Optimal. Leaf size=60 \[ -\frac {2 (b+2 c x) (3 b B-4 A c)}{3 b^3 \sqrt {b x+c x^2}}-\frac {2 A}{3 b x \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 60, 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 = {792, 613} \begin {gather*} -\frac {2 (b+2 c x) (3 b B-4 A c)}{3 b^3 \sqrt {b x+c x^2}}-\frac {2 A}{3 b x \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 A}{3 b x \sqrt {b x+c x^2}}+\frac {\left (2 \left (b B-A c+\frac {1}{2} (b B-2 A c)\right )\right ) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {2 A}{3 b x \sqrt {b x+c x^2}}-\frac {2 (3 b B-4 A c) (b+2 c x)}{3 b^3 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.87 \begin {gather*} -\frac {2 \left (A \left (b^2-4 b c x-8 c^2 x^2\right )+3 b B x (b+2 c x)\right )}{3 b^3 x \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 66, normalized size = 1.10 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (A b^2-4 A b c x-8 A c^2 x^2+3 b^2 B x+6 b B c x^2\right )}{3 b^3 x^2 (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 68, normalized size = 1.13 \begin {gather*} -\frac {2 \, {\left (A b^{2} + 2 \, {\left (3 \, B b c - 4 \, A c^{2}\right )} x^{2} + {\left (3 \, B b^{2} - 4 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{3} c x^{3} + b^{4} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 58, normalized size = 0.97 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-8 A \,c^{2} x^{2}+6 B b c \,x^{2}-4 A b c x +3 B \,b^{2} x +A \,b^{2}\right )}{3 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 96, normalized size = 1.60 \begin {gather*} -\frac {4 \, B c x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {16 \, A c^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, B}{\sqrt {c x^{2} + b x} b} + \frac {8 \, A c}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, A}{3 \, \sqrt {c x^{2} + b x} b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 62, normalized size = 1.03 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (3\,B\,b^2\,x+A\,b^2+6\,B\,b\,c\,x^2-4\,A\,b\,c\,x-8\,A\,c^2\,x^2\right )}{3\,b^3\,x^2\,\left (b+c\,x\right )} \end {gather*}
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 \begin {gather*} \int \frac {A + B x}{x \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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