Optimal. Leaf size=42 \[ \frac {(b c-a d) \left (a+b x^2\right )^3}{6 b^2}+\frac {d \left (a+b x^2\right )^4}{8 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 = {455, 45}
\begin {gather*} \frac {\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac {d \left (a+b x^2\right )^4}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 455
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int (a+b x)^2 (c+d x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {(b c-a d) (a+b x)^2}{b}+\frac {d (a+b x)^3}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d) \left (a+b x^2\right )^3}{6 b^2}+\frac {d \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 c+6 a (2 b c+a d) x^2+4 b (b c+2 a d) x^4+3 b^2 d x^6\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 52, normalized size = 1.24
method | result | size |
default | \(\frac {b^{2} d \,x^{8}}{8}+\frac {\left (2 a b d +b^{2} c \right ) x^{6}}{6}+\frac {\left (a^{2} d +2 a b c \right ) x^{4}}{4}+\frac {a^{2} c \,x^{2}}{2}\) | \(52\) |
norman | \(\frac {b^{2} d \,x^{8}}{8}+\left (\frac {1}{3} a b d +\frac {1}{6} b^{2} c \right ) x^{6}+\left (\frac {1}{4} a^{2} d +\frac {1}{2} a b c \right ) x^{4}+\frac {a^{2} c \,x^{2}}{2}\) | \(52\) |
gosper | \(\frac {1}{8} b^{2} d \,x^{8}+\frac {1}{3} x^{6} a b d +\frac {1}{6} x^{6} b^{2} c +\frac {1}{4} x^{4} a^{2} d +\frac {1}{2} x^{4} a b c +\frac {1}{2} a^{2} c \,x^{2}\) | \(54\) |
risch | \(\frac {1}{8} b^{2} d \,x^{8}+\frac {1}{3} x^{6} a b d +\frac {1}{6} x^{6} b^{2} c +\frac {1}{4} x^{4} a^{2} d +\frac {1}{2} x^{4} a b c +\frac {1}{2} a^{2} c \,x^{2}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 51, normalized size = 1.21 \begin {gather*} \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{6} \, {\left (b^{2} c + 2 \, a b d\right )} x^{6} + \frac {1}{2} \, a^{2} c x^{2} + \frac {1}{4} \, {\left (2 \, a b c + a^{2} d\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.81, size = 51, normalized size = 1.21 \begin {gather*} \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{6} \, {\left (b^{2} c + 2 \, a b d\right )} x^{6} + \frac {1}{2} \, a^{2} c x^{2} + \frac {1}{4} \, {\left (2 \, a b c + a^{2} d\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.01, size = 53, normalized size = 1.26 \begin {gather*} \frac {a^{2} c x^{2}}{2} + \frac {b^{2} d x^{8}}{8} + x^{6} \left (\frac {a b d}{3} + \frac {b^{2} c}{6}\right ) + x^{4} \left (\frac {a^{2} d}{4} + \frac {a b c}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.75, size = 53, normalized size = 1.26 \begin {gather*} \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{6} \, b^{2} c x^{6} + \frac {1}{3} \, a b d x^{6} + \frac {1}{2} \, a b c x^{4} + \frac {1}{4} \, a^{2} d x^{4} + \frac {1}{2} \, a^{2} c x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.02, size = 51, normalized size = 1.21 \begin {gather*} x^4\,\left (\frac {d\,a^2}{4}+\frac {b\,c\,a}{2}\right )+x^6\,\left (\frac {c\,b^2}{6}+\frac {a\,d\,b}{3}\right )+\frac {a^2\,c\,x^2}{2}+\frac {b^2\,d\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________