Optimal. Leaf size=33 \[ \frac {1}{4} A b x^4+\frac {1}{6} (b B+A c) x^6+\frac {1}{8} B c x^8 \]
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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1598, 457, 77}
\begin {gather*} \frac {1}{6} x^6 (A c+b B)+\frac {1}{4} A b x^4+\frac {1}{8} B c x^8 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int x \left (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx &=\int x^3 \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int x (A+B x) (b+c x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (A b x+(b B+A c) x^2+B c x^3\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4} A b x^4+\frac {1}{6} (b B+A c) x^6+\frac {1}{8} B c x^8\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {1}{4} A b x^4+\frac {1}{6} (b B+A c) x^6+\frac {1}{8} B c x^8 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 28, normalized size = 0.85
method | result | size |
default | \(\frac {A b \,x^{4}}{4}+\frac {\left (A c +B b \right ) x^{6}}{6}+\frac {B c \,x^{8}}{8}\) | \(28\) |
norman | \(\frac {B c \,x^{8}}{8}+\left (\frac {A c}{6}+\frac {B b}{6}\right ) x^{6}+\frac {A b \,x^{4}}{4}\) | \(29\) |
risch | \(\frac {1}{4} A b \,x^{4}+\frac {1}{6} x^{6} A c +\frac {1}{6} b B \,x^{6}+\frac {1}{8} B c \,x^{8}\) | \(30\) |
gosper | \(\frac {x^{4} \left (3 B c \,x^{4}+4 A c \,x^{2}+4 b B \,x^{2}+6 A b \right )}{24}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{8} \, B c x^{8} + \frac {1}{6} \, {\left (B b + A c\right )} x^{6} + \frac {1}{4} \, A b x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.16, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{8} \, B c x^{8} + \frac {1}{6} \, {\left (B b + A c\right )} x^{6} + \frac {1}{4} \, A b x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 29, normalized size = 0.88 \begin {gather*} \frac {A b x^{4}}{4} + \frac {B c x^{8}}{8} + x^{6} \left (\frac {A c}{6} + \frac {B b}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.66, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{8} \, B c x^{8} + \frac {1}{6} \, B b x^{6} + \frac {1}{6} \, A c x^{6} + \frac {1}{4} \, A b x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 28, normalized size = 0.85 \begin {gather*} \frac {B\,c\,x^8}{8}+\left (\frac {A\,c}{6}+\frac {B\,b}{6}\right )\,x^6+\frac {A\,b\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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