Optimal. Leaf size=25 \[ \frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} e (1+x)^{12} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {27, 45}
\begin {gather*} \frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{12} e (x+1)^{12} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 45
Rubi steps
\begin {align*} \int (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((d-e) (1+x)^{10}+e (1+x)^{11}\right ) \, dx\\ &=\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} e (1+x)^{12}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(25)=50\).
time = 0.01, size = 113, normalized size = 4.52 \begin {gather*} \frac {1}{132} e x^2 \left (66+440 x+1485 x^2+3168 x^3+4620 x^4+4752 x^5+3465 x^6+1760 x^7+594 x^8+120 x^9+11 x^{10}\right )+d \left (x+5 x^2+15 x^3+30 x^4+42 x^5+42 x^6+30 x^7+15 x^8+5 x^9+x^{10}+\frac {x^{11}}{11}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(126\) vs.
\(2(21)=42\).
time = 0.64, size = 127, normalized size = 5.08
method | result | size |
norman | \(\frac {x^{12} e}{12}+\left (\frac {d}{11}+\frac {10 e}{11}\right ) x^{11}+\left (d +\frac {9 e}{2}\right ) x^{10}+\left (5 d +\frac {40 e}{3}\right ) x^{9}+\left (15 d +\frac {105 e}{4}\right ) x^{8}+\left (30 d +36 e \right ) x^{7}+\left (42 d +35 e \right ) x^{6}+\left (42 d +24 e \right ) x^{5}+\left (30 d +\frac {45 e}{4}\right ) x^{4}+\left (15 d +\frac {10 e}{3}\right ) x^{3}+\left (5 d +\frac {e}{2}\right ) x^{2}+d x\) | \(119\) |
default | \(\frac {x^{12} e}{12}+\frac {\left (d +10 e \right ) x^{11}}{11}+\frac {\left (10 d +45 e \right ) x^{10}}{10}+\frac {\left (45 d +120 e \right ) x^{9}}{9}+\frac {\left (120 d +210 e \right ) x^{8}}{8}+\frac {\left (210 d +252 e \right ) x^{7}}{7}+\frac {\left (252 d +210 e \right ) x^{6}}{6}+\frac {\left (210 d +120 e \right ) x^{5}}{5}+\frac {\left (120 d +45 e \right ) x^{4}}{4}+\frac {\left (45 d +10 e \right ) x^{3}}{3}+\frac {\left (10 d +e \right ) x^{2}}{2}+d x\) | \(127\) |
gosper | \(\frac {x \left (11 e \,x^{11}+12 d \,x^{10}+120 e \,x^{10}+132 d \,x^{9}+594 e \,x^{9}+660 d \,x^{8}+1760 e \,x^{8}+1980 d \,x^{7}+3465 e \,x^{7}+3960 d \,x^{6}+4752 x^{6} e +5544 d \,x^{5}+4620 e \,x^{5}+5544 d \,x^{4}+3168 x^{4} e +3960 d \,x^{3}+1485 e \,x^{3}+1980 d \,x^{2}+440 e \,x^{2}+660 d x +66 e x +132 d \right )}{132}\) | \(130\) |
risch | \(\frac {1}{12} x^{12} e +\frac {1}{11} x^{11} d +\frac {10}{11} e \,x^{11}+d \,x^{10}+\frac {9}{2} e \,x^{10}+5 d \,x^{9}+\frac {40}{3} e \,x^{9}+15 d \,x^{8}+\frac {105}{4} e \,x^{8}+30 d \,x^{7}+36 e \,x^{7}+42 d \,x^{6}+35 x^{6} e +42 d \,x^{5}+24 e \,x^{5}+30 d \,x^{4}+\frac {45}{4} x^{4} e +15 d \,x^{3}+\frac {10}{3} e \,x^{3}+5 d \,x^{2}+\frac {1}{2} e \,x^{2}+d x\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 137 vs.
