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

Optimal. Leaf size=122 \[ a^3 c^2 x+\frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]

[Out]

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

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Rubi [A]  time = 0.167675, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ a^3 c^2 x+\frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0391154, size = 122, normalized size = 1. \[ a^3 c^2 x+\frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 125, normalized size = 1. \[{\frac{{b}^{3}{d}^{2}{x}^{11}}{11}}+{\frac{ \left ( 3\,a{b}^{2}{d}^{2}+2\,{b}^{3}cd \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}b{d}^{2}+6\,a{b}^{2}cd+{b}^{3}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}{d}^{2}+6\,{a}^{2}bcd+3\,a{b}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{3}cd+3\,{a}^{2}b{c}^{2} \right ){x}^{3}}{3}}+{a}^{3}{c}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3*(d*x^2+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.36673, size = 167, normalized size = 1.37 \[ \frac{1}{11} \, b^{3} d^{2} x^{11} + \frac{1}{9} \,{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{9} + \frac{1}{7} \,{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{7} + a^{3} c^{2} x + \frac{1}{5} \,{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{5} + \frac{1}{3} \,{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.179498, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} d^{2} b^{3} + \frac{2}{9} x^{9} d c b^{3} + \frac{1}{3} x^{9} d^{2} b^{2} a + \frac{1}{7} x^{7} c^{2} b^{3} + \frac{6}{7} x^{7} d c b^{2} a + \frac{3}{7} x^{7} d^{2} b a^{2} + \frac{3}{5} x^{5} c^{2} b^{2} a + \frac{6}{5} x^{5} d c b a^{2} + \frac{1}{5} x^{5} d^{2} a^{3} + x^{3} c^{2} b a^{2} + \frac{2}{3} x^{3} d c a^{3} + x c^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.163172, size = 136, normalized size = 1.11 \[ a^{3} c^{2} x + \frac{b^{3} d^{2} x^{11}}{11} + x^{9} \left (\frac{a b^{2} d^{2}}{3} + \frac{2 b^{3} c d}{9}\right ) + x^{7} \left (\frac{3 a^{2} b d^{2}}{7} + \frac{6 a b^{2} c d}{7} + \frac{b^{3} c^{2}}{7}\right ) + x^{5} \left (\frac{a^{3} d^{2}}{5} + \frac{6 a^{2} b c d}{5} + \frac{3 a b^{2} c^{2}}{5}\right ) + x^{3} \left (\frac{2 a^{3} c d}{3} + a^{2} b c^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225019, size = 177, normalized size = 1.45 \[ \frac{1}{11} \, b^{3} d^{2} x^{11} + \frac{2}{9} \, b^{3} c d x^{9} + \frac{1}{3} \, a b^{2} d^{2} x^{9} + \frac{1}{7} \, b^{3} c^{2} x^{7} + \frac{6}{7} \, a b^{2} c d x^{7} + \frac{3}{7} \, a^{2} b d^{2} x^{7} + \frac{3}{5} \, a b^{2} c^{2} x^{5} + \frac{6}{5} \, a^{2} b c d x^{5} + \frac{1}{5} \, a^{3} d^{2} x^{5} + a^{2} b c^{2} x^{3} + \frac{2}{3} \, a^{3} c d x^{3} + a^{3} c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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