Optimal. Leaf size=45 \[ a^2 A x+\frac {2}{3} a A c x^3+\frac {B \left (a+c x^2\right )^3}{6 c}+\frac {1}{5} A c^2 x^5 \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {641, 194} \begin {gather*} a^2 A x+\frac {2}{3} a A c x^3+\frac {B \left (a+c x^2\right )^3}{6 c}+\frac {1}{5} A c^2 x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 194
Rule 641
Rubi steps
\begin {align*} \int (A+B x) \left (a+c x^2\right )^2 \, dx &=\frac {B \left (a+c x^2\right )^3}{6 c}+A \int \left (a+c x^2\right )^2 \, dx\\ &=\frac {B \left (a+c x^2\right )^3}{6 c}+A \int \left (a^2+2 a c x^2+c^2 x^4\right ) \, dx\\ &=a^2 A x+\frac {2}{3} a A c x^3+\frac {1}{5} A c^2 x^5+\frac {B \left (a+c x^2\right )^3}{6 c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 60, normalized size = 1.33 \begin {gather*} a^2 A x+\frac {1}{2} a^2 B x^2+\frac {2}{3} a A c x^3+\frac {1}{2} a B c x^4+\frac {1}{5} A c^2 x^5+\frac {1}{6} B c^2 x^6 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) \left (a+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 50, normalized size = 1.11 \begin {gather*} \frac {1}{6} x^{6} c^{2} B + \frac {1}{5} x^{5} c^{2} A + \frac {1}{2} x^{4} c a B + \frac {2}{3} x^{3} c a A + \frac {1}{2} x^{2} a^{2} B + x a^{2} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 50, normalized size = 1.11 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{2} \, B a c x^{4} + \frac {2}{3} \, A a c x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 1.13 \begin {gather*} \frac {1}{6} B \,c^{2} x^{6}+\frac {1}{5} A \,c^{2} x^{5}+\frac {1}{2} B a c \,x^{4}+\frac {2}{3} A a c \,x^{3}+\frac {1}{2} B \,a^{2} x^{2}+A \,a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 50, normalized size = 1.11 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{2} \, B a c x^{4} + \frac {2}{3} \, A a c x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 50, normalized size = 1.11 \begin {gather*} \frac {B\,a^2\,x^2}{2}+A\,a^2\,x+\frac {B\,a\,c\,x^4}{2}+\frac {2\,A\,a\,c\,x^3}{3}+\frac {B\,c^2\,x^6}{6}+\frac {A\,c^2\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 58, normalized size = 1.29 \begin {gather*} A a^{2} x + \frac {2 A a c x^{3}}{3} + \frac {A c^{2} x^{5}}{5} + \frac {B a^{2} x^{2}}{2} + \frac {B a c x^{4}}{2} + \frac {B c^{2} x^{6}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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