Optimal. Leaf size=75 \[ \frac {\left (d+e x^2\right )^3 \left (a e^2-b d e+c d^2\right )}{6 e^3}-\frac {\left (d+e x^2\right )^4 (2 c d-b e)}{8 e^3}+\frac {c \left (d+e x^2\right )^5}{10 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1247, 698} \[ \frac {\left (d+e x^2\right )^3 \left (a e^2-b d e+c d^2\right )}{6 e^3}-\frac {\left (d+e x^2\right )^4 (2 c d-b e)}{8 e^3}+\frac {c \left (d+e x^2\right )^5}{10 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rule 1247
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (d+e x)^2 \left (a+b x+c x^2\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^2}{e^2}+\frac {(-2 c d+b e) (d+e x)^3}{e^2}+\frac {c (d+e x)^4}{e^2}\right ) \, dx,x,x^2\right )\\ &=\frac {\left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^3}{6 e^3}-\frac {(2 c d-b e) \left (d+e x^2\right )^4}{8 e^3}+\frac {c \left (d+e x^2\right )^5}{10 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 72, normalized size = 0.96 \[ \frac {1}{120} x^2 \left (20 x^4 \left (e (a e+2 b d)+c d^2\right )+30 d x^2 (2 a e+b d)+60 a d^2+15 e x^6 (b e+2 c d)+12 c e^2 x^8\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 79, normalized size = 1.05 \[ \frac {1}{10} x^{10} e^{2} c + \frac {1}{4} x^{8} e d c + \frac {1}{8} x^{8} e^{2} b + \frac {1}{6} x^{6} d^{2} c + \frac {1}{3} x^{6} e d b + \frac {1}{6} x^{6} e^{2} a + \frac {1}{4} x^{4} d^{2} b + \frac {1}{2} x^{4} e d a + \frac {1}{2} x^{2} d^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.27, size = 79, normalized size = 1.05 \[ \frac {1}{10} \, c x^{10} e^{2} + \frac {1}{4} \, c d x^{8} e + \frac {1}{8} \, b x^{8} e^{2} + \frac {1}{6} \, c d^{2} x^{6} + \frac {1}{3} \, b d x^{6} e + \frac {1}{6} \, a x^{6} e^{2} + \frac {1}{4} \, b d^{2} x^{4} + \frac {1}{2} \, a d x^{4} e + \frac {1}{2} \, a d^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 73, normalized size = 0.97 \[ \frac {c \,e^{2} x^{10}}{10}+\frac {\left (e^{2} b +2 d e c \right ) x^{8}}{8}+\frac {\left (a \,e^{2}+2 d e b +c \,d^{2}\right ) x^{6}}{6}+\frac {a \,d^{2} x^{2}}{2}+\frac {\left (2 d e a +d^{2} b \right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.21, size = 72, normalized size = 0.96 \[ \frac {1}{10} \, c e^{2} x^{10} + \frac {1}{8} \, {\left (2 \, c d e + b e^{2}\right )} x^{8} + \frac {1}{6} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{6} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (b d^{2} + 2 \, a d e\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 73, normalized size = 0.97 \[ x^6\,\left (\frac {c\,d^2}{6}+\frac {b\,d\,e}{3}+\frac {a\,e^2}{6}\right )+x^4\,\left (\frac {b\,d^2}{4}+\frac {a\,e\,d}{2}\right )+x^8\,\left (\frac {b\,e^2}{8}+\frac {c\,d\,e}{4}\right )+\frac {a\,d^2\,x^2}{2}+\frac {c\,e^2\,x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 76, normalized size = 1.01 \[ \frac {a d^{2} x^{2}}{2} + \frac {c e^{2} x^{10}}{10} + x^{8} \left (\frac {b e^{2}}{8} + \frac {c d e}{4}\right ) + x^{6} \left (\frac {a e^{2}}{6} + \frac {b d e}{3} + \frac {c d^{2}}{6}\right ) + x^{4} \left (\frac {a d e}{2} + \frac {b d^{2}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________