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

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

[Out]

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

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Rubi [A]  time = 0.104099, 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 \[ a^3 c x+\frac{1}{3} a^2 x^3 (a d+3 b c)+\frac{1}{7} b^2 x^7 (3 a d+b c)+\frac{3}{5} a b x^5 (a d+b c)+\frac{1}{9} b^3 d x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0230202, size = 70, normalized size = 1. \[ a^3 c x+\frac{1}{3} a^2 x^3 (a d+3 b c)+\frac{1}{7} b^2 x^7 (3 a d+b c)+\frac{3}{5} a b x^5 (a d+b c)+\frac{1}{9} b^3 d x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 73, normalized size = 1. \[{\frac{{b}^{3}d{x}^{9}}{9}}+{\frac{ \left ( 3\,a{b}^{2}d+{b}^{3}c \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}bd+3\,a{b}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{3}d+3\,{a}^{2}bc \right ){x}^{3}}{3}}+{a}^{3}cx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3*(d*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.35944, size = 95, normalized size = 1.36 \[ \frac{1}{9} \, b^{3} d x^{9} + \frac{1}{7} \,{\left (b^{3} c + 3 \, a b^{2} d\right )} x^{7} + \frac{3}{5} \,{\left (a b^{2} c + a^{2} b d\right )} x^{5} + a^{3} c x + \frac{1}{3} \,{\left (3 \, a^{2} b c + a^{3} d\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.177811, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} d b^{3} + \frac{1}{7} x^{7} c b^{3} + \frac{3}{7} x^{7} d b^{2} a + \frac{3}{5} x^{5} c b^{2} a + \frac{3}{5} x^{5} d b a^{2} + x^{3} c b a^{2} + \frac{1}{3} x^{3} d a^{3} + x c a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.134674, size = 76, normalized size = 1.09 \[ a^{3} c x + \frac{b^{3} d x^{9}}{9} + x^{7} \left (\frac{3 a b^{2} d}{7} + \frac{b^{3} c}{7}\right ) + x^{5} \left (\frac{3 a^{2} b d}{5} + \frac{3 a b^{2} c}{5}\right ) + x^{3} \left (\frac{a^{3} d}{3} + a^{2} b c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.2356, size = 99, normalized size = 1.41 \[ \frac{1}{9} \, b^{3} d x^{9} + \frac{1}{7} \, b^{3} c x^{7} + \frac{3}{7} \, a b^{2} d x^{7} + \frac{3}{5} \, a b^{2} c x^{5} + \frac{3}{5} \, a^{2} b d x^{5} + a^{2} b c x^{3} + \frac{1}{3} \, a^{3} d x^{3} + a^{3} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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