Optimal. Leaf size=57 \[ \frac {\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {c d (d+e x)^4}{2 e^3}+\frac {c (d+e x)^5}{5 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 = {711}
\begin {gather*} \frac {(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac {c (d+e x)^5}{5 e^3}-\frac {c d (d+e x)^4}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 711
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^2}{e^2}-\frac {2 c d (d+e x)^3}{e^2}+\frac {c (d+e x)^4}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {c d (d+e x)^4}{2 e^3}+\frac {c (d+e x)^5}{5 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 53, normalized size = 0.93 \begin {gather*} a d^2 x+a d e x^2+\frac {1}{3} \left (c d^2+a e^2\right ) x^3+\frac {1}{2} c d e x^4+\frac {1}{5} c e^2 x^5 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.40, size = 48, normalized size = 0.84
method | result | size |
default | \(\frac {c \,e^{2} x^{5}}{5}+\frac {c d e \,x^{4}}{2}+\frac {\left (e^{2} a +c \,d^{2}\right ) x^{3}}{3}+a d e \,x^{2}+a x \,d^{2}\) | \(48\) |
norman | \(\frac {c \,e^{2} x^{5}}{5}+\frac {c d e \,x^{4}}{2}+\left (\frac {e^{2} a}{3}+\frac {c \,d^{2}}{3}\right ) x^{3}+a d e \,x^{2}+a x \,d^{2}\) | \(49\) |
gosper | \(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} c d e \,x^{4}+\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} x^{3} c \,d^{2}+a d e \,x^{2}+a x \,d^{2}\) | \(50\) |
risch | \(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} c d e \,x^{4}+\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} x^{3} c \,d^{2}+a d e \,x^{2}+a x \,d^{2}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.55, size = 47, normalized size = 0.82 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, c d x^{4} e + a d x^{2} e + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.23, size = 51, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, c d^{2} x^{3} + a d^{2} x + \frac {1}{15} \, {\left (3 \, c x^{5} + 5 \, a x^{3}\right )} e^{2} + \frac {1}{2} \, {\left (c d x^{4} + 2 \, a d x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.01, size = 51, normalized size = 0.89 \begin {gather*} a d^{2} x + a d e x^{2} + \frac {c d e x^{4}}{2} + \frac {c e^{2} x^{5}}{5} + x^{3} \left (\frac {a e^{2}}{3} + \frac {c d^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.39, size = 49, normalized size = 0.86 \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}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.02, size = 48, normalized size = 0.84 \begin {gather*} x^3\,\left (\frac {c\,d^2}{3}+\frac {a\,e^2}{3}\right )+\frac {c\,e^2\,x^5}{5}+a\,d^2\,x+a\,d\,e\,x^2+\frac {c\,d\,e\,x^4}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________