Optimal. Leaf size=32 \[ \frac {2 b x^{n+1} \sqrt {b x^n} (c x)^m}{2 m+3 n+2} \]
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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, 20, 30} \[ \frac {2 b x^{n+1} \sqrt {b x^n} (c x)^m}{2 m+3 n+2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 30
Rubi steps
\begin {align*} \int (c x)^m \left (b x^n\right )^{3/2} \, dx &=\left (b x^{-n/2} \sqrt {b x^n}\right ) \int x^{3 n/2} (c x)^m \, dx\\ &=\left (b x^{-m-\frac {n}{2}} (c x)^m \sqrt {b x^n}\right ) \int x^{m+\frac {3 n}{2}} \, dx\\ &=\frac {2 b x^{1+n} (c x)^m \sqrt {b x^n}}{2+2 m+3 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.81 \[ \frac {x \left (b x^n\right )^{3/2} (c x)^m}{m+\frac {3 n}{2}+1} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b x^{n}\right )^{\frac {3}{2}} \left (c x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.81 \[ \frac {2 \left (b \,x^{n}\right )^{\frac {3}{2}} x \left (c x \right )^{m}}{2 m +3 n +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 27, normalized size = 0.84 \[ \frac {2 \, b^{\frac {3}{2}} c^{m} x x^{m} {\left (x^{n}\right )}^{\frac {3}{2}}}{2 \, m + 3 \, n + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 30, normalized size = 0.94 \[ \frac {2\,b\,x^{n+1}\,\sqrt {b\,x^n}\,{\left (c\,x\right )}^m}{2\,m+3\,n+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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