Optimal. Leaf size=109 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{f \left (a+b x^4\right )^3}{12 b}+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]
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Rubi [A] time = 0.0744559, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1582, 1657} \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{f \left (a+b x^4\right )^3}{12 b}+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1657
Rubi steps
\begin{align*} \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^2 \, dx &=\frac{f \left (a+b x^4\right )^3}{12 b}+\int \left (c+d x+e x^2\right ) \left (a+b x^4\right )^2 \, dx\\ &=\frac{f \left (a+b x^4\right )^3}{12 b}+\int \left (a^2 c+a^2 d x+a^2 e x^2+2 a b c x^4+2 a b d x^5+2 a b e x^6+b^2 c x^8+b^2 d x^9+b^2 e x^{10}\right ) \, dx\\ &=a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}+\frac{f \left (a+b x^4\right )^3}{12 b}\\ \end{align*}
Mathematica [A] time = 0.0036873, size = 124, normalized size = 1.14 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{4} a^2 f x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{4} a b f x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}+\frac{1}{12} b^2 f x^{12} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 103, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}f{x}^{12}}{12}}+{\frac{{b}^{2}e{x}^{11}}{11}}+{\frac{{b}^{2}d{x}^{10}}{10}}+{\frac{{b}^{2}c{x}^{9}}{9}}+{\frac{fab{x}^{8}}{4}}+{\frac{2\,abe{x}^{7}}{7}}+{\frac{abd{x}^{6}}{3}}+{\frac{2\,abc{x}^{5}}{5}}+{\frac{f{a}^{2}{x}^{4}}{4}}+{\frac{{a}^{2}e{x}^{3}}{3}}+{\frac{{a}^{2}d{x}^{2}}{2}}+{a}^{2}cx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.914583, size = 138, normalized size = 1.27 \begin{align*} \frac{1}{12} \, b^{2} f x^{12} + \frac{1}{11} \, b^{2} e x^{11} + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b f x^{8} + \frac{2}{7} \, a b e x^{7} + \frac{1}{3} \, a b d x^{6} + \frac{2}{5} \, a b c x^{5} + \frac{1}{4} \, a^{2} f x^{4} + \frac{1}{3} \, a^{2} e x^{3} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48887, size = 258, normalized size = 2.37 \begin{align*} \frac{1}{12} x^{12} f b^{2} + \frac{1}{11} x^{11} e b^{2} + \frac{1}{10} x^{10} d b^{2} + \frac{1}{9} x^{9} c b^{2} + \frac{1}{4} x^{8} f b a + \frac{2}{7} x^{7} e b a + \frac{1}{3} x^{6} d b a + \frac{2}{5} x^{5} c b a + \frac{1}{4} x^{4} f a^{2} + \frac{1}{3} x^{3} e a^{2} + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.074751, size = 121, normalized size = 1.11 \begin{align*} a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{a^{2} e x^{3}}{3} + \frac{a^{2} f x^{4}}{4} + \frac{2 a b c x^{5}}{5} + \frac{a b d x^{6}}{3} + \frac{2 a b e x^{7}}{7} + \frac{a b f x^{8}}{4} + \frac{b^{2} c x^{9}}{9} + \frac{b^{2} d x^{10}}{10} + \frac{b^{2} e x^{11}}{11} + \frac{b^{2} f x^{12}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08465, size = 142, normalized size = 1.3 \begin{align*} \frac{1}{12} \, b^{2} f x^{12} + \frac{1}{11} \, b^{2} x^{11} e + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b f x^{8} + \frac{2}{7} \, a b x^{7} e + \frac{1}{3} \, a b d x^{6} + \frac{2}{5} \, a b c x^{5} + \frac{1}{4} \, a^{2} f x^{4} + \frac{1}{3} \, a^{2} x^{3} e + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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