Optimal. Leaf size=103 \[ -\frac {2 a^3 A}{\sqrt {x}}+2 a^3 B \sqrt {x}+2 a^2 A c x^{3/2}+\frac {6}{5} a^2 B c x^{5/2}+\frac {6}{7} a A c^2 x^{7/2}+\frac {2}{3} a B c^2 x^{9/2}+\frac {2}{11} A c^3 x^{11/2}+\frac {2}{13} B c^3 x^{13/2} \]
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Rubi [A] time = 0.04, antiderivative size = 103, 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} \[ 2 a^2 A c x^{3/2}-\frac {2 a^3 A}{\sqrt {x}}+\frac {6}{5} a^2 B c x^{5/2}+2 a^3 B \sqrt {x}+\frac {6}{7} a A c^2 x^{7/2}+\frac {2}{3} a B c^2 x^{9/2}+\frac {2}{11} A c^3 x^{11/2}+\frac {2}{13} B c^3 x^{13/2} \]
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 )^3}{x^{3/2}} \, dx &=\int \left (\frac {a^3 A}{x^{3/2}}+\frac {a^3 B}{\sqrt {x}}+3 a^2 A c \sqrt {x}+3 a^2 B c x^{3/2}+3 a A c^2 x^{5/2}+3 a B c^2 x^{7/2}+A c^3 x^{9/2}+B c^3 x^{11/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{\sqrt {x}}+2 a^3 B \sqrt {x}+2 a^2 A c x^{3/2}+\frac {6}{5} a^2 B c x^{5/2}+\frac {6}{7} a A c^2 x^{7/2}+\frac {2}{3} a B c^2 x^{9/2}+\frac {2}{11} A c^3 x^{11/2}+\frac {2}{13} B c^3 x^{13/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 80, normalized size = 0.78 \[ \frac {2 a^3 (B x-A)}{\sqrt {x}}+\frac {2}{5} a^2 c x^{3/2} (5 A+3 B x)+\frac {2}{21} a c^2 x^{7/2} (9 A+7 B x)+\frac {2}{143} c^3 x^{11/2} (13 A+11 B x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 77, normalized size = 0.75 \[ \frac {2 \, {\left (1155 \, B c^{3} x^{7} + 1365 \, A c^{3} x^{6} + 5005 \, B a c^{2} x^{5} + 6435 \, A a c^{2} x^{4} + 9009 \, B a^{2} c x^{3} + 15015 \, A a^{2} c x^{2} + 15015 \, B a^{3} x - 15015 \, A a^{3}\right )}}{15015 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 77, normalized size = 0.75 \[ \frac {2}{13} \, B c^{3} x^{\frac {13}{2}} + \frac {2}{11} \, A c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B a c^{2} x^{\frac {9}{2}} + \frac {6}{7} \, A a c^{2} x^{\frac {7}{2}} + \frac {6}{5} \, B a^{2} c x^{\frac {5}{2}} + 2 \, A a^{2} c x^{\frac {3}{2}} + 2 \, B a^{3} \sqrt {x} - \frac {2 \, A a^{3}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 0.76 \[ -\frac {2 \left (-1155 B \,c^{3} x^{7}-1365 A \,c^{3} x^{6}-5005 B a \,c^{2} x^{5}-6435 A a \,c^{2} x^{4}-9009 B \,a^{2} c \,x^{3}-15015 A \,a^{2} c \,x^{2}-15015 B \,a^{3} x +15015 A \,a^{3}\right )}{15015 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 77, normalized size = 0.75 \[ \frac {2}{13} \, B c^{3} x^{\frac {13}{2}} + \frac {2}{11} \, A c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B a c^{2} x^{\frac {9}{2}} + \frac {6}{7} \, A a c^{2} x^{\frac {7}{2}} + \frac {6}{5} \, B a^{2} c x^{\frac {5}{2}} + 2 \, A a^{2} c x^{\frac {3}{2}} + 2 \, B a^{3} \sqrt {x} - \frac {2 \, A a^{3}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 77, normalized size = 0.75 \[ 2\,B\,a^3\,\sqrt {x}-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,A\,c^3\,x^{11/2}}{11}+\frac {2\,B\,c^3\,x^{13/2}}{13}+2\,A\,a^2\,c\,x^{3/2}+\frac {6\,A\,a\,c^2\,x^{7/2}}{7}+\frac {6\,B\,a^2\,c\,x^{5/2}}{5}+\frac {2\,B\,a\,c^2\,x^{9/2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.26, size = 109, normalized size = 1.06 \[ - \frac {2 A a^{3}}{\sqrt {x}} + 2 A a^{2} c x^{\frac {3}{2}} + \frac {6 A a c^{2} x^{\frac {7}{2}}}{7} + \frac {2 A c^{3} x^{\frac {11}{2}}}{11} + 2 B a^{3} \sqrt {x} + \frac {6 B a^{2} c x^{\frac {5}{2}}}{5} + \frac {2 B a c^{2} x^{\frac {9}{2}}}{3} + \frac {2 B c^{3} x^{\frac {13}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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