Optimal. Leaf size=70 \[ \frac {2 \sqrt {b x+c x^2} (2 b B-A c)}{b c^2 \sqrt {x}}-\frac {2 x^{3/2} (b B-A c)}{b c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {788, 648} \begin {gather*} \frac {2 \sqrt {b x+c x^2} (2 b B-A c)}{b c^2 \sqrt {x}}-\frac {2 x^{3/2} (b B-A c)}{b c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 788
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b B-A c) x^{3/2}}{b c \sqrt {b x+c x^2}}-\frac {\left (2 \left (\frac {1}{2} (b B-2 A c)+\frac {3}{2} (-b B+A c)\right )\right ) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{b c}\\ &=-\frac {2 (b B-A c) x^{3/2}}{b c \sqrt {b x+c x^2}}+\frac {2 (2 b B-A c) \sqrt {b x+c x^2}}{b c^2 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {x} (-A c+2 b B+B c x)}{c^2 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 43, normalized size = 0.61 \begin {gather*} \frac {2 \sqrt {b x+c x^2} (-A c+2 b B+B c x)}{c^2 \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 45, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (B c x + 2 \, B b - A c\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{c^{3} x^{2} + b c^{2} x} \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 = 0.73 \begin {gather*} \frac {2 \, \sqrt {c x + b} B}{c^{2}} + \frac {2 \, {\left (B b - A c\right )}}{\sqrt {c x + b} c^{2}} - \frac {2 \, {\left (2 \, B b - A c\right )}}{\sqrt {b} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 0.54 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-B c x +A c -2 b B \right ) x^{\frac {3}{2}}}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} x^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{3/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \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 {x^{\frac {3}{2}} \left (A + B x\right )}{\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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