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

Optimal. Leaf size=161 \[ a^3 A x+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{11} c x^{11} \left (a B c+A b c+b^2 B\right )+\frac{3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{9} x^9 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{7} x^7 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{13} c^2 x^{13} (A c+3 b B)+\frac{1}{15} B c^3 x^{15} \]

[Out]

a^3*A*x + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^5)/5 + ((3*
a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^7)/7 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c +
3*a*A*c^2)*x^9)/9 + (3*c*(b^2*B + A*b*c + a*B*c)*x^11)/11 + (c^2*(3*b*B + A*c)*x
^13)/13 + (B*c^3*x^15)/15

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Rubi [A]  time = 0.302797, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ a^3 A x+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{11} c x^{11} \left (a B c+A b c+b^2 B\right )+\frac{3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{9} x^9 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{7} x^7 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{13} c^2 x^{13} (A c+3 b B)+\frac{1}{15} B c^3 x^{15} \]

Antiderivative was successfully verified.

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

[Out]

a^3*A*x + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^5)/5 + ((3*
a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^7)/7 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c +
3*a*A*c^2)*x^9)/9 + (3*c*(b^2*B + A*b*c + a*B*c)*x^11)/11 + (c^2*(3*b*B + A*c)*x
^13)/13 + (B*c^3*x^15)/15

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c^{3} x^{15}}{15} + a^{3} \int A\, dx + \frac{a^{2} x^{3} \left (3 A b + B a\right )}{3} + \frac{3 a x^{5} \left (A a c + A b^{2} + B a b\right )}{5} + \frac{c^{2} x^{13} \left (A c + 3 B b\right )}{13} + \frac{3 c x^{11} \left (A b c + B a c + B b^{2}\right )}{11} + x^{9} \left (\frac{A a c^{2}}{3} + \frac{A b^{2} c}{3} + \frac{2 B a b c}{3} + \frac{B b^{3}}{9}\right ) + x^{7} \left (\frac{6 A a b c}{7} + \frac{A b^{3}}{7} + \frac{3 B a^{2} c}{7} + \frac{3 B a b^{2}}{7}\right ) \]

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)

[Out]

B*c**3*x**15/15 + a**3*Integral(A, x) + a**2*x**3*(3*A*b + B*a)/3 + 3*a*x**5*(A*
a*c + A*b**2 + B*a*b)/5 + c**2*x**13*(A*c + 3*B*b)/13 + 3*c*x**11*(A*b*c + B*a*c
+ B*b**2)/11 + x**9*(A*a*c**2/3 + A*b**2*c/3 + 2*B*a*b*c/3 + B*b**3/9) + x**7*(
6*A*a*b*c/7 + A*b**3/7 + 3*B*a**2*c/7 + 3*B*a*b**2/7)

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Mathematica [A]  time = 0.0896112, size = 161, normalized size = 1. \[ a^3 A x+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{11} c x^{11} \left (a B c+A b c+b^2 B\right )+\frac{3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{9} x^9 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{7} x^7 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{13} c^2 x^{13} (A c+3 b B)+\frac{1}{15} B c^3 x^{15} \]

Antiderivative was successfully verified.

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

[Out]

a^3*A*x + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^5)/5 + ((3*
a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^7)/7 + ((b^3*B + 3*A*b^2*c + 6*a*b*B*c +
3*a*A*c^2)*x^9)/9 + (3*c*(b^2*B + A*b*c + a*B*c)*x^11)/11 + (c^2*(3*b*B + A*c)*x
^13)/13 + (B*c^3*x^15)/15

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Maple [A]  time = 0.001, size = 223, normalized size = 1.4 \[{\frac{B{c}^{3}{x}^{15}}{15}}+{\frac{ \left ( A{c}^{3}+3\,B{c}^{2}b \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,A{c}^{2}b+B \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{11}}{11}}+{\frac{ \left ( A \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( A \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( A \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +3\,B{a}^{2}b \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){x}^{3}}{3}}+{a}^{3}Ax \]

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)

[Out]

1/15*B*c^3*x^15+1/13*(A*c^3+3*B*b*c^2)*x^13+1/11*(3*A*c^2*b+B*(a*c^2+2*b^2*c+c*(
2*a*c+b^2)))*x^11+1/9*(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^9+1/7*(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^7+1/5*(A
*(a*(2*a*c+b^2)+2*a*b^2+a^2*c)+3*B*a^2*b)*x^5+1/3*(3*A*a^2*b+B*a^3)*x^3+a^3*A*x

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Maxima [A]  time = 0.712528, size = 220, normalized size = 1.37 \[ \frac{1}{15} \, B c^{3} x^{15} + \frac{1}{13} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{13} + \frac{3}{11} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{11} + \frac{1}{9} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{9} + \frac{1}{7} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{7} + \frac{3}{5} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + A a^{3} x + \frac{1}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \]

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, algorithm="maxima")

[Out]

1/15*B*c^3*x^15 + 1/13*(3*B*b*c^2 + A*c^3)*x^13 + 3/11*(B*b^2*c + (B*a + A*b)*c^
2)*x^11 + 1/9*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^9 + 1/7*(3*B*a*b^2 +
A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^7 + 3/5*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^5 + A*
a^3*x + 1/3*(B*a^3 + 3*A*a^2*b)*x^3

