3.11 \(\int x^2 (a+b x^2)^2 (A+B x^2) \, dx\)

Optimal. Leaf size=55 \[ \frac {1}{3} a^2 A x^3+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{5} a x^5 (a B+2 A b)+\frac {1}{9} b^2 B x^9 \]

[Out]

1/3*a^2*A*x^3+1/5*a*(2*A*b+B*a)*x^5+1/7*b*(A*b+2*B*a)*x^7+1/9*b^2*B*x^9

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \[ \frac {1}{3} a^2 A x^3+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{5} a x^5 (a B+2 A b)+\frac {1}{9} b^2 B x^9 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)^2*(A + B*x^2),x]

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^9)/9

Rule 448

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 55, normalized size = 1.00 \[ \frac {1}{3} a^2 A x^3+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{5} a x^5 (a B+2 A b)+\frac {1}{9} b^2 B x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^2)^2*(A + B*x^2),x]

[Out]

(a^2*A*x^3)/3 + (a*(2*A*b + a*B)*x^5)/5 + (b*(A*b + 2*a*B)*x^7)/7 + (b^2*B*x^9)/9

________________________________________________________________________________________

fricas [A]  time = 0.38, size = 53, normalized size = 0.96 \[ \frac {1}{9} x^{9} b^{2} B + \frac {2}{7} x^{7} b a B + \frac {1}{7} x^{7} b^{2} A + \frac {1}{5} x^{5} a^{2} B + \frac {2}{5} x^{5} b a A + \frac {1}{3} x^{3} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^2+a)^2*(B*x^2+A),x, algorithm="fricas")

[Out]

1/9*x^9*b^2*B + 2/7*x^7*b*a*B + 1/7*x^7*b^2*A + 1/5*x^5*a^2*B + 2/5*x^5*b*a*A + 1/3*x^3*a^2*A

________________________________________________________________________________________

giac [A]  time = 0.31, size = 53, normalized size = 0.96 \[ \frac {1}{9} \, B b^{2} x^{9} + \frac {2}{7} \, B a b x^{7} + \frac {1}{7} \, A b^{2} x^{7} + \frac {1}{5} \, B a^{2} x^{5} + \frac {2}{5} \, A a b x^{5} + \frac {1}{3} \, A a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^2+a)^2*(B*x^2+A),x, algorithm="giac")

[Out]

1/9*B*b^2*x^9 + 2/7*B*a*b*x^7 + 1/7*A*b^2*x^7 + 1/5*B*a^2*x^5 + 2/5*A*a*b*x^5 + 1/3*A*a^2*x^3

________________________________________________________________________________________

maple [A]  time = 0.00, size = 52, normalized size = 0.95 \[ \frac {B \,b^{2} x^{9}}{9}+\frac {\left (b^{2} A +2 a b B \right ) x^{7}}{7}+\frac {A \,a^{2} x^{3}}{3}+\frac {\left (2 a b A +a^{2} B \right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^2+a)^2*(B*x^2+A),x)

[Out]

1/9*b^2*B*x^9+1/7*(A*b^2+2*B*a*b)*x^7+1/5*(2*A*a*b+B*a^2)*x^5+1/3*a^2*A*x^3

________________________________________________________________________________________

maxima [A]  time = 1.38, size = 51, normalized size = 0.93 \[ \frac {1}{9} \, B b^{2} x^{9} + \frac {1}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (B a^{2} + 2 \, A a b\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^2+a)^2*(B*x^2+A),x, algorithm="maxima")

[Out]

1/9*B*b^2*x^9 + 1/7*(2*B*a*b + A*b^2)*x^7 + 1/3*A*a^2*x^3 + 1/5*(B*a^2 + 2*A*a*b)*x^5

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 51, normalized size = 0.93 \[ x^5\,\left (\frac {B\,a^2}{5}+\frac {2\,A\,b\,a}{5}\right )+x^7\,\left (\frac {A\,b^2}{7}+\frac {2\,B\,a\,b}{7}\right )+\frac {A\,a^2\,x^3}{3}+\frac {B\,b^2\,x^9}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(A + B*x^2)*(a + b*x^2)^2,x)

[Out]

x^5*((B*a^2)/5 + (2*A*a*b)/5) + x^7*((A*b^2)/7 + (2*B*a*b)/7) + (A*a^2*x^3)/3 + (B*b^2*x^9)/9

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 56, normalized size = 1.02 \[ \frac {A a^{2} x^{3}}{3} + \frac {B b^{2} x^{9}}{9} + x^{7} \left (\frac {A b^{2}}{7} + \frac {2 B a b}{7}\right ) + x^{5} \left (\frac {2 A a b}{5} + \frac {B a^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**2+a)**2*(B*x**2+A),x)

[Out]

A*a**2*x**3/3 + B*b**2*x**9/9 + x**7*(A*b**2/7 + 2*B*a*b/7) + x**5*(2*A*a*b/5 + B*a**2/5)

________________________________________________________________________________________