Optimal. Leaf size=87 \[ \frac {1}{8} x^8 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {1}{4} a^2 c^2 x^4+\frac {1}{5} b d x^{10} (a d+b c)+\frac {1}{3} a c x^6 (a d+b c)+\frac {1}{12} b^2 d^2 x^{12} \]
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Rubi [A] time = 0.11, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 77} \[ \frac {1}{8} x^8 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {1}{4} a^2 c^2 x^4+\frac {1}{5} b d x^{10} (a d+b c)+\frac {1}{3} a c x^6 (a d+b c)+\frac {1}{12} b^2 d^2 x^{12} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^2 (c+d x)^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^2 c^2 x+2 a c (b c+a d) x^2+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^3+2 b d (b c+a d) x^4+b^2 d^2 x^5\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4} a^2 c^2 x^4+\frac {1}{3} a c (b c+a d) x^6+\frac {1}{8} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^8+\frac {1}{5} b d (b c+a d) x^{10}+\frac {1}{12} b^2 d^2 x^{12}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 0.93 \[ \frac {1}{120} x^4 \left (15 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+30 a^2 c^2+24 b d x^6 (a d+b c)+40 a c x^2 (a d+b c)+10 b^2 d^2 x^8\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 94, normalized size = 1.08 \[ \frac {1}{12} x^{12} d^{2} b^{2} + \frac {1}{5} x^{10} d c b^{2} + \frac {1}{5} x^{10} d^{2} b a + \frac {1}{8} x^{8} c^{2} b^{2} + \frac {1}{2} x^{8} d c b a + \frac {1}{8} x^{8} d^{2} a^{2} + \frac {1}{3} x^{6} c^{2} b a + \frac {1}{3} x^{6} d c a^{2} + \frac {1}{4} x^{4} c^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 94, normalized size = 1.08 \[ \frac {1}{12} \, b^{2} d^{2} x^{12} + \frac {1}{5} \, b^{2} c d x^{10} + \frac {1}{5} \, a b d^{2} x^{10} + \frac {1}{8} \, b^{2} c^{2} x^{8} + \frac {1}{2} \, a b c d x^{8} + \frac {1}{8} \, a^{2} d^{2} x^{8} + \frac {1}{3} \, a b c^{2} x^{6} + \frac {1}{3} \, a^{2} c d x^{6} + \frac {1}{4} \, a^{2} c^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 90, normalized size = 1.03 \[ \frac {b^{2} d^{2} x^{12}}{12}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{10}}{10}+\frac {a^{2} c^{2} x^{4}}{4}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{8}}{8}+\frac {\left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 85, normalized size = 0.98 \[ \frac {1}{12} \, b^{2} d^{2} x^{12} + \frac {1}{5} \, {\left (b^{2} c d + a b d^{2}\right )} x^{10} + \frac {1}{8} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{8} + \frac {1}{4} \, a^{2} c^{2} x^{4} + \frac {1}{3} \, {\left (a b c^{2} + a^{2} c d\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 78, normalized size = 0.90 \[ x^8\,\left (\frac {a^2\,d^2}{8}+\frac {a\,b\,c\,d}{2}+\frac {b^2\,c^2}{8}\right )+\frac {a^2\,c^2\,x^4}{4}+\frac {b^2\,d^2\,x^{12}}{12}+\frac {a\,c\,x^6\,\left (a\,d+b\,c\right )}{3}+\frac {b\,d\,x^{10}\,\left (a\,d+b\,c\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 92, normalized size = 1.06 \[ \frac {a^{2} c^{2} x^{4}}{4} + \frac {b^{2} d^{2} x^{12}}{12} + x^{10} \left (\frac {a b d^{2}}{5} + \frac {b^{2} c d}{5}\right ) + x^{8} \left (\frac {a^{2} d^{2}}{8} + \frac {a b c d}{2} + \frac {b^{2} c^{2}}{8}\right ) + x^{6} \left (\frac {a^{2} c d}{3} + \frac {a b c^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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