Optimal. Leaf size=35 \[ \frac{b (d x)^{m+2}}{d^2 (m+2)}+\frac{c (d x)^{m+3}}{d^3 (m+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0139718, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {14} \[ \frac{b (d x)^{m+2}}{d^2 (m+2)}+\frac{c (d x)^{m+3}}{d^3 (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rubi steps
\begin{align*} \int (d x)^m \left (b x+c x^2\right ) \, dx &=\int \left (\frac{b (d x)^{1+m}}{d}+\frac{c (d x)^{2+m}}{d^2}\right ) \, dx\\ &=\frac{b (d x)^{2+m}}{d^2 (2+m)}+\frac{c (d x)^{3+m}}{d^3 (3+m)}\\ \end{align*}
Mathematica [A] time = 0.0147581, size = 25, normalized size = 0.71 \[ x^2 (d x)^m \left (\frac{b}{m+2}+\frac{c x}{m+3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 35, normalized size = 1. \begin{align*}{\frac{ \left ( dx \right ) ^{m} \left ( cmx+bm+2\,cx+3\,b \right ){x}^{2}}{ \left ( 3+m \right ) \left ( 2+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15542, size = 45, normalized size = 1.29 \begin{align*} \frac{c d^{m} x^{3} x^{m}}{m + 3} + \frac{b d^{m} x^{2} x^{m}}{m + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.03065, size = 82, normalized size = 2.34 \begin{align*} \frac{{\left ({\left (c m + 2 \, c\right )} x^{3} +{\left (b m + 3 \, b\right )} x^{2}\right )} \left (d x\right )^{m}}{m^{2} + 5 \, m + 6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.79318, size = 112, normalized size = 3.2 \begin{align*} \begin{cases} \frac{- \frac{b}{x} + c \log{\left (x \right )}}{d^{3}} & \text{for}\: m = -3 \\\frac{b \log{\left (x \right )} + c x}{d^{2}} & \text{for}\: m = -2 \\\frac{b d^{m} m x^{2} x^{m}}{m^{2} + 5 m + 6} + \frac{3 b d^{m} x^{2} x^{m}}{m^{2} + 5 m + 6} + \frac{c d^{m} m x^{3} x^{m}}{m^{2} + 5 m + 6} + \frac{2 c d^{m} x^{3} x^{m}}{m^{2} + 5 m + 6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.33834, size = 76, normalized size = 2.17 \begin{align*} \frac{\left (d x\right )^{m} c m x^{3} + \left (d x\right )^{m} b m x^{2} + 2 \, \left (d x\right )^{m} c x^{3} + 3 \, \left (d x\right )^{m} b x^{2}}{m^{2} + 5 \, m + 6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]