Optimal. Leaf size=166 \[ \frac {1}{2} a^3 A x^2+\frac {1}{4} a^2 (3 A b+a B) x^4+\frac {1}{2} a \left (a b B+A \left (b^2+a c\right )\right ) x^6+\frac {1}{8} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^8+\frac {1}{10} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{10}+\frac {1}{4} c \left (b^2 B+A b c+a B c\right ) x^{12}+\frac {1}{14} c^2 (3 b B+A c) x^{14}+\frac {1}{16} B c^3 x^{16} \]
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Rubi [A]
time = 0.18, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1261, 645}
\begin {gather*} \frac {1}{2} a^3 A x^2+\frac {1}{4} a^2 x^4 (a B+3 A b)+\frac {1}{4} c x^{12} \left (a B c+A b c+b^2 B\right )+\frac {1}{2} a x^6 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{10} x^{10} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{8} x^8 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{14} c^2 x^{14} (A c+3 b B)+\frac {1}{16} B c^3 x^{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 645
Rule 1261
Rubi steps
\begin {align*} \int x \left (A+B x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx &=\frac {1}{2} \text {Subst}\left (\int (A+B x) \left (a+b x+c x^2\right )^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a^3 A+a^2 (3 A b+a B) x+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^2+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^3+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^4+3 c \left (b^2 B+A b c+a B c\right ) x^5+c^2 (3 b B+A c) x^6+B c^3 x^7\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} a^3 A x^2+\frac {1}{4} a^2 (3 A b+a B) x^4+\frac {1}{2} a \left (a b B+A \left (b^2+a c\right )\right ) x^6+\frac {1}{8} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^8+\frac {1}{10} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{10}+\frac {1}{4} c \left (b^2 B+A b c+a B c\right ) x^{12}+\frac {1}{14} c^2 (3 b B+A c) x^{14}+\frac {1}{16} B c^3 x^{16}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 154, normalized size = 0.93 \begin {gather*} \frac {1}{560} x^2 \left (280 a^3 A+140 a^2 (3 A b+a B) x^2+280 a \left (a b B+A \left (b^2+a c\right )\right ) x^4+70 \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^6+56 \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^8+140 c \left (b^2 B+A b c+a B c\right ) x^{10}+40 c^2 (3 b B+A c) x^{12}+35 B c^3 x^{14}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 226, normalized size = 1.36
method | result | size |
norman | \(\frac {a^{3} A \,x^{2}}{2}+\left (\frac {3}{4} a^{2} b A +\frac {1}{4} a^{3} B \right ) x^{4}+\left (\frac {1}{2} a^{2} c A +\frac {1}{2} A a \,b^{2}+\frac {1}{2} B \,a^{2} b \right ) x^{6}+\left (\frac {3}{4} A a b c +\frac {1}{8} A \,b^{3}+\frac {3}{8} a^{2} c B +\frac {3}{8} B a \,b^{2}\right ) x^{8}+\left (\frac {3}{10} c^{2} a A +\frac {3}{10} A \,b^{2} c +\frac {3}{5} a b B c +\frac {1}{10} b^{3} B \right ) x^{10}+\left (\frac {1}{4} b \,c^{2} A +\frac {1}{4} c^{2} a B +\frac {1}{4} B \,b^{2} c \right ) x^{12}+\left (\frac {1}{14} c^{3} A +\frac {3}{14} B b \,c^{2}\right ) x^{14}+\frac {B \,c^{3} x^{16}}{16}\) | \(170\) |
gosper | \(\frac {1}{2} a^{3} A \,x^{2}+\frac {3}{4} x^{4} a^{2} b A +\frac {1}{4} x^{4} a^{3} B +\frac {1}{2} x^{6} a^{2} c A +\frac {1}{2} x^{6} A a \,b^{2}+\frac {1}{2} x^{6} B \,a^{2} b +\frac {3}{4} x^{8} A a b c +\frac {1}{8} x^{8} A \,b^{3}+\frac {3}{8} x^{8} a^{2} c B +\frac {3}{8} x^{8} B a \,b^{2}+\frac {3}{10} x^{10} c^{2} a A +\frac {3}{10} x^{10} A \,b^{2} c +\frac {3}{5} x^{10} a b B c +\frac {1}{10} x^{10} b^{3} B +\frac {1}{4} x^{12} b \,c^{2} A +\frac {1}{4} x^{12} c^{2} a B +\frac {1}{4} x^{12} B \,b^{2} c +\frac {1}{14} x^{14} c^{3} A +\frac {3}{14} x^{14} B b \,c^{2}+\frac {1}{16} B \,c^{3} x^{16}\) | \(194\) |
risch | \(\frac {1}{2} a^{3} A \,x^{2}+\frac {3}{4} x^{4} a^{2} b A +\frac {1}{4} x^{4} a^{3} B +\frac {1}{2} x^{6} a^{2} c A +\frac {1}{2} x^{6} A a \,b^{2}+\frac {1}{2} x^{6} B \,a^{2} b +\frac {3}{4} x^{8} A a b c +\frac {1}{8} x^{8} A \,b^{3}+\frac {3}{8} x^{8} a^{2} c B +\frac {3}{8} x^{8} B a \,b^{2}+\frac {3}{10} x^{10} c^{2} a A +\frac {3}{10} x^{10} A \,b^{2} c +\frac {3}{5} x^{10} a b B c +\frac {1}{10} x^{10} b^{3} B +\frac {1}{4} x^{12} b \,c^{2} A +\frac {1}{4} x^{12} c^{2} a B +\frac {1}{4} x^{12} B \,b^{2} c +\frac {1}{14} x^{14} c^{3} A +\frac {3}{14} x^{14} B b \,c^{2}+\frac {1}{16} B \,c^{3} x^{16}\) | \(194\) |