\(2 (23) = 46\).
time = 0.26, size = 137, normalized size = 5.48 \begin {gather*} \frac {1}{12} \, x^{12} e + \frac {1}{11} \, {\left (d + 10 \, e\right )} x^{11} + \frac {1}{2} \, {\left (2 \, d + 9 \, e\right )} x^{10} + \frac {5}{3} \, {\left (3 \, d + 8 \, e\right )} x^{9} + \frac {15}{4} \, {\left (4 \, d + 7 \, e\right )} x^{8} + 6 \, {\left (5 \, d + 6 \, e\right )} x^{7} + 7 \, {\left (6 \, d + 5 \, e\right )} x^{6} + 6 \, {\left (7 \, d + 4 \, e\right )} x^{5} + \frac {15}{4} \, {\left (8 \, d + 3 \, e\right )} x^{4} + \frac {5}{3} \, {\left (9 \, d + 2 \, e\right )} x^{3} + \frac {1}{2} \, {\left (10 \, d + e\right )} x^{2} + d x \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. 123 vs.
\(2 (23) = 46\).
time = 3.44, size = 123, normalized size = 4.92 \begin {gather*} \frac {1}{11} \, d x^{11} + d x^{10} + 5 \, d x^{9} + 15 \, d x^{8} + 30 \, d x^{7} + 42 \, d x^{6} + 42 \, d x^{5} + 30 \, d x^{4} + 15 \, d x^{3} + 5 \, d x^{2} + d x + \frac {1}{132} \, {\left (11 \, x^{12} + 120 \, x^{11} + 594 \, x^{10} + 1760 \, x^{9} + 3465 \, x^{8} + 4752 \, x^{7} + 4620 \, x^{6} + 3168 \, x^{5} + 1485 \, x^{4} + 440 \, x^{3} + 66 \, x^{2}\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. 119 vs.
\(2 (19) = 38\).
time = 0.03, size = 119, normalized size = 4.76 \begin {gather*} d x + \frac {e x^{12}}{12} + x^{11} \left (\frac {d}{11} + \frac {10 e}{11}\right ) + x^{10} \left (d + \frac {9 e}{2}\right ) + x^{9} \cdot \left (5 d + \frac {40 e}{3}\right ) + x^{8} \cdot \left (15 d + \frac {105 e}{4}\right ) + x^{7} \cdot \left (30 d + 36 e\right ) + x^{6} \cdot \left (42 d + 35 e\right ) + x^{5} \cdot \left (42 d + 24 e\right ) + x^{4} \cdot \left (30 d + \frac {45 e}{4}\right ) + x^{3} \cdot \left (15 d + \frac {10 e}{3}\right ) + x^{2} \cdot \left (5 d + \frac {e}{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. 140 vs.
\(2 (23) = 46\).
time = 1.08, size = 140, normalized size = 5.60 \begin {gather*} \frac {1}{12} \, x^{12} e + \frac {1}{11} \, d x^{11} + \frac {10}{11} \, x^{11} e + d x^{10} + \frac {9}{2} \, x^{10} e + 5 \, d x^{9} + \frac {40}{3} \, x^{9} e + 15 \, d x^{8} + \frac {105}{4} \, x^{8} e + 30 \, d x^{7} + 36 \, x^{7} e + 42 \, d x^{6} + 35 \, x^{6} e + 42 \, d x^{5} + 24 \, x^{5} e + 30 \, d x^{4} + \frac {45}{4} \, x^{4} e + 15 \, d x^{3} + \frac {10}{3} \, x^{3} e + 5 \, d x^{2} + \frac {1}{2} \, x^{2} e + d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.08, size = 118, normalized size = 4.72 \begin {gather*} \frac {e\,x^{12}}{12}+\left (\frac {d}{11}+\frac {10\,e}{11}\right )\,x^{11}+\left (d+\frac {9\,e}{2}\right )\,x^{10}+\left (5\,d+\frac {40\,e}{3}\right )\,x^9+\left (15\,d+\frac {105\,e}{4}\right )\,x^8+\left (30\,d+36\,e\right )\,x^7+\left (42\,d+35\,e\right )\,x^6+\left (42\,d+24\,e\right )\,x^5+\left (30\,d+\frac {45\,e}{4}\right )\,x^4+\left (15\,d+\frac {10\,e}{3}\right )\,x^3+\left (5\,d+\frac {e}{2}\right )\,x^2+d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________