Optimal. Leaf size=75 \[ 2 a^2 A \sqrt {x}+\frac {2}{3} a^2 B x^{3/2}+\frac {4}{5} a A c x^{5/2}+\frac {4}{7} a B c x^{7/2}+\frac {2}{9} A c^2 x^{9/2}+\frac {2}{11} B c^2 x^{11/2} \]
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Rubi [A] time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {766} \[ 2 a^2 A \sqrt {x}+\frac {2}{3} a^2 B x^{3/2}+\frac {4}{5} a A c x^{5/2}+\frac {4}{7} a B c x^{7/2}+\frac {2}{9} A c^2 x^{9/2}+\frac {2}{11} B c^2 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {x}} \, dx &=\int \left (\frac {a^2 A}{\sqrt {x}}+a^2 B \sqrt {x}+2 a A c x^{3/2}+2 a B c x^{5/2}+A c^2 x^{7/2}+B c^2 x^{9/2}\right ) \, dx\\ &=2 a^2 A \sqrt {x}+\frac {2}{3} a^2 B x^{3/2}+\frac {4}{5} a A c x^{5/2}+\frac {4}{7} a B c x^{7/2}+\frac {2}{9} A c^2 x^{9/2}+\frac {2}{11} B c^2 x^{11/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.79 \[ \frac {2}{3} a^2 \sqrt {x} (3 A+B x)+\frac {4}{35} a c x^{5/2} (7 A+5 B x)+\frac {2}{99} c^2 x^{9/2} (11 A+9 B x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 53, normalized size = 0.71 \[ \frac {2}{3465} \, {\left (315 \, B c^{2} x^{5} + 385 \, A c^{2} x^{4} + 990 \, B a c x^{3} + 1386 \, A a c x^{2} + 1155 \, B a^{2} x + 3465 \, A a^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.71 \[ \frac {2}{11} \, B c^{2} x^{\frac {11}{2}} + \frac {2}{9} \, A c^{2} x^{\frac {9}{2}} + \frac {4}{7} \, B a c x^{\frac {7}{2}} + \frac {4}{5} \, A a c x^{\frac {5}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + 2 \, A a^{2} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 0.72 \[ \frac {2 \left (315 B \,c^{2} x^{5}+385 A \,c^{2} x^{4}+990 B a c \,x^{3}+1386 A a c \,x^{2}+1155 B \,a^{2} x +3465 A \,a^{2}\right ) \sqrt {x}}{3465} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 53, normalized size = 0.71 \[ \frac {2}{11} \, B c^{2} x^{\frac {11}{2}} + \frac {2}{9} \, A c^{2} x^{\frac {9}{2}} + \frac {4}{7} \, B a c x^{\frac {7}{2}} + \frac {4}{5} \, A a c x^{\frac {5}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + 2 \, A a^{2} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 53, normalized size = 0.71 \[ 2\,A\,a^2\,\sqrt {x}+\frac {2\,B\,a^2\,x^{3/2}}{3}+\frac {2\,A\,c^2\,x^{9/2}}{9}+\frac {2\,B\,c^2\,x^{11/2}}{11}+\frac {4\,A\,a\,c\,x^{5/2}}{5}+\frac {4\,B\,a\,c\,x^{7/2}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.44, size = 78, normalized size = 1.04 \[ 2 A a^{2} \sqrt {x} + \frac {4 A a c x^{\frac {5}{2}}}{5} + \frac {2 A c^{2} x^{\frac {9}{2}}}{9} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + \frac {4 B a c x^{\frac {7}{2}}}{7} + \frac {2 B c^{2} x^{\frac {11}{2}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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