3.161 \(\int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx\)

Optimal. Leaf size=127 \[ \frac{1}{9} d x^9 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{7} c x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^3 x^3+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{11} b d^2 x^{11} (2 a d+3 b c)+\frac{1}{13} b^2 d^3 x^{13} \]

[Out]

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

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Rubi [A]  time = 0.232067, antiderivative size = 127, 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 \[ \frac{1}{9} d x^9 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{7} c x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^3 x^3+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{11} b d^2 x^{11} (2 a d+3 b c)+\frac{1}{13} b^2 d^3 x^{13} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.8853, size = 124, normalized size = 0.98 \[ \frac{a^{2} c^{3} x^{3}}{3} + \frac{a c^{2} x^{5} \left (3 a d + 2 b c\right )}{5} + \frac{b^{2} d^{3} x^{13}}{13} + \frac{b d^{2} x^{11} \left (2 a d + 3 b c\right )}{11} + \frac{c x^{7} \left (3 a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{7} + \frac{d x^{9} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x**2+a)**2*(d*x**2+c)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0401486, size = 127, normalized size = 1. \[ \frac{1}{9} d x^9 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{7} c x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 c^3 x^3+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{11} b d^2 x^{11} (2 a d+3 b c)+\frac{1}{13} b^2 d^3 x^{13} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 128, normalized size = 1. \[{\frac{{b}^{2}{d}^{3}{x}^{13}}{13}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{5}}{5}}+{\frac{{a}^{2}{c}^{3}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^2*d^3*x^13+1/11*(2*a*b*d^3+3*b^2*c*d^2)*x^11+1/9*(a^2*d^3+6*a*b*c*d^2+3*b
^2*c^2*d)*x^9+1/7*(3*a^2*c*d^2+6*a*b*c^2*d+b^2*c^3)*x^7+1/5*(3*a^2*c^2*d+2*a*b*c
^3)*x^5+1/3*a^2*c^3*x^3

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Maxima [A]  time = 1.36089, size = 171, normalized size = 1.35 \[ \frac{1}{13} \, b^{2} d^{3} x^{13} + \frac{1}{11} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{11} + \frac{1}{9} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{9} + \frac{1}{3} \, a^{2} c^{3} x^{3} + \frac{1}{7} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + \frac{1}{5} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^2*d^3*x^13 + 1/11*(3*b^2*c*d^2 + 2*a*b*d^3)*x^11 + 1/9*(3*b^2*c^2*d + 6*a
*b*c*d^2 + a^2*d^3)*x^9 + 1/3*a^2*c^3*x^3 + 1/7*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c
*d^2)*x^7 + 1/5*(2*a*b*c^3 + 3*a^2*c^2*d)*x^5

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Fricas [A]  time = 0.207578, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} d^{3} b^{2} + \frac{3}{11} x^{11} d^{2} c b^{2} + \frac{2}{11} x^{11} d^{3} b a + \frac{1}{3} x^{9} d c^{2} b^{2} + \frac{2}{3} x^{9} d^{2} c b a + \frac{1}{9} x^{9} d^{3} a^{2} + \frac{1}{7} x^{7} c^{3} b^{2} + \frac{6}{7} x^{7} d c^{2} b a + \frac{3}{7} x^{7} d^{2} c a^{2} + \frac{2}{5} x^{5} c^{3} b a + \frac{3}{5} x^{5} d c^{2} a^{2} + \frac{1}{3} x^{3} c^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.169082, size = 143, normalized size = 1.13 \[ \frac{a^{2} c^{3} x^{3}}{3} + \frac{b^{2} d^{3} x^{13}}{13} + x^{11} \left (\frac{2 a b d^{3}}{11} + \frac{3 b^{2} c d^{2}}{11}\right ) + x^{9} \left (\frac{a^{2} d^{3}}{9} + \frac{2 a b c d^{2}}{3} + \frac{b^{2} c^{2} d}{3}\right ) + x^{7} \left (\frac{3 a^{2} c d^{2}}{7} + \frac{6 a b c^{2} d}{7} + \frac{b^{2} c^{3}}{7}\right ) + x^{5} \left (\frac{3 a^{2} c^{2} d}{5} + \frac{2 a b c^{3}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x**2+a)**2*(d*x**2+c)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.221822, size = 182, normalized size = 1.43 \[ \frac{1}{13} \, b^{2} d^{3} x^{13} + \frac{3}{11} \, b^{2} c d^{2} x^{11} + \frac{2}{11} \, a b d^{3} x^{11} + \frac{1}{3} \, b^{2} c^{2} d x^{9} + \frac{2}{3} \, a b c d^{2} x^{9} + \frac{1}{9} \, a^{2} d^{3} x^{9} + \frac{1}{7} \, b^{2} c^{3} x^{7} + \frac{6}{7} \, a b c^{2} d x^{7} + \frac{3}{7} \, a^{2} c d^{2} x^{7} + \frac{2}{5} \, a b c^{3} x^{5} + \frac{3}{5} \, a^{2} c^{2} d x^{5} + \frac{1}{3} \, a^{2} c^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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