Optimal. Leaf size=103 \[ -\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^6}{6 e^4}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^7}{7 e^4}-\frac {(3 c d-b e) (d+e x)^8}{8 e^4}+\frac {c (d+e x)^9}{9 e^4} \]
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Rubi [A]
time = 0.11, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {785}
\begin {gather*} \frac {(d+e x)^7 \left (3 c d^2-e (2 b d-a e)\right )}{7 e^4}-\frac {d (d+e x)^6 \left (a e^2-b d e+c d^2\right )}{6 e^4}-\frac {(d+e x)^8 (3 c d-b e)}{8 e^4}+\frac {c (d+e x)^9}{9 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int x (d+e x)^5 \left (a+b x+c x^2\right ) \, dx &=\int \left (-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{e^3}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^6}{e^3}+\frac {(-3 c d+b e) (d+e x)^7}{e^3}+\frac {c (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac {d \left (c d^2-b d e+a e^2\right ) (d+e x)^6}{6 e^4}+\frac {\left (3 c d^2-e (2 b d-a e)\right ) (d+e x)^7}{7 e^4}-\frac {(3 c d-b e) (d+e x)^8}{8 e^4}+\frac {c (d+e x)^9}{9 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 166, normalized size = 1.61 \begin {gather*} \frac {1}{2} a d^5 x^2+\frac {1}{3} d^4 (b d+5 a e) x^3+\frac {1}{4} d^3 \left (c d^2+5 b d e+10 a e^2\right ) x^4+d^2 e \left (c d^2+2 b d e+2 a e^2\right ) x^5+\frac {5}{6} d e^2 \left (2 c d^2+2 b d e+a e^2\right ) x^6+\frac {1}{7} e^3 \left (10 c d^2+5 b d e+a e^2\right ) x^7+\frac {1}{8} e^4 (5 c d+b e) x^8+\frac {1}{9} c e^5 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 172, normalized size = 1.67
method | result | size |
norman | \(\frac {e^{5} c \,x^{9}}{9}+\left (\frac {1}{8} e^{5} b +\frac {5}{8} d \,e^{4} c \right ) x^{8}+\left (\frac {1}{7} e^{5} a +\frac {5}{7} d \,e^{4} b +\frac {10}{7} d^{2} e^{3} c \right ) x^{7}+\left (\frac {5}{6} d \,e^{4} a +\frac {5}{3} d^{2} e^{3} b +\frac {5}{3} d^{3} e^{2} c \right ) x^{6}+\left (2 d^{2} e^{3} a +2 d^{3} e^{2} b +d^{4} e c \right ) x^{5}+\left (\frac {5}{2} d^{3} e^{2} a +\frac {5}{4} d^{4} e b +\frac {1}{4} d^{5} c \right ) x^{4}+\left (\frac {5}{3} d^{4} e a +\frac {1}{3} d^{5} b \right ) x^{3}+\frac {d^{5} a \,x^{2}}{2}\) | \(169\) |
default | \(\frac {e^{5} c \,x^{9}}{9}+\frac {\left (e^{5} b +5 d \,e^{4} c \right ) x^{8}}{8}+\frac {\left (e^{5} a +5 d \,e^{4} b +10 d^{2} e^{3} c \right ) x^{7}}{7}+\frac {\left (5 d \,e^{4} a +10 d^{2} e^{3} b +10 d^{3} e^{2} c \right ) x^{6}}{6}+\frac {\left (10 d^{2} e^{3} a +10 d^{3} e^{2} b +5 d^{4} e c \right ) x^{5}}{5}+\frac {\left (10 d^{3} e^{2} a +5 d^{4} e b +d^{5} c \right ) x^{4}}{4}+\frac {\left (5 d^{4} e a +d^{5} b \right ) x^{3}}{3}+\frac {d^{5} a \,x^{2}}{2}\) | \(172\) |
gosper | \(\frac {1}{9} e^{5} c \,x^{9}+\frac {1}{8} x^{8} e^{5} b +\frac {5}{8} x^{8} d \,e^{4} c +\frac {1}{7} x^{7} e^{5} a +\frac {5}{7} x^{7} d \,e^{4} b +\frac {10}{7} x^{7} d^{2} e^{3} c +\frac {5}{6} x^{6} d \,e^{4} a +\frac {5}{3} x^{6} d^{2} e^{3} b +\frac {5}{3} x^{6} d^{3} e^{2} c +2 a \,d^{2} e^{3} x^{5}+2 b \,d^{3} e^{2} x^{5}+c \,d^{4} e \,x^{5}+\frac {5}{2} x^{4} d^{3} e^{2} a +\frac {5}{4} x^{4} d^{4} e b +\frac {1}{4} x^{4} d^{5} c +\frac {5}{3} x^{3} d^{4} e a +\frac {1}{3} x^{3} d^{5} b +\frac {1}{2} d^{5} a \,x^{2}\) | \(187\) |
