Optimal. Leaf size=161 \[ a^3 A x+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {3}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^5+\frac {1}{7} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^7+\frac {1}{9} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^9+\frac {3}{11} c \left (b^2 B+A b c+a B c\right ) x^{11}+\frac {1}{13} c^2 (3 b B+A c) x^{13}+\frac {1}{15} B c^3 x^{15} \]
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Rubi [A]
time = 0.07, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1167}
\begin {gather*} a^3 A x+\frac {1}{3} a^2 x^3 (a B+3 A b)+\frac {3}{11} c x^{11} \left (a B c+A b c+b^2 B\right )+\frac {3}{5} a x^5 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{9} x^9 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{7} x^7 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{13} c^2 x^{13} (A c+3 b B)+\frac {1}{15} B c^3 x^{15} \end {gather*}
Antiderivative was successfully verified.
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Rule 1167
Rubi steps
\begin {align*} \int \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 A+a^2 (3 A b+a B) x^2+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^4+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^6+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^8+3 c \left (b^2 B+A b c+a B c\right ) x^{10}+c^2 (3 b B+A c) x^{12}+B c^3 x^{14}\right ) \, dx\\ &=a^3 A x+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {3}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^5+\frac {1}{7} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^7+\frac {1}{9} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^9+\frac {3}{11} c \left (b^2 B+A b c+a B c\right ) x^{11}+\frac {1}{13} c^2 (3 b B+A c) x^{13}+\frac {1}{15} B c^3 x^{15}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 161, normalized size = 1.00 \begin {gather*} a^3 A x+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {3}{5} a \left (a b B+A \left (b^2+a c\right )\right ) x^5+\frac {1}{7} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^7+\frac {1}{9} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^9+\frac {3}{11} c \left (b^2 B+A b c+a B c\right ) x^{11}+\frac {1}{13} c^2 (3 b B+A c) x^{13}+\frac {1}{15} B c^3 x^{15} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 223, normalized size = 1.39
method | result | size |
norman | \(a^{3} A x +\left (a^{2} b A +\frac {1}{3} a^{3} B \right ) x^{3}+\left (\frac {3}{5} a^{2} c A +\frac {3}{5} A a \,b^{2}+\frac {3}{5} B \,a^{2} b \right ) x^{5}+\left (\frac {6}{7} A a b c +\frac {1}{7} A \,b^{3}+\frac {3}{7} a^{2} c B +\frac {3}{7} B a \,b^{2}\right ) x^{7}+\left (\frac {1}{3} c^{2} a A +\frac {1}{3} A \,b^{2} c +\frac {2}{3} a b B c +\frac {1}{9} b^{3} B \right ) x^{9}+\left (\frac {3}{11} b \,c^{2} A +\frac {3}{11} c^{2} a B +\frac {3}{11} B \,b^{2} c \right ) x^{11}+\left (\frac {1}{13} c^{3} A +\frac {3}{13} B b \,c^{2}\right ) x^{13}+\frac {B \,c^{3} x^{15}}{15}\) | \(166\) |
gosper | \(a^{3} A x +x^{3} a^{2} b A +\frac {1}{3} x^{3} a^{3} B +\frac {3}{5} x^{5} a^{2} c A +\frac {3}{5} x^{5} A a \,b^{2}+\frac {3}{5} x^{5} B \,a^{2} b +\frac {6}{7} x^{7} A a b c +\frac {1}{7} x^{7} A \,b^{3}+\frac {3}{7} x^{7} a^{2} c B +\frac {3}{7} x^{7} B a \,b^{2}+\frac {1}{3} x^{9} c^{2} a A +\frac {1}{3} x^{9} A \,b^{2} c +\frac {2}{3} x^{9} a b B c +\frac {1}{9} x^{9} b^{3} B +\frac {3}{11} x^{11} b \,c^{2} A +\frac {3}{11} x^{11} c^{2} a B +\frac {3}{11} x^{11} B \,b^{2} c +\frac {1}{13} x^{13} c^{3} A +\frac {3}{13} x^{13} B b \,c^{2}+\frac {1}{15} B \,c^{3} x^{15}\) | \(190\) |
risch | \(a^{3} A x +x^{3} a^{2} b A +\frac {1}{3} x^{3} a^{3} B +\frac {3}{5} x^{5} a^{2} c A +\frac {3}{5} x^{5} A a \,b^{2}+\frac {3}{5} x^{5} B \,a^{2} b +\frac {6}{7} x^{7} A a b c +\frac {1}{7} x^{7} A \,b^{3}+\frac {3}{7} x^{7} a^{2} c B +\frac {3}{7} x^{7} B a \,b^{2}+\frac {1}{3} x^{9} c^{2} a A +\frac {1}{3} x^{9} A \,b^{2} c +\frac {2}{3} x^{9} a b B c +\frac {1}{9} x^{9} b^{3} B +\frac {3}{11} x^{11} b \,c^{2} A +\frac {3}{11} x^{11} c^{2} a B +\frac {3}{11} x^{11} B \,b^{2} c +\frac {1}{13} x^{13} c^{3} A +\frac {3}{13} x^{13} B b \,c^{2}+\frac {1}{15} B \,c^{3} x^{15}\) | \(190\) |
