Optimal. Leaf size=67 \[ \frac {2 x (2 A c+b B)}{3 b^2 c \sqrt {b x+c x^2}}-\frac {2 x^2 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {788, 636} \[ \frac {2 x (2 A c+b B)}{3 b^2 c \sqrt {b x+c x^2}}-\frac {2 x^2 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 636
Rule 788
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^2}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {(b B+2 A c) \int \frac {x}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b c}\\ &=-\frac {2 (b B-A c) x^2}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (b B+2 A c) x}{3 b^2 c \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.52 \[ \frac {2 x^2 (3 A b+2 A c x+b B x)}{3 b^2 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 51, normalized size = 0.76 \[ \frac {2 \, \sqrt {c x^{2} + b x} {\left (3 \, A b + {\left (B b + 2 \, A c\right )} x\right )}}{3 \, {\left (b^{2} c^{2} x^{2} + 2 \, b^{3} c x + b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 119, normalized size = 1.78 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b \sqrt {c} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A c^{\frac {3}{2}} + B b^{2} + 2 \, A b c\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b\right )}^{3} c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 0.58 \[ \frac {2 \left (c x +b \right ) \left (2 A c x +B b x +3 A b \right ) x^{3}}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 134, normalized size = 2.00 \[ -\frac {B x^{2}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {4 \, A x}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {B b x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c^{2}} - \frac {2 \, A x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} c} + \frac {2 \, B x}{3 \, \sqrt {c x^{2} + b x} b c} + \frac {B}{3 \, \sqrt {c x^{2} + b x} c^{2}} + \frac {2 \, A}{3 \, \sqrt {c x^{2} + b x} b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 37, normalized size = 0.55 \[ \frac {2\,\sqrt {c\,x^2+b\,x}\,\left (3\,A\,b+2\,A\,c\,x+B\,b\,x\right )}{3\,b^2\,{\left (b+c\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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