risch | \(\frac {1}{9} e^{5} c \,x^{9}+\frac {1}{8} x^{8} e^{5} b +\frac {5}{8} x^{8} d \,e^{4} c +\frac {1}{7} x^{7} e^{5} a +\frac {5}{7} x^{7} d \,e^{4} b +\frac {10}{7} x^{7} d^{2} e^{3} c +\frac {5}{6} x^{6} d \,e^{4} a +\frac {5}{3} x^{6} d^{2} e^{3} b +\frac {5}{3} x^{6} d^{3} e^{2} c +2 a \,d^{2} e^{3} x^{5}+2 b \,d^{3} e^{2} x^{5}+c \,d^{4} e \,x^{5}+\frac {5}{2} x^{4} d^{3} e^{2} a +\frac {5}{4} x^{4} d^{4} e b +\frac {1}{4} x^{4} d^{5} c +\frac {5}{3} x^{3} d^{4} e a +\frac {1}{3} x^{3} d^{5} b +\frac {1}{2} d^{5} a \,x^{2}\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 159, normalized size = 1.54 \begin {gather*} \frac {1}{9} \, c x^{9} e^{5} + \frac {1}{8} \, {\left (5 \, c d e^{4} + b e^{5}\right )} x^{8} + \frac {1}{2} \, a d^{5} x^{2} + \frac {1}{7} \, {\left (10 \, c d^{2} e^{3} + 5 \, b d e^{4} + a e^{5}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, c d^{3} e^{2} + 2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{6} + {\left (c d^{4} e + 2 \, b d^{3} e^{2} + 2 \, a d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (c d^{5} + 5 \, b d^{4} e + 10 \, a d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b d^{5} + 5 \, a d^{4} e\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.06, size = 173, normalized size = 1.68 \begin {gather*} \frac {1}{4} \, c d^{5} x^{4} + \frac {1}{3} \, b d^{5} x^{3} + \frac {1}{2} \, a d^{5} x^{2} + \frac {1}{504} \, {\left (56 \, c x^{9} + 63 \, b x^{8} + 72 \, a x^{7}\right )} e^{5} + \frac {5}{168} \, {\left (21 \, c d x^{8} + 24 \, b d x^{7} + 28 \, a d x^{6}\right )} e^{4} + \frac {1}{21} \, {\left (30 \, c d^{2} x^{7} + 35 \, b d^{2} x^{6} + 42 \, a d^{2} x^{5}\right )} e^{3} + \frac {1}{6} \, {\left (10 \, c d^{3} x^{6} + 12 \, b d^{3} x^{5} + 15 \, a d^{3} x^{4}\right )} e^{2} + \frac {1}{12} \, {\left (12 \, c d^{4} x^{5} + 15 \, b d^{4} x^{4} + 20 \, a d^{4} x^{3}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 192 vs.
\(2 (94) = 188\).
time = 0.02, size = 192, normalized size = 1.86 \begin {gather*} \frac {a d^{5} x^{2}}{2} + \frac {c e^{5} x^{9}}{9} + x^{8} \left (\frac {b e^{5}}{8} + \frac {5 c d e^{4}}{8}\right ) + x^{7} \left (\frac {a e^{5}}{7} + \frac {5 b d e^{4}}{7} + \frac {10 c d^{2} e^{3}}{7}\right ) + x^{6} \cdot \left (\frac {5 a d e^{4}}{6} + \frac {5 b d^{2} e^{3}}{3} + \frac {5 c d^{3} e^{2}}{3}\right ) + x^{5} \cdot \left (2 a d^{2} e^{3} + 2 b d^{3} e^{2} + c d^{4} e\right ) + x^{4} \cdot \left (\frac {5 a d^{3} e^{2}}{2} + \frac {5 b d^{4} e}{4} + \frac {c d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {5 a d^{4} e}{3} + \frac {b d^{5}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.49, size = 177, normalized size = 1.72 \begin {gather*} \frac {1}{9} \, c x^{9} e^{5} + \frac {5}{8} \, c d x^{8} e^{4} + \frac {10}{7} \, c d^{2} x^{7} e^{3} + \frac {5}{3} \, c d^{3} x^{6} e^{2} + c d^{4} x^{5} e + \frac {1}{4} \, c d^{5} x^{4} + \frac {1}{8} \, b x^{8} e^{5} + \frac {5}{7} \, b d x^{7} e^{4} + \frac {5}{3} \, b d^{2} x^{6} e^{3} + 2 \, b d^{3} x^{5} e^{2} + \frac {5}{4} \, b d^{4} x^{4} e + \frac {1}{3} \, b d^{5} x^{3} + \frac {1}{7} \, a x^{7} e^{5} + \frac {5}{6} \, a d x^{6} e^{4} + 2 \, a d^{2} x^{5} e^{3} + \frac {5}{2} \, a d^{3} x^{4} e^{2} + \frac {5}{3} \, a d^{4} x^{3} e + \frac {1}{2} \, a d^{5} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.36, size = 160, normalized size = 1.55 \begin {gather*} x^3\,\left (\frac {b\,d^5}{3}+\frac {5\,a\,e\,d^4}{3}\right )+x^8\,\left (\frac {b\,e^5}{8}+\frac {5\,c\,d\,e^4}{8}\right )+x^4\,\left (\frac {c\,d^5}{4}+\frac {5\,b\,d^4\,e}{4}+\frac {5\,a\,d^3\,e^2}{2}\right )+x^7\,\left (\frac {10\,c\,d^2\,e^3}{7}+\frac {5\,b\,d\,e^4}{7}+\frac {a\,e^5}{7}\right )+\frac {a\,d^5\,x^2}{2}+\frac {c\,e^5\,x^9}{9}+d^2\,e\,x^5\,\left (c\,d^2+2\,b\,d\,e+2\,a\,e^2\right )+\frac {5\,d\,e^2\,x^6\,\left (2\,c\,d^2+2\,b\,d\,e+a\,e^2\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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