\(\int x^2 (a+b x^2) (A+B x^2) \, dx\) [1]

Optimal result

Integrand size = 18, antiderivative size = 33 \[ \int x^2 \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{3} a A x^3+\frac {1}{5} (A b+a B) x^5+\frac {1}{7} b B x^7 \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {459} \[ \int x^2 \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{5} x^5 (a B+A b)+\frac {1}{3} a A x^3+\frac {1}{7} b B x^7 \]

[In]

Int[x^2*(a + b*x^2)*(A + B*x^2),x]

[Out]

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

Rule 459

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{3} a A x^3+\frac {1}{5} (A b+a B) x^5+\frac {1}{7} b B x^7 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {a A \,x^{3}}{3}+\frac {\left (A b +B a \right ) x^{5}}{5}+\frac {b B \,x^{7}}{7}\)	\(28\)
norman	\(\frac {b B \,x^{7}}{7}+\left (\frac {A b}{5}+\frac {B a}{5}\right ) x^{5}+\frac {a A \,x^{3}}{3}\)	\(29\)
gosper	\(\frac {1}{7} b B \,x^{7}+\frac {1}{5} x^{5} A b +\frac {1}{5} x^{5} B a +\frac {1}{3} a A \,x^{3}\)	\(30\)
risch	\(\frac {1}{7} b B \,x^{7}+\frac {1}{5} x^{5} A b +\frac {1}{5} x^{5} B a +\frac {1}{3} a A \,x^{3}\)	\(30\)
parallelrisch	\(\frac {1}{7} b B \,x^{7}+\frac {1}{5} x^{5} A b +\frac {1}{5} x^{5} B a +\frac {1}{3} a A \,x^{3}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int x^2 \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{7} \, B b x^{7} + \frac {1}{5} \, {\left (B a + A b\right )} x^{5} + \frac {1}{3} \, A a x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {A a x^{3}}{3} + \frac {B b x^{7}}{7} + x^{5} \left (\frac {A b}{5} + \frac {B a}{5}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int x^2 \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{7} \, B b x^{7} + \frac {1}{5} \, {\left (B a + A b\right )} x^{5} + \frac {1}{3} \, A a x^{3} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {1}{7} \, B b x^{7} + \frac {1}{5} \, B a x^{5} + \frac {1}{5} \, A b x^{5} + \frac {1}{3} \, A a x^{3} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.81 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {B\,b\,x^7}{7}+\left (\frac {A\,b}{5}+\frac {B\,a}{5}\right )\,x^5+\frac {A\,a\,x^3}{3} \]

[In]

int(x^2*(A + B*x^2)*(a + b*x^2),x)

[Out]

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