Optimal. Leaf size=29 \[ \frac {b \sqrt {b x^2} (c x)^{m+4}}{c^4 (m+4) x} \]
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Rubi [A] time = 0.01, antiderivative size = 29, 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} \[ \frac {b \sqrt {b x^2} (c x)^{m+4}}{c^4 (m+4) x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 32
Rubi steps
\begin {align*} \int (c x)^m \left (b x^2\right )^{3/2} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int x^3 (c x)^m \, dx}{x}\\ &=\frac {\left (b \sqrt {b x^2}\right ) \int (c x)^{3+m} \, dx}{c^3 x}\\ &=\frac {b (c x)^{4+m} \sqrt {b x^2}}{c^4 (4+m) x}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 0.72 \[ \frac {x \left (b x^2\right )^{3/2} (c x)^m}{m+4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 22, normalized size = 0.76 \[ \frac {\sqrt {b x^{2}} \left (c x\right )^{m} b x^{3}}{m + 4} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.69 \[ \frac {\left (b \,x^{2}\right )^{\frac {3}{2}} x \left (c x \right )^{m}}{m +4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 18, normalized size = 0.62 \[ \frac {b^{\frac {3}{2}} c^{m} x^{4} x^{m}}{m + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 22, normalized size = 0.76 \[ \frac {b^{3/2}\,x^3\,{\left (c\,x\right )}^m\,\sqrt {x^2}}{m+4} \]
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 \[ \begin {cases} \frac {b^{\frac {3}{2}} c^{m} x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m + 4} & \text {for}\: m \neq -4 \\\frac {\int \frac {\left (b x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx}{c^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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