default | \(\frac {B \,c^{3} x^{16}}{16}+\frac {\left (c^{3} A +3 B b \,c^{2}\right ) x^{14}}{14}+\frac {\left (3 b \,c^{2} A +B \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{12}}{12}+\frac {\left (\left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) A +B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{10}}{10}+\frac {\left (\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) A +B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{8}}{8}+\frac {\left (\left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right ) A +3 B \,a^{2} b \right ) x^{6}}{6}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{4}}{4}+\frac {a^{3} A \,x^{2}}{2}\) | \(226\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 166, normalized size = 1.00 \begin {gather*} \frac {1}{16} \, B c^{3} x^{16} + \frac {1}{14} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{14} + \frac {1}{4} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{12} + \frac {1}{10} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{10} + \frac {1}{8} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{8} + \frac {1}{2} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{6} + \frac {1}{2} \, A a^{3} x^{2} + \frac {1}{4} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 166, normalized size = 1.00 \begin {gather*} \frac {1}{16} \, B c^{3} x^{16} + \frac {1}{14} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{14} + \frac {1}{4} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{12} + \frac {1}{10} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{10} + \frac {1}{8} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{8} + \frac {1}{2} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{6} + \frac {1}{2} \, A a^{3} x^{2} + \frac {1}{4} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \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 = 1.20 \begin {gather*} \frac {A a^{3} x^{2}}{2} + \frac {B c^{3} x^{16}}{16} + x^{14} \left (\frac {A c^{3}}{14} + \frac {3 B b c^{2}}{14}\right ) + x^{12} \left (\frac {A b c^{2}}{4} + \frac {B a c^{2}}{4} + \frac {B b^{2} c}{4}\right ) + x^{10} \cdot \left (\frac {3 A a c^{2}}{10} + \frac {3 A b^{2} c}{10} + \frac {3 B a b c}{5} + \frac {B b^{3}}{10}\right ) + x^{8} \cdot \left (\frac {3 A a b c}{4} + \frac {A b^{3}}{8} + \frac {3 B a^{2} c}{8} + \frac {3 B a b^{2}}{8}\right ) + x^{6} \left (\frac {A a^{2} c}{2} + \frac {A a b^{2}}{2} + \frac {B a^{2} b}{2}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} b}{4} + \frac {B a^{3}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.05, size = 193, normalized size = 1.16 \begin {gather*} \frac {1}{16} \, B c^{3} x^{16} + \frac {3}{14} \, B b c^{2} x^{14} + \frac {1}{14} \, A c^{3} x^{14} + \frac {1}{4} \, B b^{2} c x^{12} + \frac {1}{4} \, B a c^{2} x^{12} + \frac {1}{4} \, A b c^{2} x^{12} + \frac {1}{10} \, B b^{3} x^{10} + \frac {3}{5} \, B a b c x^{10} + \frac {3}{10} \, A b^{2} c x^{10} + \frac {3}{10} \, A a c^{2} x^{10} + \frac {3}{8} \, B a b^{2} x^{8} + \frac {1}{8} \, A b^{3} x^{8} + \frac {3}{8} \, B a^{2} c x^{8} + \frac {3}{4} \, A a b c x^{8} + \frac {1}{2} \, B a^{2} b x^{6} + \frac {1}{2} \, A a b^{2} x^{6} + \frac {1}{2} \, A a^{2} c x^{6} + \frac {1}{4} \, B a^{3} x^{4} + \frac {3}{4} \, A a^{2} b x^{4} + \frac {1}{2} \, A a^{3} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 169, normalized size = 1.02 \begin {gather*} x^8\,\left (\frac {3\,B\,c\,a^2}{8}+\frac {3\,B\,a\,b^2}{8}+\frac {3\,A\,c\,a\,b}{4}+\frac {A\,b^3}{8}\right )+x^{10}\,\left (\frac {B\,b^3}{10}+\frac {3\,A\,b^2\,c}{10}+\frac {3\,B\,a\,b\,c}{5}+\frac {3\,A\,a\,c^2}{10}\right )+x^4\,\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )+x^{14}\,\left (\frac {A\,c^3}{14}+\frac {3\,B\,b\,c^2}{14}\right )+x^6\,\left (\frac {B\,a^2\,b}{2}+\frac {A\,c\,a^2}{2}+\frac {A\,a\,b^2}{2}\right )+x^{12}\,\left (\frac {B\,b^2\,c}{4}+\frac {A\,b\,c^2}{4}+\frac {B\,a\,c^2}{4}\right )+\frac {A\,a^3\,x^2}{2}+\frac {B\,c^3\,x^{16}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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