Optimal. Leaf size=73 \[ -\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a^2 B}{3 x^{3/2}}-\frac {4 a A c}{\sqrt {x}}+4 a B c \sqrt {x}+\frac {2}{3} A c^2 x^{3/2}+\frac {2}{5} B c^2 x^{5/2} \]
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Rubi [A] time = 0.03, antiderivative size = 73, 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} \[ -\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a^2 B}{3 x^{3/2}}-\frac {4 a A c}{\sqrt {x}}+4 a B c \sqrt {x}+\frac {2}{3} A c^2 x^{3/2}+\frac {2}{5} B c^2 x^{5/2} \]
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}{x^{7/2}} \, dx &=\int \left (\frac {a^2 A}{x^{7/2}}+\frac {a^2 B}{x^{5/2}}+\frac {2 a A c}{x^{3/2}}+\frac {2 a B c}{\sqrt {x}}+A c^2 \sqrt {x}+B c^2 x^{3/2}\right ) \, dx\\ &=-\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a^2 B}{3 x^{3/2}}-\frac {4 a A c}{\sqrt {x}}+4 a B c \sqrt {x}+\frac {2}{3} A c^2 x^{3/2}+\frac {2}{5} B c^2 x^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.73 \[ \frac {-2 a^2 (3 A+5 B x)+60 a c x^2 (B x-A)+2 c^2 x^4 (5 A+3 B x)}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 53, normalized size = 0.73 \[ \frac {2 \, {\left (3 \, B c^{2} x^{5} + 5 \, A c^{2} x^{4} + 30 \, B a c x^{3} - 30 \, A a c x^{2} - 5 \, B a^{2} x - 3 \, A a^{2}\right )}}{15 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 54, normalized size = 0.74 \[ \frac {2}{5} \, B c^{2} x^{\frac {5}{2}} + \frac {2}{3} \, A c^{2} x^{\frac {3}{2}} + 4 \, B a c \sqrt {x} - \frac {2 \, {\left (30 \, A a c x^{2} + 5 \, B a^{2} x + 3 \, A a^{2}\right )}}{15 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 0.74 \[ -\frac {2 \left (-3 B \,c^{2} x^{5}-5 A \,c^{2} x^{4}-30 B a c \,x^{3}+30 A a c \,x^{2}+5 B \,a^{2} x +3 A \,a^{2}\right )}{15 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 54, normalized size = 0.74 \[ \frac {2}{5} \, B c^{2} x^{\frac {5}{2}} + \frac {2}{3} \, A c^{2} x^{\frac {3}{2}} + 4 \, B a c \sqrt {x} - \frac {2 \, {\left (30 \, A a c x^{2} + 5 \, B a^{2} x + 3 \, A a^{2}\right )}}{15 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 54, normalized size = 0.74 \[ \frac {2\,A\,c^2\,x^{3/2}}{3}-\frac {\frac {2\,B\,a^2\,x}{3}+\frac {2\,A\,a^2}{5}+4\,A\,c\,a\,x^2}{x^{5/2}}+\frac {2\,B\,c^2\,x^{5/2}}{5}+4\,B\,a\,c\,\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.90, size = 76, normalized size = 1.04 \[ - \frac {2 A a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 A a c}{\sqrt {x}} + \frac {2 A c^{2} x^{\frac {3}{2}}}{3} - \frac {2 B a^{2}}{3 x^{\frac {3}{2}}} + 4 B a c \sqrt {x} + \frac {2 B c^{2} x^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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