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Fricas [A]  time = 0.239825, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} c^{3} B + \frac{3}{13} x^{13} c^{2} b B + \frac{1}{13} x^{13} c^{3} A + \frac{3}{11} x^{11} c b^{2} B + \frac{3}{11} x^{11} c^{2} a B + \frac{3}{11} x^{11} c^{2} b A + \frac{1}{9} x^{9} b^{3} B + \frac{2}{3} x^{9} c b a B + \frac{1}{3} x^{9} c b^{2} A + \frac{1}{3} x^{9} c^{2} a A + \frac{3}{7} x^{7} b^{2} a B + \frac{3}{7} x^{7} c a^{2} B + \frac{1}{7} x^{7} b^{3} A + \frac{6}{7} x^{7} c b a A + \frac{3}{5} x^{5} b a^{2} B + \frac{3}{5} x^{5} b^{2} a A + \frac{3}{5} x^{5} c a^{2} A + \frac{1}{3} x^{3} a^{3} B + x^{3} b a^{2} A + x a^{3} A \]

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, algorithm="fricas")

[Out]

1/15*x^15*c^3*B + 3/13*x^13*c^2*b*B + 1/13*x^13*c^3*A + 3/11*x^11*c*b^2*B + 3/11
*x^11*c^2*a*B + 3/11*x^11*c^2*b*A + 1/9*x^9*b^3*B + 2/3*x^9*c*b*a*B + 1/3*x^9*c*
b^2*A + 1/3*x^9*c^2*a*A + 3/7*x^7*b^2*a*B + 3/7*x^7*c*a^2*B + 1/7*x^7*b^3*A + 6/
7*x^7*c*b*a*A + 3/5*x^5*b*a^2*B + 3/5*x^5*b^2*a*A + 3/5*x^5*c*a^2*A + 1/3*x^3*a^
3*B + x^3*b*a^2*A + x*a^3*A

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Sympy [A]  time = 0.186605, size = 199, normalized size = 1.24 \[ A a^{3} x + \frac{B c^{3} x^{15}}{15} + x^{13} \left (\frac{A c^{3}}{13} + \frac{3 B b c^{2}}{13}\right ) + x^{11} \left (\frac{3 A b c^{2}}{11} + \frac{3 B a c^{2}}{11} + \frac{3 B b^{2} c}{11}\right ) + x^{9} \left (\frac{A a c^{2}}{3} + \frac{A b^{2} c}{3} + \frac{2 B a b c}{3} + \frac{B b^{3}}{9}\right ) + x^{7} \left (\frac{6 A a b c}{7} + \frac{A b^{3}}{7} + \frac{3 B a^{2} c}{7} + \frac{3 B a b^{2}}{7}\right ) + x^{5} \left (\frac{3 A a^{2} c}{5} + \frac{3 A a b^{2}}{5} + \frac{3 B a^{2} b}{5}\right ) + x^{3} \left (A a^{2} b + \frac{B a^{3}}{3}\right ) \]

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)

[Out]

A*a**3*x + B*c**3*x**15/15 + x**13*(A*c**3/13 + 3*B*b*c**2/13) + x**11*(3*A*b*c*
*2/11 + 3*B*a*c**2/11 + 3*B*b**2*c/11) + x**9*(A*a*c**2/3 + A*b**2*c/3 + 2*B*a*b
*c/3 + B*b**3/9) + x**7*(6*A*a*b*c/7 + A*b**3/7 + 3*B*a**2*c/7 + 3*B*a*b**2/7) +
x**5*(3*A*a**2*c/5 + 3*A*a*b**2/5 + 3*B*a**2*b/5) + x**3*(A*a**2*b + B*a**3/3)

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GIAC/XCAS [A]  time = 0.262109, size = 255, normalized size = 1.58 \[ \frac{1}{15} \, B c^{3} x^{15} + \frac{3}{13} \, B b c^{2} x^{13} + \frac{1}{13} \, A c^{3} x^{13} + \frac{3}{11} \, B b^{2} c x^{11} + \frac{3}{11} \, B a c^{2} x^{11} + \frac{3}{11} \, A b c^{2} x^{11} + \frac{1}{9} \, B b^{3} x^{9} + \frac{2}{3} \, B a b c x^{9} + \frac{1}{3} \, A b^{2} c x^{9} + \frac{1}{3} \, A a c^{2} x^{9} + \frac{3}{7} \, B a b^{2} x^{7} + \frac{1}{7} \, A b^{3} x^{7} + \frac{3}{7} \, B a^{2} c x^{7} + \frac{6}{7} \, A a b c x^{7} + \frac{3}{5} \, B a^{2} b x^{5} + \frac{3}{5} \, A a b^{2} x^{5} + \frac{3}{5} \, A a^{2} c x^{5} + \frac{1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + A a^{3} x \]

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, algorithm="giac")

[Out]

1/15*B*c^3*x^15 + 3/13*B*b*c^2*x^13 + 1/13*A*c^3*x^13 + 3/11*B*b^2*c*x^11 + 3/11
*B*a*c^2*x^11 + 3/11*A*b*c^2*x^11 + 1/9*B*b^3*x^9 + 2/3*B*a*b*c*x^9 + 1/3*A*b^2*
c*x^9 + 1/3*A*a*c^2*x^9 + 3/7*B*a*b^2*x^7 + 1/7*A*b^3*x^7 + 3/7*B*a^2*c*x^7 + 6/
7*A*a*b*c*x^7 + 3/5*B*a^2*b*x^5 + 3/5*A*a*b^2*x^5 + 3/5*A*a^2*c*x^5 + 1/3*B*a^3*
x^3 + A*a^2*b*x^3 + A*a^3*x