Optimal. Leaf size=39 \[ \frac {1}{4} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^4+\frac {c d (d+e x)^5}{5 e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 45}
\begin {gather*} \frac {1}{4} (d+e x)^4 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^5}{5 e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 640
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^3 \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^3}{e}+\frac {c d (d+e x)^4}{e}\right ) \, dx\\ &=\frac {1}{4} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^4+\frac {c d (d+e x)^5}{5 e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 73, normalized size = 1.87 \begin {gather*} \frac {1}{20} x \left (5 a e \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+c d x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs.
\(2(35)=70\).
time = 0.54, size = 112, normalized size = 2.87
method | result | size |
norman | \(\frac {d \,e^{3} c \,x^{5}}{5}+\left (\frac {1}{4} e^{4} a +\frac {3}{4} d^{2} e^{2} c \right ) x^{4}+\left (a d \,e^{3}+c \,d^{3} e \right ) x^{3}+\left (\frac {3}{2} a \,d^{2} e^{2}+\frac {1}{2} c \,d^{4}\right ) x^{2}+a e \,d^{3} x\) | \(76\) |
risch | \(\frac {1}{5} d \,e^{3} c \,x^{5}+\frac {1}{4} x^{4} e^{4} a +\frac {3}{4} x^{4} d^{2} e^{2} c +a d \,e^{3} x^{3}+c \,d^{3} e \,x^{3}+\frac {3}{2} x^{2} a \,d^{2} e^{2}+\frac {1}{2} x^{2} c \,d^{4}+a e \,d^{3} x\) | \(79\) |
gosper | \(\frac {x \left (4 d \,e^{3} c \,x^{4}+5 x^{3} e^{4} a +15 x^{3} d^{2} e^{2} c +20 a d \,e^{3} x^{2}+20 c \,d^{3} e \,x^{2}+30 x a \,d^{2} e^{2}+10 x c \,d^{4}+20 a e \,d^{3}\right )}{20}\) | \(80\) |
default | \(\frac {d \,e^{3} c \,x^{5}}{5}+\frac {\left (2 d^{2} e^{2} c +e^{2} \left (e^{2} a +c \,d^{2}\right )\right ) x^{4}}{4}+\frac {\left (c \,d^{3} e +2 d e \left (e^{2} a +c \,d^{2}\right )+a d \,e^{3}\right ) x^{3}}{3}+\frac {\left (d^{2} \left (e^{2} a +c \,d^{2}\right )+2 a \,d^{2} e^{2}\right ) x^{2}}{2}+a e \,d^{3} x\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 72, normalized size = 1.85 \begin {gather*} \frac {1}{5} \, c d x^{5} e^{3} + a d^{3} x e + \frac {1}{4} \, {\left (3 \, c d^{2} e^{2} + a e^{4}\right )} x^{4} + {\left (c d^{3} e + a d e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{4} + 3 \, a d^{2} e^{2}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (36) = 72\).
time = 2.52, size = 76, normalized size = 1.95 \begin {gather*} \frac {1}{2} \, c d^{4} x^{2} + \frac {1}{4} \, a x^{4} e^{4} + \frac {1}{5} \, {\left (c d x^{5} + 5 \, a d x^{3}\right )} e^{3} + \frac {3}{4} \, {\left (c d^{2} x^{4} + 2 \, a d^{2} x^{2}\right )} e^{2} + {\left (c d^{3} x^{3} + a d^{3} x\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (34) = 68\).
time = 0.01, size = 80, normalized size = 2.05 \begin {gather*} a d^{3} e x + \frac {c d e^{3} x^{5}}{5} + x^{4} \left (\frac {a e^{4}}{4} + \frac {3 c d^{2} e^{2}}{4}\right ) + x^{3} \left (a d e^{3} + c d^{3} e\right ) + x^{2} \cdot \left (\frac {3 a d^{2} e^{2}}{2} + \frac {c d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs.
\(2 (36) = 72\).
time = 0.85, size = 75, normalized size = 1.92 \begin {gather*} \frac {1}{5} \, c d x^{5} e^{3} + \frac {3}{4} \, c d^{2} x^{4} e^{2} + c d^{3} x^{3} e + \frac {1}{2} \, c d^{4} x^{2} + \frac {1}{4} \, a x^{4} e^{4} + a d x^{3} e^{3} + \frac {3}{2} \, a d^{2} x^{2} e^{2} + a d^{3} x e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.55, size = 75, normalized size = 1.92 \begin {gather*} x^2\,\left (\frac {c\,d^4}{2}+\frac {3\,a\,d^2\,e^2}{2}\right )+x^4\,\left (\frac {3\,c\,d^2\,e^2}{4}+\frac {a\,e^4}{4}\right )+x^3\,\left (c\,d^3\,e+a\,d\,e^3\right )+a\,d^3\,e\,x+\frac {c\,d\,e^3\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________