3.4 \(\int (a+b x^2) (c+d x^2) \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5 \]

[Out]

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

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Rubi [A]  time = 0.0128864, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {373} \[ \frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(c + d*x^2),x]

[Out]

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

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \left (a+b x^2\right ) \left (c+d x^2\right ) \, dx &=\int \left (a c+(b c+a d) x^2+b d x^4\right ) \, dx\\ &=a c x+\frac{1}{3} (b c+a d) x^3+\frac{1}{5} b d x^5\\ \end{align*}

Mathematica [A]  time = 0.005004, size = 28, normalized size = 1. \[ \frac{1}{3} x^3 (a d+b c)+a c x+\frac{1}{5} b d x^5 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)*(c + d*x^2),x]

[Out]

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

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Maple [A]  time = 0.001, size = 25, normalized size = 0.9 \begin{align*} acx+{\frac{ \left ( ad+bc \right ){x}^{3}}{3}}+{\frac{bd{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(d*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.46598, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{5} \, b d x^{5} + \frac{1}{3} \,{\left (b c + a d\right )} x^{3} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.51667, size = 66, normalized size = 2.36 \begin{align*} \frac{1}{5} x^{5} d b + \frac{1}{3} x^{3} c b + \frac{1}{3} x^{3} d a + x c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.054436, size = 26, normalized size = 0.93 \begin{align*} a c x + \frac{b d x^{5}}{5} + x^{3} \left (\frac{a d}{3} + \frac{b c}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08404, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{5} \, b d x^{5} + \frac{1}{3} \, b c x^{3} + \frac{1}{3} \, a d x^{3} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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