Optimal. Leaf size=32 \[ \frac{2 b x^{n+1} \sqrt{b x^n} (c x)^m}{2 m+3 n+2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0068314, antiderivative size = 32, normalized size of antiderivative = 1., 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} \[ \frac{2 b x^{n+1} \sqrt{b x^n} (c x)^m}{2 m+3 n+2} \]
Antiderivative was successfully verified.
[In]
[Out]
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*}
Mathematica [A] time = 0.0058748, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 26, normalized size = 0.8 \begin{align*} 2\,{\frac{x \left ( cx \right ) ^{m} \left ( b{x}^{n} \right ) ^{3/2}}{2+2\,m+3\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03371, size = 36, normalized size = 1.12 \begin{align*} \frac{2 \, b^{\frac{3}{2}} c^{m} x x^{m}{\left (x^{n}\right )}^{\frac{3}{2}}}{2 \, m + 3 \, n + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16829, size = 42, normalized size = 1.31 \begin{align*} \frac{2 \, b^{\frac{3}{2}} x x^{\frac{3}{2} \, n} e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{2 \, m + 3 \, n + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]