Optimal. Leaf size=19 \[ \frac {x (c x)^m}{m \sqrt {b x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 19, 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 {x (c x)^m}{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}{\sqrt {b x^2}} \, dx &=\frac {x \int \frac {(c x)^m}{x} \, dx}{\sqrt {b x^2}}\\ &=\frac {(c x) \int (c x)^{-1+m} \, dx}{\sqrt {b x^2}}\\ &=\frac {x (c x)^m}{m \sqrt {b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {x (c x)^m}{m \sqrt {b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 24, normalized size = 1.26 \begin {gather*} \frac {\sqrt {b x^2} (c x)^m}{b m x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 22, normalized size = 1.16 \begin {gather*} \frac {\sqrt {b x^{2}} \left (c x\right )^{m}}{b m x} \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*} \int \frac {\left (c x\right )^{m}}{\sqrt {b x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.95 \begin {gather*} \frac {x \left (c x \right )^{m}}{\sqrt {b \,x^{2}}\, m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 13, normalized size = 0.68 \begin {gather*} \frac {c^{m} x^{m}}{\sqrt {b} m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 18, normalized size = 0.95 \begin {gather*} \frac {x\,{\left (c\,x\right )}^m}{\sqrt {b}\,m\,\sqrt {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*} \begin {cases} \frac {c^{m} x x^{m}}{\sqrt {b} m \sqrt {x^{2}}} & \text {for}\: m \neq 0 \\\int \frac {1}{\sqrt {b x^{2}}}\, dx & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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