Optimal. Leaf size=23 \[ -\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \]
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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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {662}
\begin {gather*} -\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2 \sqrt {x}}{c \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 25, normalized size = 1.09
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) x^{\frac {3}{2}}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\) | \(25\) |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}}{\sqrt {x}\, \left (c x +b \right ) c}\) | \(25\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.60, size = 30, normalized size = 1.30 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} \sqrt {x}}{c^{2} x^{2} + b c 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 (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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Giac [A]
time = 1.40, size = 21, normalized size = 0.91 \begin {gather*} -\frac {2}{\sqrt {c x + b} c} + \frac {2}{\sqrt {b} c} \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 = -0.04 \begin {gather*} \int \frac {x^{3/2}}{{\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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