Optimal. Leaf size=55 \[ \frac {1}{4} A b^2 x^4+\frac {1}{8} c x^8 (A c+2 b B)+\frac {1}{6} b x^6 (2 A c+b B)+\frac {1}{10} B c^2 x^{10} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1584, 446, 76} \[ \frac {1}{4} A b^2 x^4+\frac {1}{8} c x^8 (A c+2 b B)+\frac {1}{6} b x^6 (2 A c+b B)+\frac {1}{10} B c^2 x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x} \, dx &=\int x^3 \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (A+B x) (b+c x)^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (A b^2 x+b (b B+2 A c) x^2+c (2 b B+A c) x^3+B c^2 x^4\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4} A b^2 x^4+\frac {1}{6} b (b B+2 A c) x^6+\frac {1}{8} c (2 b B+A c) x^8+\frac {1}{10} B c^2 x^{10}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 55, normalized size = 1.00 \[ \frac {1}{4} A b^2 x^4+\frac {1}{8} c x^8 (A c+2 b B)+\frac {1}{6} b x^6 (2 A c+b B)+\frac {1}{10} B c^2 x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 51, normalized size = 0.93 \[ \frac {1}{10} \, B c^{2} x^{10} + \frac {1}{8} \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + \frac {1}{4} \, A b^{2} x^{4} + \frac {1}{6} \, {\left (B b^{2} + 2 \, A b c\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 53, normalized size = 0.96 \[ \frac {1}{10} \, B c^{2} x^{10} + \frac {1}{4} \, B b c x^{8} + \frac {1}{8} \, A c^{2} x^{8} + \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, A b c x^{6} + \frac {1}{4} \, A b^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 52, normalized size = 0.95 \[ \frac {B \,c^{2} x^{10}}{10}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{8}}{8}+\frac {A \,b^{2} x^{4}}{4}+\frac {\left (2 A b c +B \,b^{2}\right ) x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 51, normalized size = 0.93 \[ \frac {1}{10} \, B c^{2} x^{10} + \frac {1}{8} \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + \frac {1}{4} \, A b^{2} x^{4} + \frac {1}{6} \, {\left (B b^{2} + 2 \, A b c\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 51, normalized size = 0.93 \[ x^6\,\left (\frac {B\,b^2}{6}+\frac {A\,c\,b}{3}\right )+x^8\,\left (\frac {A\,c^2}{8}+\frac {B\,b\,c}{4}\right )+\frac {A\,b^2\,x^4}{4}+\frac {B\,c^2\,x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 53, normalized size = 0.96 \[ \frac {A b^{2} x^{4}}{4} + \frac {B c^{2} x^{10}}{10} + x^{8} \left (\frac {A c^{2}}{8} + \frac {B b c}{4}\right ) + x^{6} \left (\frac {A b c}{3} + \frac {B b^{2}}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________