Optimal. Leaf size=46 \[ \frac {1}{3} x^3 (a C+A c)+a A x+\frac {1}{2} a B x^2+\frac {1}{4} B c x^4+\frac {1}{5} c C x^5 \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1657} \[ \frac {1}{3} x^3 (a C+A c)+a A x+\frac {1}{2} a B x^2+\frac {1}{4} B c x^4+\frac {1}{5} c C x^5 \]
Antiderivative was successfully verified.
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Rule 1657
Rubi steps
\begin {align*} \int \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx &=\int \left (a A+a B x+(A c+a C) x^2+B c x^3+c C x^4\right ) \, dx\\ &=a A x+\frac {1}{2} a B x^2+\frac {1}{3} (A c+a C) x^3+\frac {1}{4} B c x^4+\frac {1}{5} c C x^5\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \[ \frac {1}{3} x^3 (a C+A c)+a A x+\frac {1}{2} a B x^2+\frac {1}{4} B c x^4+\frac {1}{5} c C x^5 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 40, normalized size = 0.87 \[ \frac {1}{5} x^{5} c C + \frac {1}{4} x^{4} c B + \frac {1}{3} x^{3} a C + \frac {1}{3} x^{3} c A + \frac {1}{2} x^{2} a B + x a A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 40, normalized size = 0.87 \[ \frac {1}{5} \, C c x^{5} + \frac {1}{4} \, B c x^{4} + \frac {1}{3} \, C a x^{3} + \frac {1}{3} \, A c x^{3} + \frac {1}{2} \, B a x^{2} + A a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 0.85 \[ \frac {C c \,x^{5}}{5}+\frac {B c \,x^{4}}{4}+\frac {B a \,x^{2}}{2}+A a x +\frac {\left (A c +a C \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 38, normalized size = 0.83 \[ \frac {1}{5} \, C c x^{5} + \frac {1}{4} \, B c x^{4} + \frac {1}{2} \, B a x^{2} + \frac {1}{3} \, {\left (C a + A c\right )} x^{3} + A a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 39, normalized size = 0.85 \[ \frac {C\,c\,x^5}{5}+\frac {B\,c\,x^4}{4}+\left (\frac {A\,c}{3}+\frac {C\,a}{3}\right )\,x^3+\frac {B\,a\,x^2}{2}+A\,a\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 42, normalized size = 0.91 \[ A a x + \frac {B a x^{2}}{2} + \frac {B c x^{4}}{4} + \frac {C c x^{5}}{5} + x^{3} \left (\frac {A c}{3} + \frac {C a}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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