Optimal. Leaf size=55 \[ \frac {1}{2} b d^2 x^2+\frac {1}{3} d (c d+2 b e) x^3+\frac {1}{4} e (2 c d+b e) x^4+\frac {1}{5} c e^2 x^5 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {712}
\begin {gather*} \frac {1}{4} e x^4 (b e+2 c d)+\frac {1}{3} d x^3 (2 b e+c d)+\frac {1}{2} b d^2 x^2+\frac {1}{5} c e^2 x^5 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 712
Rubi steps
\begin {align*} \int (d+e x)^2 \left (b x+c x^2\right ) \, dx &=\int \left (b d^2 x+d (c d+2 b e) x^2+e (2 c d+b e) x^3+c e^2 x^4\right ) \, dx\\ &=\frac {1}{2} b d^2 x^2+\frac {1}{3} d (c d+2 b e) x^3+\frac {1}{4} e (2 c d+b e) x^4+\frac {1}{5} c e^2 x^5\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 49, normalized size = 0.89 \begin {gather*} \frac {1}{60} x^2 \left (30 b d^2+20 d (c d+2 b e) x+15 e (2 c d+b e) x^2+12 c e^2 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 52, normalized size = 0.95
method | result | size |
gosper | \(\frac {x^{2} \left (12 c \,e^{2} x^{3}+15 b \,e^{2} x^{2}+30 c d e \,x^{2}+40 b d e x +20 c \,d^{2} x +30 b \,d^{2}\right )}{60}\) | \(52\) |
default | \(\frac {c \,e^{2} x^{5}}{5}+\frac {\left (b \,e^{2}+2 d e c \right ) x^{4}}{4}+\frac {\left (2 b d e +c \,d^{2}\right ) x^{3}}{3}+\frac {b \,d^{2} x^{2}}{2}\) | \(52\) |
norman | \(\frac {c \,e^{2} x^{5}}{5}+\left (\frac {1}{4} b \,e^{2}+\frac {1}{2} d e c \right ) x^{4}+\left (\frac {2}{3} b d e +\frac {1}{3} c \,d^{2}\right ) x^{3}+\frac {b \,d^{2} x^{2}}{2}\) | \(52\) |
risch | \(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{4} x^{4} b \,e^{2}+\frac {1}{2} x^{4} d e c +\frac {2}{3} x^{3} b d e +\frac {1}{3} x^{3} c \,d^{2}+\frac {1}{2} b \,d^{2} x^{2}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, b d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, c d e + b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{2} + 2 \, b d e\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.55, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{2} \, b d^{2} x^{2} + \frac {1}{20} \, {\left (4 \, c x^{5} + 5 \, b x^{4}\right )} e^{2} + \frac {1}{6} \, {\left (3 \, c d x^{4} + 4 \, b d x^{3}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.01, size = 54, normalized size = 0.98 \begin {gather*} \frac {b d^{2} x^{2}}{2} + \frac {c e^{2} x^{5}}{5} + x^{4} \left (\frac {b e^{2}}{4} + \frac {c d e}{2}\right ) + x^{3} \cdot \left (\frac {2 b d e}{3} + \frac {c d^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.04, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, c d x^{4} e + \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{4} \, b x^{4} e^{2} + \frac {2}{3} \, b d x^{3} e + \frac {1}{2} \, b d^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 51, normalized size = 0.93 \begin {gather*} x^3\,\left (\frac {c\,d^2}{3}+\frac {2\,b\,e\,d}{3}\right )+x^4\,\left (\frac {b\,e^2}{4}+\frac {c\,d\,e}{2}\right )+\frac {b\,d^2\,x^2}{2}+\frac {c\,e^2\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________