Optimal. Leaf size=68 \[ -\frac {\left (b+c x^2\right )^5 (2 b B-A c)}{10 c^3}+\frac {b \left (b+c x^2\right )^4 (b B-A c)}{8 c^3}+\frac {B \left (b+c x^2\right )^6}{12 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 68, 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} \[ -\frac {\left (b+c x^2\right )^5 (2 b B-A c)}{10 c^3}+\frac {b \left (b+c x^2\right )^4 (b B-A c)}{8 c^3}+\frac {B \left (b+c x^2\right )^6}{12 c^3} \]
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 )^3}{x^3} \, dx &=\int x^3 \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (A+B x) (b+c x)^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b (b B-A c) (b+c x)^3}{c^2}+\frac {(-2 b B+A c) (b+c x)^4}{c^2}+\frac {B (b+c x)^5}{c^2}\right ) \, dx,x,x^2\right )\\ &=\frac {b (b B-A c) \left (b+c x^2\right )^4}{8 c^3}-\frac {(2 b B-A c) \left (b+c x^2\right )^5}{10 c^3}+\frac {B \left (b+c x^2\right )^6}{12 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 69, normalized size = 1.01 \[ \frac {1}{120} x^4 \left (30 A b^3+20 b^2 x^2 (3 A c+b B)+12 c^2 x^6 (A c+3 b B)+45 b c x^4 (A c+b B)+10 B c^3 x^8\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.93, size = 73, normalized size = 1.07 \[ \frac {1}{12} \, B c^{3} x^{12} + \frac {1}{10} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + \frac {3}{8} \, {\left (B b^{2} c + A b c^{2}\right )} x^{8} + \frac {1}{4} \, A b^{3} x^{4} + \frac {1}{6} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 77, normalized size = 1.13 \[ \frac {1}{12} \, B c^{3} x^{12} + \frac {3}{10} \, B b c^{2} x^{10} + \frac {1}{10} \, A c^{3} x^{10} + \frac {3}{8} \, B b^{2} c x^{8} + \frac {3}{8} \, A b c^{2} x^{8} + \frac {1}{6} \, B b^{3} x^{6} + \frac {1}{2} \, A b^{2} c x^{6} + \frac {1}{4} \, A b^{3} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 76, normalized size = 1.12 \[ \frac {B \,c^{3} x^{12}}{12}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{10}}{10}+\frac {A \,b^{3} x^{4}}{4}+\frac {\left (3 A b \,c^{2}+3 B c \,b^{2}\right ) x^{8}}{8}+\frac {\left (3 A c \,b^{2}+B \,b^{3}\right ) x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.32, size = 73, normalized size = 1.07 \[ \frac {1}{12} \, B c^{3} x^{12} + \frac {1}{10} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + \frac {3}{8} \, {\left (B b^{2} c + A b c^{2}\right )} x^{8} + \frac {1}{4} \, A b^{3} x^{4} + \frac {1}{6} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 69, normalized size = 1.01 \[ x^6\,\left (\frac {B\,b^3}{6}+\frac {A\,c\,b^2}{2}\right )+x^{10}\,\left (\frac {A\,c^3}{10}+\frac {3\,B\,b\,c^2}{10}\right )+\frac {A\,b^3\,x^4}{4}+\frac {B\,c^3\,x^{12}}{12}+\frac {3\,b\,c\,x^8\,\left (A\,c+B\,b\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 82, normalized size = 1.21 \[ \frac {A b^{3} x^{4}}{4} + \frac {B c^{3} x^{12}}{12} + x^{10} \left (\frac {A c^{3}}{10} + \frac {3 B b c^{2}}{10}\right ) + x^{8} \left (\frac {3 A b c^{2}}{8} + \frac {3 B b^{2} c}{8}\right ) + x^{6} \left (\frac {A b^{2} c}{2} + \frac {B b^{3}}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________