Optimal. Leaf size=166 \[ \frac{1}{5} a^2 x^5 (a B+3 A b)+\frac{1}{3} a^3 A x^3+\frac{1}{11} x^{11} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{13} c x^{13} \left (a B c+A b c+b^2 B\right )+\frac{1}{9} x^9 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{7} a x^7 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{15} c^2 x^{15} (A c+3 b B)+\frac{1}{17} B c^3 x^{17} \]
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Rubi [A] time = 0.153507, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1261} \[ \frac{1}{5} a^2 x^5 (a B+3 A b)+\frac{1}{3} a^3 A x^3+\frac{1}{11} x^{11} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{13} c x^{13} \left (a B c+A b c+b^2 B\right )+\frac{1}{9} x^9 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{7} a x^7 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{15} c^2 x^{15} (A c+3 b B)+\frac{1}{17} B c^3 x^{17} \]
Antiderivative was successfully verified.
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Rule 1261
Rubi steps
\begin{align*} \int x^2 \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 A x^2+a^2 (3 A b+a B) x^4+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^6+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^8+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{10}+3 c \left (b^2 B+A b c+a B c\right ) x^{12}+c^2 (3 b B+A c) x^{14}+B c^3 x^{16}\right ) \, dx\\ &=\frac{1}{3} a^3 A x^3+\frac{1}{5} a^2 (3 A b+a B) x^5+\frac{3}{7} a \left (a b B+A \left (b^2+a c\right )\right ) x^7+\frac{1}{9} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^9+\frac{1}{11} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{11}+\frac{3}{13} c \left (b^2 B+A b c+a B c\right ) x^{13}+\frac{1}{15} c^2 (3 b B+A c) x^{15}+\frac{1}{17} B c^3 x^{17}\\ \end{align*}
Mathematica [A] time = 0.05703, size = 166, normalized size = 1. \[ \frac{1}{5} a^2 x^5 (a B+3 A b)+\frac{1}{3} a^3 A x^3+\frac{1}{11} x^{11} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{3}{13} c x^{13} \left (a B c+A b c+b^2 B\right )+\frac{1}{9} x^9 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{3}{7} a x^7 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{15} c^2 x^{15} (A c+3 b B)+\frac{1}{17} B c^3 x^{17} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 226, normalized size = 1.4 \begin{align*}{\frac{B{c}^{3}{x}^{17}}{17}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{15}}{15}}+{\frac{ \left ( 3\,Ab{c}^{2}+B \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{13}}{13}}+{\frac{ \left ( A \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{11}}{11}}+{\frac{ \left ( A \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( A \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\,B{a}^{2}b \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){x}^{5}}{5}}+{\frac{{a}^{3}A{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988011, size = 224, normalized size = 1.35 \begin{align*} \frac{1}{17} \, B c^{3} x^{17} + \frac{1}{15} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{15} + \frac{3}{13} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{13} + \frac{1}{11} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{11} + \frac{1}{9} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{9} + \frac{3}{7} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{7} + \frac{1}{3} \, A a^{3} x^{3} + \frac{1}{5} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25964, size = 494, normalized size = 2.98 \begin{align*} \frac{1}{17} x^{17} c^{3} B + \frac{1}{5} x^{15} c^{2} b B + \frac{1}{15} x^{15} c^{3} A + \frac{3}{13} x^{13} c b^{2} B + \frac{3}{13} x^{13} c^{2} a B + \frac{3}{13} x^{13} c^{2} b A + \frac{1}{11} x^{11} b^{3} B + \frac{6}{11} x^{11} c b a B + \frac{3}{11} x^{11} c b^{2} A + \frac{3}{11} x^{11} c^{2} a A + \frac{1}{3} x^{9} b^{2} a B + \frac{1}{3} x^{9} c a^{2} B + \frac{1}{9} x^{9} b^{3} A + \frac{2}{3} x^{9} c b a A + \frac{3}{7} x^{7} b a^{2} B + \frac{3}{7} x^{7} b^{2} a A + \frac{3}{7} x^{7} c a^{2} A + \frac{1}{5} x^{5} a^{3} B + \frac{3}{5} x^{5} b a^{2} A + \frac{1}{3} x^{3} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098198, size = 204, normalized size = 1.23 \begin{align*} \frac{A a^{3} x^{3}}{3} + \frac{B c^{3} x^{17}}{17} + x^{15} \left (\frac{A c^{3}}{15} + \frac{B b c^{2}}{5}\right ) + x^{13} \left (\frac{3 A b c^{2}}{13} + \frac{3 B a c^{2}}{13} + \frac{3 B b^{2} c}{13}\right ) + x^{11} \left (\frac{3 A a c^{2}}{11} + \frac{3 A b^{2} c}{11} + \frac{6 B a b c}{11} + \frac{B b^{3}}{11}\right ) + x^{9} \left (\frac{2 A a b c}{3} + \frac{A b^{3}}{9} + \frac{B a^{2} c}{3} + \frac{B a b^{2}}{3}\right ) + x^{7} \left (\frac{3 A a^{2} c}{7} + \frac{3 A a b^{2}}{7} + \frac{3 B a^{2} b}{7}\right ) + x^{5} \left (\frac{3 A a^{2} b}{5} + \frac{B a^{3}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1507, size = 261, normalized size = 1.57 \begin{align*} \frac{1}{17} \, B c^{3} x^{17} + \frac{1}{5} \, B b c^{2} x^{15} + \frac{1}{15} \, A c^{3} x^{15} + \frac{3}{13} \, B b^{2} c x^{13} + \frac{3}{13} \, B a c^{2} x^{13} + \frac{3}{13} \, A b c^{2} x^{13} + \frac{1}{11} \, B b^{3} x^{11} + \frac{6}{11} \, B a b c x^{11} + \frac{3}{11} \, A b^{2} c x^{11} + \frac{3}{11} \, A a c^{2} x^{11} + \frac{1}{3} \, B a b^{2} x^{9} + \frac{1}{9} \, A b^{3} x^{9} + \frac{1}{3} \, B a^{2} c x^{9} + \frac{2}{3} \, A a b c x^{9} + \frac{3}{7} \, B a^{2} b x^{7} + \frac{3}{7} \, A a b^{2} x^{7} + \frac{3}{7} \, A a^{2} c x^{7} + \frac{1}{5} \, B a^{3} x^{5} + \frac{3}{5} \, A a^{2} b x^{5} + \frac{1}{3} \, A a^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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