Optimal. Leaf size=73 \[ -\frac {2 \sqrt {x} (A c+2 b B)}{3 b c^2 \sqrt {b x+c x^2}}-\frac {2 x^{5/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 73, 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} \[ -\frac {2 \sqrt {x} (A c+2 b B)}{3 b c^2 \sqrt {b x+c x^2}}-\frac {2 x^{5/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 648
Rule 788
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^{5/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {(2 b B+A c) \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b c}\\ &=-\frac {2 (b B-A c) x^{5/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {2 (2 b B+A c) \sqrt {x}}{3 b c^2 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.49 \[ -\frac {2 x^{3/2} (c (A+3 B x)+2 b B)}{3 c^2 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 56, normalized size = 0.77 \[ -\frac {2 \, {\left (3 \, B c x + 2 \, B b + A c\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3 \, {\left (c^{4} x^{3} + 2 \, b c^{3} x^{2} + b^{2} c^{2} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 45, normalized size = 0.62 \[ -\frac {2 \, {\left (3 \, {\left (c x + b\right )} B - B b + A c\right )}}{3 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{2}} + \frac {2 \, {\left (2 \, B b + A c\right )}}{3 \, b^{\frac {3}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 0.52 \[ -\frac {2 \left (c x +b \right ) \left (3 B c x +A c +2 b B \right ) x^{\frac {5}{2}}}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} c^{2}} \]
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 \[ \int \frac {{\left (B x + A\right )} x^{\frac {5}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x^{5/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{\frac {5}{2}} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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