Optimal. Leaf size=23 \[ -\frac {2 \left (b x+c x^2\right )^{5/2}}{5 b x^5} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {650} \begin {gather*} -\frac {2 \left (b x+c x^2\right )^{5/2}}{5 b x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 b x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2 (x (b+c x))^{5/2}}{5 b x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 39, normalized size = 1.70 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (b^2+2 b c x+c^2 x^2\right )}{5 b x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 35, normalized size = 1.52 \begin {gather*} -\frac {2 \, {\left (c^{2} x^{2} + 2 \, b c x + b^{2}\right )} \sqrt {c x^{2} + b x}}{5 \, b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 134, normalized size = 5.83 \begin {gather*} \frac {2 \, {\left (5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} c^{2} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c^{\frac {3}{2}} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} c + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} \sqrt {c} + b^{4}\right )}}{5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.09 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 73, normalized size = 3.17 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} c^{2}}{5 \, b x} + \frac {\sqrt {c x^{2} + b x} c}{5 \, x^{2}} + \frac {3 \, \sqrt {c x^{2} + b x} b}{5 \, x^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 26, normalized size = 1.13 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,{\left (b+c\,x\right )}^2}{5\,b\,x^3} \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 {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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