Optimal. Leaf size=59 \[ -\frac {2 A b^2}{3 x^{3/2}}-\frac {2 b (2 A c+b B)}{\sqrt {x}}+2 c \sqrt {x} (A c+2 b B)+\frac {2}{3} B c^2 x^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {765} \begin {gather*} -\frac {2 A b^2}{3 x^{3/2}}-\frac {2 b (2 A c+b B)}{\sqrt {x}}+2 c \sqrt {x} (A c+2 b B)+\frac {2}{3} B c^2 x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{x^{9/2}} \, dx &=\int \left (\frac {A b^2}{x^{5/2}}+\frac {b (b B+2 A c)}{x^{3/2}}+\frac {c (2 b B+A c)}{\sqrt {x}}+B c^2 \sqrt {x}\right ) \, dx\\ &=-\frac {2 A b^2}{3 x^{3/2}}-\frac {2 b (b B+2 A c)}{\sqrt {x}}+2 c (2 b B+A c) \sqrt {x}+\frac {2}{3} B c^2 x^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.88 \begin {gather*} \frac {2 B x \left (-3 b^2+6 b c x+c^2 x^2\right )-2 A \left (b^2+6 b c x-3 c^2 x^2\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 54, normalized size = 0.92 \begin {gather*} \frac {2 \left (-A b^2-6 A b c x+3 A c^2 x^2-3 b^2 B x+6 b B c x^2+B c^2 x^3\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 50, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (B c^{2} x^{3} - A b^{2} + 3 \, {\left (2 \, B b c + A c^{2}\right )} x^{2} - 3 \, {\left (B b^{2} + 2 \, A b c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 51, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 4 \, B b c \sqrt {x} + 2 \, A c^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B b^{2} x + 6 \, A b c x + A b^{2}\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 51, normalized size = 0.86 \begin {gather*} -\frac {2 \left (-B \,c^{2} x^{3}-3 A \,c^{2} x^{2}-6 B b c \,x^{2}+6 A b c x +3 B \,b^{2} x +A \,b^{2}\right )}{3 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 51, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 2 \, {\left (2 \, B b c + A c^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A b^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.86 \begin {gather*} -\frac {6\,B\,b^2\,x+2\,A\,b^2-12\,B\,b\,c\,x^2+12\,A\,b\,c\,x-2\,B\,c^2\,x^3-6\,A\,c^2\,x^2}{3\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.96, size = 73, normalized size = 1.24 \begin {gather*} - \frac {2 A b^{2}}{3 x^{\frac {3}{2}}} - \frac {4 A b c}{\sqrt {x}} + 2 A c^{2} \sqrt {x} - \frac {2 B b^{2}}{\sqrt {x}} + 4 B b c \sqrt {x} + \frac {2 B c^{2} x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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