Optimal. Leaf size=51 \[ -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{3 x^{3/2}}+2 \sqrt {x} (A c+b B)+\frac {2}{3} B c x^{3/2} \]
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Rubi [A] time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \[ -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{3 x^{3/2}}+2 \sqrt {x} (A c+b B)+\frac {2}{3} B c x^{3/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac {a A}{x^{5/2}}+\frac {A b+a B}{x^{3/2}}+\frac {b B+A c}{\sqrt {x}}+B c \sqrt {x}\right ) \, dx\\ &=-\frac {2 a A}{3 x^{3/2}}-\frac {2 (A b+a B)}{\sqrt {x}}+2 (b B+A c) \sqrt {x}+\frac {2}{3} B c x^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 42, normalized size = 0.82 \[ \frac {2 x (B x (3 b+c x)-3 A (b-c x))-2 a (A+3 B x)}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 38, normalized size = 0.75 \[ \frac {2 \, {\left (B c x^{3} + 3 \, {\left (B b + A c\right )} x^{2} - A a - 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 41, normalized size = 0.80 \[ \frac {2}{3} \, B c x^{\frac {3}{2}} + 2 \, B b \sqrt {x} + 2 \, A c \sqrt {x} - \frac {2 \, {\left (3 \, B a x + 3 \, A b x + A a\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 41, normalized size = 0.80 \[ -\frac {2 \left (-B c \,x^{3}-3 A c \,x^{2}-3 B b \,x^{2}+3 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 39, normalized size = 0.76 \[ \frac {2}{3} \, B c x^{\frac {3}{2}} + 2 \, {\left (B b + A c\right )} \sqrt {x} - \frac {2 \, {\left (A a + 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 41, normalized size = 0.80 \[ -\frac {2\,A\,a+6\,A\,b\,x+6\,B\,a\,x-6\,A\,c\,x^2-6\,B\,b\,x^2-2\,B\,c\,x^3}{3\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 63, normalized size = 1.24 \[ - \frac {2 A a}{3 x^{\frac {3}{2}}} - \frac {2 A b}{\sqrt {x}} + 2 A c \sqrt {x} - \frac {2 B a}{\sqrt {x}} + 2 B b \sqrt {x} + \frac {2 B c x^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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