Optimal. Leaf size=32 \[ -\frac {c^2 x (c x)^{-2+m}}{b (2-m) \sqrt {b x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 16, 32}
\begin {gather*} -\frac {c^2 x (c x)^{m-2}}{b (2-m) \sqrt {b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 32
Rubi steps
\begin {align*} \int \frac {(c x)^m}{\left (b x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(c x)^m}{x^3} \, dx}{b \sqrt {b x^2}}\\ &=\frac {\left (c^3 x\right ) \int (c x)^{-3+m} \, dx}{b \sqrt {b x^2}}\\ &=-\frac {c^2 x (c x)^{-2+m}}{b (2-m) \sqrt {b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.66 \begin {gather*} \frac {x (c x)^m}{(-2+m) \left (b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 20, normalized size = 0.62
method | result | size |
gosper | \(\frac {x \left (c x \right )^{m}}{\left (-2+m \right ) \left (b \,x^{2}\right )^{\frac {3}{2}}}\) | \(20\) |
risch | \(\frac {\left (c x \right )^{m}}{b x \sqrt {b \,x^{2}}\, \left (-2+m \right )}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 0.56 \begin {gather*} \frac {c^{m} x^{m}}{b^{\frac {3}{2}} {\left (m - 2\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 29, normalized size = 0.91 \begin {gather*} \frac {\sqrt {b x^{2}} \left (c x\right )^{m}}{{\left (b^{2} m - 2 \, b^{2}\right )} 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*} \begin {cases} \frac {x \left (c x\right )^{m}}{m \left (b x^{2}\right )^{\frac {3}{2}} - 2 \left (b x^{2}\right )^{\frac {3}{2}}} & \text {for}\: m \neq 2 \\c^{2} \int \frac {x^{2}}{\left (b x^{2}\right )^{\frac {3}{2}}}\, dx & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.00, size = 22, normalized size = 0.69 \begin {gather*} \frac {{\left (c\,x\right )}^m}{b^{3/2}\,x\,\left (m-2\right )\,\sqrt {x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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