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 = 0.0398129, 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 \[ \frac{b (d x)^{m+2}}{d^2 (m+2)}+\frac{c (d x)^{m+3}}{d^3 (m+3)} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*(b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.34904, size = 29, normalized size = 0.83 \[ \frac{b \left (d x\right )^{m + 2}}{d^{2} \left (m + 2\right )} + \frac{c \left (d x\right )^{m + 3}}{d^{3} \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(c*x**2+b*x),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0237165, size = 27, normalized size = 0.77 \[ (d x)^m \left (\frac{b x^2}{m+2}+\frac{c x^3}{m+3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*(b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 35, normalized size = 1. \[{\frac{ \left ( dx \right ) ^{m} \left ( cmx+bm+2\,cx+3\,b \right ){x}^{2}}{ \left ( 3+m \right ) \left ( 2+m \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(c*x^2+b*x),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.711024, size = 45, normalized size = 1.29 \[ \frac{c d^{m} x^{3} x^{m}}{m + 3} + \frac{b d^{m} x^{2} x^{m}}{m + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(d*x)^m,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.232397, size = 53, normalized size = 1.51 \[ \frac{{\left ({\left (c m + 2 \, c\right )} x^{3} +{\left (b m + 3 \, b\right )} x^{2}\right )} \left (d x\right )^{m}}{m^{2} + 5 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(d*x)^m,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.13931, size = 112, normalized size = 3.2 \[ \begin{cases} \frac{- \frac{b}{x} + c \log{\left (x \right )}}{d^{3}} & \text{for}\: m = -3 \\\frac{b \log{\left (x \right )} + c x}{d^{2}} & \text{for}\: m = -2 \\\frac{b d^{m} m x^{2} x^{m}}{m^{2} + 5 m + 6} + \frac{3 b d^{m} x^{2} x^{m}}{m^{2} + 5 m + 6} + \frac{c d^{m} m x^{3} x^{m}}{m^{2} + 5 m + 6} + \frac{2 c d^{m} x^{3} x^{m}}{m^{2} + 5 m + 6} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(c*x**2+b*x),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.210961, size = 86, normalized size = 2.46 \[ \frac{c m x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + b m x^{2} e^{\left (m{\rm ln}\left (d x\right )\right )} + 2 \, c x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 3 \, b x^{2} e^{\left (m{\rm ln}\left (d x\right )\right )}}{m^{2} + 5 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(d*x)^m,x, algorithm="giac")
[Out]