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

Optimal. Leaf size=50 \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]

[Out]

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

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Rubi [A]  time = 0.0715088, antiderivative size = 50, 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 \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.017418, size = 50, normalized size = 1. \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 49, normalized size = 1. \[{\frac{b{d}^{2}{x}^{7}}{7}}+{\frac{ \left ( a{d}^{2}+2\,bcd \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,acd+{c}^{2}b \right ){x}^{3}}{3}}+a{c}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(d*x^2+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.39718, size = 65, normalized size = 1.3 \[ \frac{1}{7} \, b d^{2} x^{7} + \frac{1}{5} \,{\left (2 \, b c d + a d^{2}\right )} x^{5} + a c^{2} x + \frac{1}{3} \,{\left (b c^{2} + 2 \, a c d\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.177117, size = 1, normalized size = 0.02 \[ \frac{1}{7} x^{7} d^{2} b + \frac{2}{5} x^{5} d c b + \frac{1}{5} x^{5} d^{2} a + \frac{1}{3} x^{3} c^{2} b + \frac{2}{3} x^{3} d c a + x c^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.110744, size = 53, normalized size = 1.06 \[ a c^{2} x + \frac{b d^{2} x^{7}}{7} + x^{5} \left (\frac{a d^{2}}{5} + \frac{2 b c d}{5}\right ) + x^{3} \left (\frac{2 a c d}{3} + \frac{b c^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.227377, size = 68, normalized size = 1.36 \[ \frac{1}{7} \, b d^{2} x^{7} + \frac{2}{5} \, b c d x^{5} + \frac{1}{5} \, a d^{2} x^{5} + \frac{1}{3} \, b c^{2} x^{3} + \frac{2}{3} \, a c d x^{3} + a c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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