3.141 \(\int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{5} a^2 c x^5+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{7} a x^7 (a d+2 b c)+\frac{1}{11} b^2 d x^{11} \]

[Out]

(a^2*c*x^5)/5 + (a*(2*b*c + a*d)*x^7)/7 + (b*(b*c + 2*a*d)*x^9)/9 + (b^2*d*x^11)
/11

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Rubi [A]  time = 0.11334, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{1}{5} a^2 c x^5+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{7} a x^7 (a d+2 b c)+\frac{1}{11} b^2 d x^{11} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*c*x^5)/5 + (a*(2*b*c + a*d)*x^7)/7 + (b*(b*c + 2*a*d)*x^9)/9 + (b^2*d*x^11)
/11

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Rubi in Sympy [A]  time = 13.9745, size = 49, normalized size = 0.89 \[ \frac{a^{2} c x^{5}}{5} + \frac{a x^{7} \left (a d + 2 b c\right )}{7} + \frac{b^{2} d x^{11}}{11} + \frac{b x^{9} \left (2 a d + b c\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)**2*(d*x**2+c),x)

[Out]

a**2*c*x**5/5 + a*x**7*(a*d + 2*b*c)/7 + b**2*d*x**11/11 + b*x**9*(2*a*d + b*c)/
9

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Mathematica [A]  time = 0.0149234, size = 55, normalized size = 1. \[ \frac{1}{5} a^2 c x^5+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{7} a x^7 (a d+2 b c)+\frac{1}{11} b^2 d x^{11} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*c*x^5)/5 + (a*(2*b*c + a*d)*x^7)/7 + (b*(b*c + 2*a*d)*x^9)/9 + (b^2*d*x^11)
/11

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \[{\frac{{b}^{2}d{x}^{11}}{11}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{7}}{7}}+{\frac{{a}^{2}c{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^2*d*x^11+1/9*(2*a*b*d+b^2*c)*x^9+1/7*(a^2*d+2*a*b*c)*x^7+1/5*a^2*c*x^5

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Maxima [A]  time = 1.34111, size = 69, normalized size = 1.25 \[ \frac{1}{11} \, b^{2} d x^{11} + \frac{1}{9} \,{\left (b^{2} c + 2 \, a b d\right )} x^{9} + \frac{1}{5} \, a^{2} c x^{5} + \frac{1}{7} \,{\left (2 \, a b c + a^{2} d\right )} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^2*d*x^11 + 1/9*(b^2*c + 2*a*b*d)*x^9 + 1/5*a^2*c*x^5 + 1/7*(2*a*b*c + a^2
*d)*x^7

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Fricas [A]  time = 0.19699, size = 1, normalized size = 0.02 \[ \frac{1}{11} x^{11} d b^{2} + \frac{1}{9} x^{9} c b^{2} + \frac{2}{9} x^{9} d b a + \frac{2}{7} x^{7} c b a + \frac{1}{7} x^{7} d a^{2} + \frac{1}{5} x^{5} c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*x^11*d*b^2 + 1/9*x^9*c*b^2 + 2/9*x^9*d*b*a + 2/7*x^7*c*b*a + 1/7*x^7*d*a^2
+ 1/5*x^5*c*a^2

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Sympy [A]  time = 0.114571, size = 56, normalized size = 1.02 \[ \frac{a^{2} c x^{5}}{5} + \frac{b^{2} d x^{11}}{11} + x^{9} \left (\frac{2 a b d}{9} + \frac{b^{2} c}{9}\right ) + x^{7} \left (\frac{a^{2} d}{7} + \frac{2 a b c}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**2+a)**2*(d*x**2+c),x)

[Out]

a**2*c*x**5/5 + b**2*d*x**11/11 + x**9*(2*a*b*d/9 + b**2*c/9) + x**7*(a**2*d/7 +
2*a*b*c/7)

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GIAC/XCAS [A]  time = 0.221495, size = 72, normalized size = 1.31 \[ \frac{1}{11} \, b^{2} d x^{11} + \frac{1}{9} \, b^{2} c x^{9} + \frac{2}{9} \, a b d x^{9} + \frac{2}{7} \, a b c x^{7} + \frac{1}{7} \, a^{2} d x^{7} + \frac{1}{5} \, a^{2} c x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^2*d*x^11 + 1/9*b^2*c*x^9 + 2/9*a*b*d*x^9 + 2/7*a*b*c*x^7 + 1/7*a^2*d*x^7
+ 1/5*a^2*c*x^5