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

Optimal. Leaf size=70 \[ \frac{1}{3} c^2 x^3 (3 a d+b c)+\frac{1}{7} d^2 x^7 (a d+3 b c)+\frac{3}{5} c d x^5 (a d+b c)+a c^3 x+\frac{1}{9} b d^3 x^9 \]

[Out]

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

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Rubi [A]  time = 0.101858, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{1}{3} c^2 x^3 (3 a d+b c)+\frac{1}{7} d^2 x^7 (a d+3 b c)+\frac{3}{5} c d x^5 (a d+b c)+a c^3 x+\frac{1}{9} b d^3 x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.023338, size = 70, normalized size = 1. \[ \frac{1}{3} c^2 x^3 (3 a d+b c)+\frac{1}{7} d^2 x^7 (a d+3 b c)+\frac{3}{5} c d x^5 (a d+b c)+a c^3 x+\frac{1}{9} b d^3 x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 73, normalized size = 1. \[{\frac{b{d}^{3}{x}^{9}}{9}}+{\frac{ \left ( a{d}^{3}+3\,bc{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){x}^{3}}{3}}+a{c}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(d*x^2+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.33996, size = 95, normalized size = 1.36 \[ \frac{1}{9} \, b d^{3} x^{9} + \frac{1}{7} \,{\left (3 \, b c d^{2} + a d^{3}\right )} x^{7} + \frac{3}{5} \,{\left (b c^{2} d + a c d^{2}\right )} x^{5} + a c^{3} x + \frac{1}{3} \,{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.178077, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} d^{3} b + \frac{3}{7} x^{7} d^{2} c b + \frac{1}{7} x^{7} d^{3} a + \frac{3}{5} x^{5} d c^{2} b + \frac{3}{5} x^{5} d^{2} c a + \frac{1}{3} x^{3} c^{3} b + x^{3} d c^{2} a + x c^{3} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.129599, size = 76, normalized size = 1.09 \[ a c^{3} x + \frac{b d^{3} x^{9}}{9} + x^{7} \left (\frac{a d^{3}}{7} + \frac{3 b c d^{2}}{7}\right ) + x^{5} \left (\frac{3 a c d^{2}}{5} + \frac{3 b c^{2} d}{5}\right ) + x^{3} \left (a c^{2} d + \frac{b c^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235981, size = 99, normalized size = 1.41 \[ \frac{1}{9} \, b d^{3} x^{9} + \frac{3}{7} \, b c d^{2} x^{7} + \frac{1}{7} \, a d^{3} x^{7} + \frac{3}{5} \, b c^{2} d x^{5} + \frac{3}{5} \, a c d^{2} x^{5} + \frac{1}{3} \, b c^{3} x^{3} + a c^{2} d x^{3} + a c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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