Optimal. Leaf size=28 \[ \frac {\sqrt {b x^2} (c x)^{m+2}}{c^2 (m+2) x} \]
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Rubi [A] time = 0.01, antiderivative size = 28, 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 {\sqrt {b x^2} (c x)^{m+2}}{c^2 (m+2) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 32
Rubi steps
\begin {align*} \int (c x)^m \sqrt {b x^2} \, dx &=\frac {\sqrt {b x^2} \int x (c x)^m \, dx}{x}\\ &=\frac {\sqrt {b x^2} \int (c x)^{1+m} \, dx}{c x}\\ &=\frac {(c x)^{2+m} \sqrt {b x^2}}{c^2 (2+m) x}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 0.75 \begin {gather*} \frac {x \sqrt {b x^2} (c x)^m}{m+2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int (c x)^m \sqrt {b x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.64, size = 19, normalized size = 0.68 \begin {gather*} \frac {\sqrt {b x^{2}} \left (c x\right )^{m} x}{m + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.71 \begin {gather*} \frac {\sqrt {b \,x^{2}}\, x \left (c x \right )^{m}}{m +2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 18, normalized size = 0.64 \begin {gather*} \frac {\sqrt {b} c^{m} x^{2} x^{m}}{m + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 20, normalized size = 0.71 \begin {gather*} \frac {\sqrt {b}\,x\,{\left (c\,x\right )}^m\,\sqrt {x^2}}{m+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 {\sqrt {b} c^{m} x x^{m} \sqrt {x^{2}}}{m + 2} & \text {for}\: m \neq -2 \\\frac {\int \frac {\sqrt {b x^{2}}}{x^{2}}\, dx}{c^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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