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

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

________________________________________________________________________________________

Rubi [A]  time = 0.0274234, 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, Rules used = {373} \[ \frac{1}{5} d x^5 (a d+2 b c)+\frac{1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac{1}{7} b d^2 x^7 \]

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

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 )^2 \, dx &=\int \left (a c^2+c (b c+2 a d) x^2+d (2 b c+a d) x^4+b d^2 x^6\right ) \, dx\\ &=a c^2 x+\frac{1}{3} c (b c+2 a d) x^3+\frac{1}{5} d (2 b c+a d) x^5+\frac{1}{7} b d^2 x^7\\ \end{align*}

Mathematica [A]  time = 0.0108334, 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

________________________________________________________________________________________

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

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

________________________________________________________________________________________

Maxima [A]  time = 1.01954, size = 65, normalized size = 1.3 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.5615, size = 120, normalized size = 2.4 \begin{align*} \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 \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 0.064517, size = 53, normalized size = 1.06 \begin{align*} 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 ) \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.07553, size = 68, normalized size = 1.36 \begin{align*} \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 \end{align*}

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