Optimal. Leaf size=73 \[ a^4 A x+\frac {4}{3} a^3 A c x^3+\frac {6}{5} a^2 A c^2 x^5+\frac {4}{7} a A c^3 x^7+\frac {B \left (a+c x^2\right )^5}{10 c}+\frac {1}{9} A c^4 x^9 \]
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Rubi [A] time = 0.03, antiderivative size = 73, 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} \[ \frac {6}{5} a^2 A c^2 x^5+\frac {4}{3} a^3 A c x^3+a^4 A x+\frac {4}{7} a A c^3 x^7+\frac {B \left (a+c x^2\right )^5}{10 c}+\frac {1}{9} A c^4 x^9 \]
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 )^4 \, dx &=\frac {B \left (a+c x^2\right )^5}{10 c}+A \int \left (a+c x^2\right )^4 \, dx\\ &=\frac {B \left (a+c x^2\right )^5}{10 c}+A \int \left (a^4+4 a^3 c x^2+6 a^2 c^2 x^4+4 a c^3 x^6+c^4 x^8\right ) \, dx\\ &=a^4 A x+\frac {4}{3} a^3 A c x^3+\frac {6}{5} a^2 A c^2 x^5+\frac {4}{7} a A c^3 x^7+\frac {1}{9} A c^4 x^9+\frac {B \left (a+c x^2\right )^5}{10 c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 110, normalized size = 1.51 \[ a^4 A x+\frac {1}{2} a^4 B x^2+\frac {4}{3} a^3 A c x^3+a^3 B c x^4+\frac {6}{5} a^2 A c^2 x^5+a^2 B c^2 x^6+\frac {4}{7} a A c^3 x^7+\frac {1}{2} a B c^3 x^8+\frac {1}{9} A c^4 x^9+\frac {1}{10} B c^4 x^{10} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 96, normalized size = 1.32 \[ \frac {1}{10} x^{10} c^{4} B + \frac {1}{9} x^{9} c^{4} A + \frac {1}{2} x^{8} c^{3} a B + \frac {4}{7} x^{7} c^{3} a A + x^{6} c^{2} a^{2} B + \frac {6}{5} x^{5} c^{2} a^{2} A + x^{4} c a^{3} B + \frac {4}{3} x^{3} c a^{3} A + \frac {1}{2} x^{2} a^{4} B + x a^{4} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 96, normalized size = 1.32 \[ \frac {1}{10} \, B c^{4} x^{10} + \frac {1}{9} \, A c^{4} x^{9} + \frac {1}{2} \, B a c^{3} x^{8} + \frac {4}{7} \, A a c^{3} x^{7} + B a^{2} c^{2} x^{6} + \frac {6}{5} \, A a^{2} c^{2} x^{5} + B a^{3} c x^{4} + \frac {4}{3} \, A a^{3} c x^{3} + \frac {1}{2} \, B a^{4} x^{2} + A a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 97, normalized size = 1.33 \[ \frac {1}{10} B \,c^{4} x^{10}+\frac {1}{9} A \,c^{4} x^{9}+\frac {1}{2} B a \,c^{3} x^{8}+\frac {4}{7} A a \,c^{3} x^{7}+B \,a^{2} c^{2} x^{6}+\frac {6}{5} A \,a^{2} c^{2} x^{5}+B \,a^{3} c \,x^{4}+\frac {4}{3} A \,a^{3} c \,x^{3}+\frac {1}{2} B \,a^{4} x^{2}+A \,a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 96, normalized size = 1.32 \[ \frac {1}{10} \, B c^{4} x^{10} + \frac {1}{9} \, A c^{4} x^{9} + \frac {1}{2} \, B a c^{3} x^{8} + \frac {4}{7} \, A a c^{3} x^{7} + B a^{2} c^{2} x^{6} + \frac {6}{5} \, A a^{2} c^{2} x^{5} + B a^{3} c x^{4} + \frac {4}{3} \, A a^{3} c x^{3} + \frac {1}{2} \, B a^{4} x^{2} + A a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 96, normalized size = 1.32 \[ \frac {B\,a^4\,x^2}{2}+A\,a^4\,x+B\,a^3\,c\,x^4+\frac {4\,A\,a^3\,c\,x^3}{3}+B\,a^2\,c^2\,x^6+\frac {6\,A\,a^2\,c^2\,x^5}{5}+\frac {B\,a\,c^3\,x^8}{2}+\frac {4\,A\,a\,c^3\,x^7}{7}+\frac {B\,c^4\,x^{10}}{10}+\frac {A\,c^4\,x^9}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 112, normalized size = 1.53 \[ A a^{4} x + \frac {4 A a^{3} c x^{3}}{3} + \frac {6 A a^{2} c^{2} x^{5}}{5} + \frac {4 A a c^{3} x^{7}}{7} + \frac {A c^{4} x^{9}}{9} + \frac {B a^{4} x^{2}}{2} + B a^{3} c x^{4} + B a^{2} c^{2} x^{6} + \frac {B a c^{3} x^{8}}{2} + \frac {B c^{4} x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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