Optimal. Leaf size=89 \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^2}{2}+\frac{3}{10} c x^{10} \left (a c+b^2\right )+\frac{1}{8} b x^8 \left (6 a c+b^2\right )+\frac{1}{2} a x^6 \left (a c+b^2\right )+\frac{1}{4} b c^2 x^{12}+\frac{c^3 x^{14}}{14} \]
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Rubi [A] time = 0.0830906, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1107, 611} \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^2}{2}+\frac{3}{10} c x^{10} \left (a c+b^2\right )+\frac{1}{8} b x^8 \left (6 a c+b^2\right )+\frac{1}{2} a x^6 \left (a c+b^2\right )+\frac{1}{4} b c^2 x^{12}+\frac{c^3 x^{14}}{14} \]
Antiderivative was successfully verified.
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Rule 1107
Rule 611
Rubi steps
\begin{align*} \int x \left (a+b x^2+c x^4\right )^3 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^3 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^3+3 a^2 b x+3 a b^2 \left (1+\frac{a c}{b^2}\right ) x^2+b^3 \left (1+\frac{6 a c}{b^2}\right ) x^3+3 b^2 c \left (1+\frac{a c}{b^2}\right ) x^4+3 b c^2 x^5+c^3 x^6\right ) \, dx,x,x^2\right )\\ &=\frac{a^3 x^2}{2}+\frac{3}{4} a^2 b x^4+\frac{1}{2} a \left (b^2+a c\right ) x^6+\frac{1}{8} b \left (b^2+6 a c\right ) x^8+\frac{3}{10} c \left (b^2+a c\right ) x^{10}+\frac{1}{4} b c^2 x^{12}+\frac{c^3 x^{14}}{14}\\ \end{align*}
Mathematica [A] time = 0.0163152, size = 79, normalized size = 0.89 \[ \frac{1}{280} x^2 \left (210 a^2 b x^2+140 a^3+84 c x^8 \left (a c+b^2\right )+35 b x^6 \left (6 a c+b^2\right )+140 a x^4 \left (a c+b^2\right )+70 b c^2 x^{10}+20 c^3 x^{12}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 111, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{x}^{14}}{14}}+{\frac{b{c}^{2}{x}^{12}}{4}}+{\frac{ \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{10}}{10}}+{\frac{ \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ){x}^{6}}{6}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{{x}^{2}{a}^{3}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980865, size = 109, normalized size = 1.22 \begin{align*} \frac{1}{14} \, c^{3} x^{14} + \frac{1}{4} \, b c^{2} x^{12} + \frac{3}{10} \,{\left (b^{2} c + a c^{2}\right )} x^{10} + \frac{1}{8} \,{\left (b^{3} + 6 \, a b c\right )} x^{8} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{2} \,{\left (a b^{2} + a^{2} c\right )} x^{6} + \frac{1}{2} \, a^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49572, size = 216, normalized size = 2.43 \begin{align*} \frac{1}{14} x^{14} c^{3} + \frac{1}{4} x^{12} c^{2} b + \frac{3}{10} x^{10} c b^{2} + \frac{3}{10} x^{10} c^{2} a + \frac{1}{8} x^{8} b^{3} + \frac{3}{4} x^{8} c b a + \frac{1}{2} x^{6} b^{2} a + \frac{1}{2} x^{6} c a^{2} + \frac{3}{4} x^{4} b a^{2} + \frac{1}{2} x^{2} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.08057, size = 92, normalized size = 1.03 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{3 a^{2} b x^{4}}{4} + \frac{b c^{2} x^{12}}{4} + \frac{c^{3} x^{14}}{14} + x^{10} \left (\frac{3 a c^{2}}{10} + \frac{3 b^{2} c}{10}\right ) + x^{8} \left (\frac{3 a b c}{4} + \frac{b^{3}}{8}\right ) + x^{6} \left (\frac{a^{2} c}{2} + \frac{a b^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12805, size = 117, normalized size = 1.31 \begin{align*} \frac{1}{14} \, c^{3} x^{14} + \frac{1}{4} \, b c^{2} x^{12} + \frac{3}{10} \, b^{2} c x^{10} + \frac{3}{10} \, a c^{2} x^{10} + \frac{1}{8} \, b^{3} x^{8} + \frac{3}{4} \, a b c x^{8} + \frac{1}{2} \, a b^{2} x^{6} + \frac{1}{2} \, a^{2} c x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{2} \, a^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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