Optimal. Leaf size=23 \[ -\frac {2 \left (b x+c x^2\right )^{3/2}}{3 b x^3} \]
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Rubi [A]
time = 0.00, 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 = {664}
\begin {gather*} -\frac {2 \left (b x+c x^2\right )^{3/2}}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{x^3} \, dx &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{3 b x^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2 (x (b+c x))^{3/2}}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 20, normalized size = 0.87
method | result | size |
default | \(-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 b \,x^{3}}\) | \(20\) |
gosper | \(-\frac {2 \left (c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 x^{2} b}\) | \(25\) |
trager | \(-\frac {2 \left (c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 x^{2} b}\) | \(25\) |
risch | \(-\frac {2 \left (c x +b \right )^{2}}{3 x \sqrt {x \left (c x +b \right )}\, b}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 37, normalized size = 1.61 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} c}{3 \, b x} - \frac {2 \, \sqrt {c x^{2} + b x}}{3 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.97, size = 24, normalized size = 1.04 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} {\left (c x + b\right )}}{3 \, b x^{2}} \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 {\sqrt {x \left (b + c x\right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (19) = 38\).
time = 2.33, size = 76, normalized size = 3.30 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b \sqrt {c} + b^{2}\right )}}{3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 24, normalized size = 1.04 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (b+c\,x\right )}{3\,b\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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