Optimal. Leaf size=29 \[ \frac{2 \sqrt{b x^n} (c x)^{m+1}}{c (2 m+n+2)} \]
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Rubi [A] time = 0.005874, antiderivative size = 25, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 20, 30} \[ \frac{2 x \sqrt{b x^n} (c x)^m}{2 m+n+2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 30
Rubi steps
\begin{align*} \int (c x)^m \sqrt{b x^n} \, dx &=\left (x^{-n/2} \sqrt{b x^n}\right ) \int x^{n/2} (c x)^m \, dx\\ &=\left (x^{-m-\frac{n}{2}} (c x)^m \sqrt{b x^n}\right ) \int x^{m+\frac{n}{2}} \, dx\\ &=\frac{2 x (c x)^m \sqrt{b x^n}}{2+2 m+n}\\ \end{align*}
Mathematica [A] time = 0.004673, size = 26, normalized size = 0.9 \[ \frac{x \sqrt{b x^n} (c x)^m}{m+\frac{n}{2}+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 24, normalized size = 0.8 \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m}\sqrt{b{x}^{n}}}{2+2\,m+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00467, size = 34, normalized size = 1.17 \begin{align*} \frac{2 \, \sqrt{b} c^{m} x x^{m} \sqrt{x^{n}}}{2 \, m + n + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18557, size = 39, normalized size = 1.34 \begin{align*} \frac{2 \, \sqrt{b} x x^{\frac{1}{2} \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{2 \, m + n + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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