default | \(\frac {B \,c^{3} x^{15}}{15}+\frac {\left (c^{3} A +3 B b \,c^{2}\right ) x^{13}}{13}+\frac {\left (3 b \,c^{2} A +B \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{11}}{11}+\frac {\left (\left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) A +B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{9}}{9}+\frac {\left (\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) A +B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{7}}{7}+\frac {\left (\left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right ) A +3 B \,a^{2} b \right ) x^{5}}{5}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{3}}{3}+a^{3} A x\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 163, normalized size = 1.01 \begin {gather*} \frac {1}{15} \, B c^{3} x^{15} + \frac {1}{13} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{13} + \frac {3}{11} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{9} + \frac {1}{7} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{7} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + A a^{3} x + \frac {1}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 163, normalized size = 1.01 \begin {gather*} \frac {1}{15} \, B c^{3} x^{15} + \frac {1}{13} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{13} + \frac {3}{11} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{9} + \frac {1}{7} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{7} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + A a^{3} x + \frac {1}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 199, normalized size = 1.24 \begin {gather*} A a^{3} x + \frac {B c^{3} x^{15}}{15} + x^{13} \left (\frac {A c^{3}}{13} + \frac {3 B b c^{2}}{13}\right ) + x^{11} \cdot \left (\frac {3 A b c^{2}}{11} + \frac {3 B a c^{2}}{11} + \frac {3 B b^{2} c}{11}\right ) + x^{9} \left (\frac {A a c^{2}}{3} + \frac {A b^{2} c}{3} + \frac {2 B a b c}{3} + \frac {B b^{3}}{9}\right ) + x^{7} \cdot \left (\frac {6 A a b c}{7} + \frac {A b^{3}}{7} + \frac {3 B a^{2} c}{7} + \frac {3 B a b^{2}}{7}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} c}{5} + \frac {3 A a b^{2}}{5} + \frac {3 B a^{2} b}{5}\right ) + x^{3} \left (A a^{2} b + \frac {B a^{3}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.13, size = 189, normalized size = 1.17 \begin {gather*} \frac {1}{15} \, B c^{3} x^{15} + \frac {3}{13} \, B b c^{2} x^{13} + \frac {1}{13} \, A c^{3} x^{13} + \frac {3}{11} \, B b^{2} c x^{11} + \frac {3}{11} \, B a c^{2} x^{11} + \frac {3}{11} \, A b c^{2} x^{11} + \frac {1}{9} \, B b^{3} x^{9} + \frac {2}{3} \, B a b c x^{9} + \frac {1}{3} \, A b^{2} c x^{9} + \frac {1}{3} \, A a c^{2} x^{9} + \frac {3}{7} \, B a b^{2} x^{7} + \frac {1}{7} \, A b^{3} x^{7} + \frac {3}{7} \, B a^{2} c x^{7} + \frac {6}{7} \, A a b c x^{7} + \frac {3}{5} \, B a^{2} b x^{5} + \frac {3}{5} \, A a b^{2} x^{5} + \frac {3}{5} \, A a^{2} c x^{5} + \frac {1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + A a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 165, normalized size = 1.02 \begin {gather*} x^7\,\left (\frac {3\,B\,c\,a^2}{7}+\frac {3\,B\,a\,b^2}{7}+\frac {6\,A\,c\,a\,b}{7}+\frac {A\,b^3}{7}\right )+x^9\,\left (\frac {B\,b^3}{9}+\frac {A\,b^2\,c}{3}+\frac {2\,B\,a\,b\,c}{3}+\frac {A\,a\,c^2}{3}\right )+x^3\,\left (\frac {B\,a^3}{3}+A\,b\,a^2\right )+x^{13}\,\left (\frac {A\,c^3}{13}+\frac {3\,B\,b\,c^2}{13}\right )+x^5\,\left (\frac {3\,B\,a^2\,b}{5}+\frac {3\,A\,c\,a^2}{5}+\frac {3\,A\,a\,b^2}{5}\right )+x^{11}\,\left (\frac {3\,B\,b^2\,c}{11}+\frac {3\,A\,b\,c^2}{11}+\frac {3\,B\,a\,c^2}{11}\right )+\frac {B\,c^3\,x^{15}}{15}+A\,a^3\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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