2*a*c+b^2))+B*(a*(2*a*c+b^2)+2*a*b^2+a^2*c))*x^8+1/6*
(A*(a*(2*a*c+b^2)+2*a*b^2+a^2*c)+3*B*a^2*b)*x^6+1/4*(3*A*a^2*b+B*a^3)*x^4+1/2*a^
3*A*x^2

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Maxima [A]  time = 0.695589, size = 224, normalized size = 1.35 \[ \frac{1}{16} \, B c^{3} x^{16} + \frac{1}{14} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{14} + \frac{1}{4} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{12} + \frac{1}{10} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{10} + \frac{1}{8} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{8} + \frac{1}{2} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{6} + \frac{1}{2} \, A a^{3} x^{2} + \frac{1}{4} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \]

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]

1/16*B*c^3*x^16 + 1/14*(3*B*b*c^2 + A*c^3)*x^14 + 1/4*(B*b^2*c + (B*a + A*b)*c^2
)*x^12 + 1/10*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^10 + 1/8*(3*B*a*b^2
+ A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^8 + 1/2*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^6 + 1
/2*A*a^3*x^2 + 1/4*(B*a^3 + 3*A*a^2*b)*x^4

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Fricas [A]  time = 0.228335, size = 1, normalized size = 0.01 \[ \frac{1}{16} x^{16} c^{3} B + \frac{3}{14} x^{14} c^{2} b B + \frac{1}{14} x^{14} c^{3} A + \frac{1}{4} x^{12} c b^{2} B + \frac{1}{4} x^{12} c^{2} a B + \frac{1}{4} x^{12} c^{2} b A + \frac{1}{10} x^{10} b^{3} B + \frac{3}{5} x^{10} c b a B + \frac{3}{10} x^{10} c b^{2} A + \frac{3}{10} x^{10} c^{2} a A + \frac{3}{8} x^{8} b^{2} a B + \frac{3}{8} x^{8} c a^{2} B + \frac{1}{8} x^{8} b^{3} A + \frac{3}{4} x^{8} c b a A + \frac{1}{2} x^{6} b a^{2} B + \frac{1}{2} x^{6} b^{2} a A + \frac{1}{2} x^{6} c a^{2} A + \frac{1}{4} x^{4} a^{3} B + \frac{3}{4} x^{4} b a^{2} A + \frac{1}{2} x^{2} a^{3} A \]

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]

1/16*x^16*c^3*B + 3/14*x^14*c^2*b*B + 1/14*x^14*c^3*A + 1/4*x^12*c*b^2*B + 1/4*x
^12*c^2*a*B + 1/4*x^12*c^2*b*A + 1/10*x^10*b^3*B + 3/5*x^10*c*b*a*B + 3/10*x^10*
c*b^2*A + 3/10*x^10*c^2*a*A + 3/8*x^8*b^2*a*B + 3/8*x^8*c*a^2*B + 1/8*x^8*b^3*A
+ 3/4*x^8*c*b*a*A + 1/2*x^6*b*a^2*B + 1/2*x^6*b^2*a*A + 1/2*x^6*c*a^2*A + 1/4*x^
4*a^3*B + 3/4*x^4*b*a^2*A + 1/2*x^2*a^3*A

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Sympy [A]  time = 0.187363, size = 199, normalized size = 1.2 \[ \frac{A a^{3} x^{2}}{2} + \frac{B c^{3} x^{16}}{16} + x^{14} \left (\frac{A c^{3}}{14} + \frac{3 B b c^{2}}{14}\right ) + x^{12} \left (\frac{A b c^{2}}{4} + \frac{B a c^{2}}{4} + \frac{B b^{2} c}{4}\right ) + x^{10} \left (\frac{3 A a c^{2}}{10} + \frac{3 A b^{2} c}{10} + \frac{3 B a b c}{5} + \frac{B b^{3}}{10}\right ) + x^{8} \left (\frac{3 A a b c}{4} + \frac{A b^{3}}{8} + \frac{3 B a^{2} c}{8} + \frac{3 B a b^{2}}{8}\right ) + x^{6} \left (\frac{A a^{2} c}{2} + \frac{A a b^{2}}{2} + \frac{B a^{2} b}{2}\right ) + x^{4} \left (\frac{3 A a^{2} b}{4} + \frac{B a^{3}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**3*x**2/2 + B*c**3*x**16/16 + x**14*(A*c**3/14 + 3*B*b*c**2/14) + x**12*(A*b
*c**2/4 + B*a*c**2/4 + B*b**2*c/4) + x**10*(3*A*a*c**2/10 + 3*A*b**2*c/10 + 3*B*
a*b*c/5 + B*b**3/10) + x**8*(3*A*a*b*c/4 + A*b**3/8 + 3*B*a**2*c/8 + 3*B*a*b**2/
8) + x**6*(A*a**2*c/2 + A*a*b**2/2 + B*a**2*b/2) + x**4*(3*A*a**2*b/4 + B*a**3/4
)

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GIAC/XCAS [A]  time = 0.262962, size = 261, normalized size = 1.57 \[ \frac{1}{16} \, B c^{3} x^{16} + \frac{3}{14} \, B b c^{2} x^{14} + \frac{1}{14} \, A c^{3} x^{14} + \frac{1}{4} \, B b^{2} c x^{12} + \frac{1}{4} \, B a c^{2} x^{12} + \frac{1}{4} \, A b c^{2} x^{12} + \frac{1}{10} \, B b^{3} x^{10} + \frac{3}{5} \, B a b c x^{10} + \frac{3}{10} \, A b^{2} c x^{10} + \frac{3}{10} \, A a c^{2} x^{10} + \frac{3}{8} \, B a b^{2} x^{8} + \frac{1}{8} \, A b^{3} x^{8} + \frac{3}{8} \, B a^{2} c x^{8} + \frac{3}{4} \, A a b c x^{8} + \frac{1}{2} \, B a^{2} b x^{6} + \frac{1}{2} \, A a b^{2} x^{6} + \frac{1}{2} \, A a^{2} c x^{6} + \frac{1}{4} \, B a^{3} x^{4} + \frac{3}{4} \, A a^{2} b x^{4} + \frac{1}{2} \, A a^{3} x^{2} \]

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="giac")

[Out]

1/16*B*c^3*x^16 + 3/14*B*b*c^2*x^14 + 1/14*A*c^3*x^14 + 1/4*B*b^2*c*x^12 + 1/4*B
*a*c^2*x^12 + 1/4*A*b*c^2*x^12 + 1/10*B*b^3*x^10 + 3/5*B*a*b*c*x^10 + 3/10*A*b^2
*c*x^10 + 3/10*A*a*c^2*x^10 + 3/8*B*a*b^2*x^8 + 1/8*A*b^3*x^8 + 3/8*B*a^2*c*x^8
+ 3/4*A*a*b*c*x^8 + 1/2*B*a^2*b*x^6 + 1/2*A*a*b^2*x^6 + 1/2*A*a^2*c*x^6 + 1/4*B*
a^3*x^4 + 3/4*A*a^2*b*x^4 + 1/2*A*a^3*x^2