Optimal. Leaf size=71 \[ \frac {d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac {\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac {d^2 \left (a+b x^2\right )^5}{10 b^3} \]
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Rubi [A] time = 0.11, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 43} \[ \frac {d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac {\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac {d^2 \left (a+b x^2\right )^5}{10 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b x)^2 (c+d x)^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(b c-a d)^2 (a+b x)^2}{b^2}+\frac {2 d (b c-a d) (a+b x)^3}{b^2}+\frac {d^2 (a+b x)^4}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2 \left (a+b x^2\right )^3}{6 b^3}+\frac {d (b c-a d) \left (a+b x^2\right )^4}{4 b^3}+\frac {d^2 \left (a+b x^2\right )^5}{10 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 1.14 \[ \frac {1}{60} x^2 \left (10 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+30 a^2 c^2+15 b d x^6 (a d+b c)+30 a c x^2 (a d+b c)+6 b^2 d^2 x^8\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.36, size = 94, normalized size = 1.32 \[ \frac {1}{10} x^{10} d^{2} b^{2} + \frac {1}{4} x^{8} d c b^{2} + \frac {1}{4} x^{8} d^{2} b a + \frac {1}{6} x^{6} c^{2} b^{2} + \frac {2}{3} x^{6} d c b a + \frac {1}{6} x^{6} d^{2} a^{2} + \frac {1}{2} x^{4} c^{2} b a + \frac {1}{2} x^{4} d c a^{2} + \frac {1}{2} x^{2} c^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 94, normalized size = 1.32 \[ \frac {1}{10} \, b^{2} d^{2} x^{10} + \frac {1}{4} \, b^{2} c d x^{8} + \frac {1}{4} \, a b d^{2} x^{8} + \frac {1}{6} \, b^{2} c^{2} x^{6} + \frac {2}{3} \, a b c d x^{6} + \frac {1}{6} \, a^{2} d^{2} x^{6} + \frac {1}{2} \, a b c^{2} x^{4} + \frac {1}{2} \, a^{2} c d x^{4} + \frac {1}{2} \, a^{2} c^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 90, normalized size = 1.27 \[ \frac {b^{2} d^{2} x^{10}}{10}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{8}}{8}+\frac {a^{2} c^{2} x^{2}}{2}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (2 a^{2} c d +2 a b \,c^{2}\right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 85, normalized size = 1.20 \[ \frac {1}{10} \, b^{2} d^{2} x^{10} + \frac {1}{4} \, {\left (b^{2} c d + a b d^{2}\right )} x^{8} + \frac {1}{6} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + \frac {1}{2} \, a^{2} c^{2} x^{2} + \frac {1}{2} \, {\left (a b c^{2} + a^{2} c d\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 78, normalized size = 1.10 \[ x^6\,\left (\frac {a^2\,d^2}{6}+\frac {2\,a\,b\,c\,d}{3}+\frac {b^2\,c^2}{6}\right )+\frac {a^2\,c^2\,x^2}{2}+\frac {b^2\,d^2\,x^{10}}{10}+\frac {a\,c\,x^4\,\left (a\,d+b\,c\right )}{2}+\frac {b\,d\,x^8\,\left (a\,d+b\,c\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 94, normalized size = 1.32 \[ \frac {a^{2} c^{2} x^{2}}{2} + \frac {b^{2} d^{2} x^{10}}{10} + x^{8} \left (\frac {a b d^{2}}{4} + \frac {b^{2} c d}{4}\right ) + x^{6} \left (\frac {a^{2} d^{2}}{6} + \frac {2 a b c d}{3} + \frac {b^{2} c^{2}}{6}\right ) + x^{4} \left (\frac {a^{2} c d}{2} + \frac {a b c^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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