Optimal. Leaf size=54 \[ \frac {16 c (b+2 c x)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (b+2 c x)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {614, 613} \[ \frac {16 c (b+2 c x)}{3 b^4 \sqrt {b x+c x^2}}-\frac {2 (b+2 c x)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rubi steps
\begin {align*} \int \frac {1}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b+2 c x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {(8 c) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (b+2 c x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {16 c (b+2 c x)}{3 b^4 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 48, normalized size = 0.89 \[ \frac {-2 b^3+12 b^2 c x+48 b c^2 x^2+32 c^3 x^3}{3 b^4 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 72, normalized size = 1.33 \[ \frac {2 \, {\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} + 6 \, b^{2} c x - b^{3}\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 50, normalized size = 0.93 \[ \frac {2 \, {\left (2 \, {\left (4 \, x {\left (\frac {2 \, c^{3} x}{b^{4}} + \frac {3 \, c^{2}}{b^{3}}\right )} + \frac {3 \, c}{b^{2}}\right )} x - \frac {1}{b}\right )}}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 0.94 \[ -\frac {2 \left (c x +b \right ) \left (-16 x^{3} c^{3}-24 b \,x^{2} c^{2}-6 b^{2} x c +b^{3}\right ) x}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 72, normalized size = 1.33 \[ -\frac {4 \, c x}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {32 \, c^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {2}{3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b} + \frac {16 \, c}{3 \, \sqrt {c x^{2} + b x} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 43, normalized size = 0.80 \[ \frac {\left (2\,b+4\,c\,x\right )\,\left (-b^2+8\,b\,c\,x+8\,c^2\,x^2\right )}{3\,b^4\,{\left (c\,x^2+b\,x\right )}^{3/2}} \]
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 \[ \int \frac {1}{\left (b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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