Optimal. Leaf size=42 \[ \frac {\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac {d \left (a+b x^2\right )^4}{8 b^2} \]
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Rubi [A] time = 0.06, 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 = {444, 43} \[ \frac {\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac {d \left (a+b x^2\right )^4}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b x)^2 (c+d x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {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*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 1.21 \[ \frac {1}{24} x^2 \left (12 a^2 c+4 b x^4 (2 a d+b c)+6 a x^2 (a d+2 b c)+3 b^2 d x^6\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 53, normalized size = 1.26 \[ \frac {1}{8} x^{8} d b^{2} + \frac {1}{6} x^{6} c b^{2} + \frac {1}{3} x^{6} d b a + \frac {1}{2} x^{4} c b a + \frac {1}{4} x^{4} d a^{2} + \frac {1}{2} x^{2} c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 53, normalized size = 1.26 \[ \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{6} \, b^{2} c x^{6} + \frac {1}{3} \, a b d x^{6} + \frac {1}{2} \, a b c x^{4} + \frac {1}{4} \, a^{2} d x^{4} + \frac {1}{2} \, a^{2} c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 1.24 \[ \frac {b^{2} d \,x^{8}}{8}+\frac {\left (2 a b d +b^{2} c \right ) x^{6}}{6}+\frac {a^{2} c \,x^{2}}{2}+\frac {\left (a^{2} d +2 a b c \right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 51, normalized size = 1.21 \[ \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{6} \, {\left (b^{2} c + 2 \, a b d\right )} x^{6} + \frac {1}{2} \, a^{2} c x^{2} + \frac {1}{4} \, {\left (2 \, a b c + a^{2} d\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 51, normalized size = 1.21 \[ x^4\,\left (\frac {d\,a^2}{4}+\frac {b\,c\,a}{2}\right )+x^6\,\left (\frac {c\,b^2}{6}+\frac {a\,d\,b}{3}\right )+\frac {a^2\,c\,x^2}{2}+\frac {b^2\,d\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 53, normalized size = 1.26 \[ \frac {a^{2} c x^{2}}{2} + \frac {b^{2} d x^{8}}{8} + x^{6} \left (\frac {a b d}{3} + \frac {b^{2} c}{6}\right ) + x^{4} \left (\frac {a^{2} d}{4} + \frac {a b c}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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