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

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

________________________________________________________________________________________

Rubi [A]  time = 0.0420841, 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, Rules used = {373} \[ \frac{1}{3} c^2 x^3 (3 a d+b c)+\frac{1}{7} d^2 x^7 (a d+3 b c)+\frac{3}{5} c d x^5 (a d+b c)+a c^3 x+\frac{1}{9} b d^3 x^9 \]

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

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 )^3 \, dx &=\int \left (a c^3+c^2 (b c+3 a d) x^2+3 c d (b c+a d) x^4+d^2 (3 b c+a d) x^6+b d^3 x^8\right ) \, dx\\ &=a c^3 x+\frac{1}{3} c^2 (b c+3 a d) x^3+\frac{3}{5} c d (b c+a d) x^5+\frac{1}{7} d^2 (3 b c+a d) x^7+\frac{1}{9} b d^3 x^9\\ \end{align*}

Mathematica [A]  time = 0.0139783, 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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 73, normalized size = 1. \begin{align*}{\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 \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 0.961674, size = 95, normalized size = 1.36 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.55559, size = 169, normalized size = 2.41 \begin{align*} \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 \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 0.068978, size = 76, normalized size = 1.09 \begin{align*} 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 ) \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.14304, size = 99, normalized size = 1.41 \begin{align*} \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 \end{align*}

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