Optimal. Leaf size=42 \[ \frac {\left (a+b x^2\right )^3 (A b-a B)}{6 b^2}+\frac {B \left (a+b x^2\right )^4}{8 b^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {444, 43} \begin {gather*} \frac {\left (a+b x^2\right )^3 (A b-a B)}{6 b^2}+\frac {B \left (a+b x^2\right )^4}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 444
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b x)^2 (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(A b-a B) (a+b x)^2}{b}+\frac {B (a+b x)^3}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {(A b-a B) \left (a+b x^2\right )^3}{6 b^2}+\frac {B \left (a+b x^2\right )^4}{8 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 51, normalized size = 1.21 \begin {gather*} \frac {1}{24} x^2 \left (12 a^2 A+4 b x^4 (2 a B+A b)+6 a x^2 (a B+2 A b)+3 b^2 B x^6\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 53, normalized size = 1.26 \begin {gather*} \frac {1}{8} x^{8} b^{2} B + \frac {1}{3} x^{6} b a B + \frac {1}{6} x^{6} b^{2} A + \frac {1}{4} x^{4} a^{2} B + \frac {1}{2} x^{4} b a A + \frac {1}{2} x^{2} a^{2} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.28, size = 53, normalized size = 1.26 \begin {gather*} \frac {1}{8} \, B b^{2} x^{8} + \frac {1}{3} \, B a b x^{6} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{2} \, A a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 52, normalized size = 1.24 \begin {gather*} \frac {B \,b^{2} x^{8}}{8}+\frac {\left (b^{2} A +2 a b B \right ) x^{6}}{6}+\frac {A \,a^{2} x^{2}}{2}+\frac {\left (2 a b A +a^{2} B \right ) x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 51, normalized size = 1.21 \begin {gather*} \frac {1}{8} \, B b^{2} x^{8} + \frac {1}{6} \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + \frac {1}{2} \, A a^{2} x^{2} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 51, normalized size = 1.21 \begin {gather*} x^4\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^6\,\left (\frac {A\,b^2}{6}+\frac {B\,a\,b}{3}\right )+\frac {A\,a^2\,x^2}{2}+\frac {B\,b^2\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 53, normalized size = 1.26 \begin {gather*} \frac {A a^{2} x^{2}}{2} + \frac {B b^{2} x^{8}}{8} + x^{6} \left (\frac {A b^{2}}{6} + \frac {B a b}{3}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {B a^{2}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________