Optimal. Leaf size=441 \[ \frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^3}{3 e^9}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^9}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{5 e^9}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^6}{3 e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^7}{7 e^9}-\frac {c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^8}{2 e^9}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^9}{9 e^9}-\frac {2 c^3 (2 c d-b e) (d+e x)^{10}}{5 e^9}+\frac {c^4 (d+e x)^{11}}{11 e^9} \]
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Rubi [A]
time = 0.34, antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712}
\begin {gather*} \frac {(d+e x)^7 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^9}+\frac {2 c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9}-\frac {c (d+e x)^8 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9}-\frac {2 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {2 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^4}{3 e^9}-\frac {2 c^3 (d+e x)^{10} (2 c d-b e)}{5 e^9}+\frac {c^4 (d+e x)^{11}}{11 e^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^4 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^2}{e^8}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^3}{e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^4}{e^8}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^5}{e^8}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^6}{e^8}+\frac {4 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^7}{e^8}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^8}{e^8}-\frac {4 c^3 (2 c d-b e) (d+e x)^9}{e^8}+\frac {c^4 (d+e x)^{10}}{e^8}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^3}{3 e^9}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^9}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^5}{5 e^9}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^6}{3 e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^7}{7 e^9}-\frac {c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^8}{2 e^9}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^9}{9 e^9}-\frac {2 c^3 (2 c d-b e) (d+e x)^{10}}{5 e^9}+\frac {c^4 (d+e x)^{11}}{11 e^9}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 428, normalized size = 0.97 \begin {gather*} a^4 d^2 x+a^3 d (2 b d+a e) x^2+\frac {1}{3} a^2 \left (6 b^2 d^2+8 a b d e+a \left (4 c d^2+a e^2\right )\right ) x^3+a \left (b^3 d^2+3 a b^2 d e+2 a^2 c d e+a b \left (3 c d^2+a e^2\right )\right ) x^4+\frac {1}{5} \left (b^4 d^2+8 a b^3 d e+24 a^2 b c d e+6 a b^2 \left (2 c d^2+a e^2\right )+2 a^2 c \left (3 c d^2+2 a e^2\right )\right ) x^5+\frac {1}{3} \left (b^4 d e+12 a b^2 c d e+6 a^2 c^2 d e+2 b^3 \left (c d^2+a e^2\right )+6 a b c \left (c d^2+a e^2\right )\right ) x^6+\frac {1}{7} \left (8 b^3 c d e+24 a b c^2 d e+b^4 e^2+6 b^2 c \left (c d^2+2 a e^2\right )+2 a c^2 \left (2 c d^2+3 a e^2\right )\right ) x^7+\frac {1}{2} c \left (3 b^2 c d e+2 a c^2 d e+b^3 e^2+b c \left (c d^2+3 a e^2\right )\right ) x^8+\frac {1}{9} c^2 \left (c^2 d^2+6 b^2 e^2+4 c e (2 b d+a e)\right ) x^9+\frac {1}{5} c^3 e (c d+2 b e) x^{10}+\frac {1}{11} c^4 e^2 x^{11} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 545, normalized size = 1.24 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 450, normalized size = 1.02 \begin {gather*} \frac {1}{11} \, c^{4} x^{11} e^{2} + \frac {1}{5} \, {\left (c^{4} d e + 2 \, b c^{3} e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{2} + 8 \, b c^{3} d e + 6 \, b^{2} c^{2} e^{2} + 4 \, a c^{3} e^{2}\right )} x^{9} + \frac {1}{2} \, {\left (b c^{3} d^{2} + b^{3} c e^{2} + 3 \, a b c^{2} e^{2} + {\left (3 \, b^{2} c^{2} e + 2 \, a c^{3} e\right )} d\right )} x^{8} + \frac {1}{7} \, {\left (b^{4} e^{2} + 12 \, a b^{2} c e^{2} + 6 \, a^{2} c^{2} e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} + 8 \, {\left (b^{3} c e + 3 \, a b c^{2} e\right )} d\right )} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (2 \, a b^{3} e^{2} + 6 \, a^{2} b c e^{2} + 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} + {\left (b^{4} e + 12 \, a b^{2} c e + 6 \, a^{2} c^{2} e\right )} d\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} b^{2} e^{2} + 4 \, a^{3} c e^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} + 8 \, {\left (a b^{3} e + 3 \, a^{2} b c e\right )} d\right )} x^{5} + {\left (a^{3} b e^{2} + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} + {\left (3 \, a^{2} b^{2} e + 2 \, a^{3} c e\right )} d\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a^{3} b d e + a^{4} e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.23, size = 445, normalized size = 1.01 \begin {gather*} \frac {1}{9} \, c^{4} d^{2} x^{9} + \frac {1}{2} \, b c^{3} d^{2} x^{8} + \frac {2}{7} \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} x^{7} + \frac {2}{3} \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} x^{6} + 2 \, a^{3} b d^{2} x^{2} + \frac {1}{5} \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} x^{5} + a^{4} d^{2} x + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} x^{3} + \frac {1}{6930} \, {\left (630 \, c^{4} x^{11} + 2772 \, b c^{3} x^{10} + 1540 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{9} + 3465 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} x^{8} + 6930 \, a^{3} b x^{4} + 990 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{7} + 2310 \, a^{4} x^{3} + 4620 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} x^{6} + 2772 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} x^{5}\right )} e^{2} + \frac {1}{630} \, {\left (126 \, c^{4} d x^{10} + 560 \, b c^{3} d x^{9} + 315 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{8} + 720 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{7} + 1680 \, a^{3} b d x^{3} + 210 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{6} + 630 \, a^{4} d x^{2} + 1008 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{5} + 630 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x^{4}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 537, normalized size = 1.22 \begin {gather*} a^{4} d^{2} x + \frac {c^{4} e^{2} x^{11}}{11} + x^{10} \cdot \left (\frac {2 b c^{3} e^{2}}{5} + \frac {c^{4} d e}{5}\right ) + x^{9} \cdot \left (\frac {4 a c^{3} e^{2}}{9} + \frac {2 b^{2} c^{2} e^{2}}{3} + \frac {8 b c^{3} d e}{9} + \frac {c^{4} d^{2}}{9}\right ) + x^{8} \cdot \left (\frac {3 a b c^{2} e^{2}}{2} + a c^{3} d e + \frac {b^{3} c e^{2}}{2} + \frac {3 b^{2} c^{2} d e}{2} + \frac {b c^{3} d^{2}}{2}\right ) + x^{7} \cdot \left (\frac {6 a^{2} c^{2} e^{2}}{7} + \frac {12 a b^{2} c e^{2}}{7} + \frac {24 a b c^{2} d e}{7} + \frac {4 a c^{3} d^{2}}{7} + \frac {b^{4} e^{2}}{7} + \frac {8 b^{3} c d e}{7} + \frac {6 b^{2} c^{2} d^{2}}{7}\right ) + x^{6} \cdot \left (2 a^{2} b c e^{2} + 2 a^{2} c^{2} d e + \frac {2 a b^{3} e^{2}}{3} + 4 a b^{2} c d e + 2 a b c^{2} d^{2} + \frac {b^{4} d e}{3} + \frac {2 b^{3} c d^{2}}{3}\right ) + x^{5} \cdot \left (\frac {4 a^{3} c e^{2}}{5} + \frac {6 a^{2} b^{2} e^{2}}{5} + \frac {24 a^{2} b c d e}{5} + \frac {6 a^{2} c^{2} d^{2}}{5} + \frac {8 a b^{3} d e}{5} + \frac {12 a b^{2} c d^{2}}{5} + \frac {b^{4} d^{2}}{5}\right ) + x^{4} \left (a^{3} b e^{2} + 2 a^{3} c d e + 3 a^{2} b^{2} d e + 3 a^{2} b c d^{2} + a b^{3} d^{2}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {8 a^{3} b d e}{3} + \frac {4 a^{3} c d^{2}}{3} + 2 a^{2} b^{2} d^{2}\right ) + x^{2} \left (a^{4} d e + 2 a^{3} b d^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.10, size = 537, normalized size = 1.22 \begin {gather*} \frac {1}{11} \, c^{4} x^{11} e^{2} + \frac {1}{5} \, c^{4} d x^{10} e + \frac {1}{9} \, c^{4} d^{2} x^{9} + \frac {2}{5} \, b c^{3} x^{10} e^{2} + \frac {8}{9} \, b c^{3} d x^{9} e + \frac {1}{2} \, b c^{3} d^{2} x^{8} + \frac {2}{3} \, b^{2} c^{2} x^{9} e^{2} + \frac {4}{9} \, a c^{3} x^{9} e^{2} + \frac {3}{2} \, b^{2} c^{2} d x^{8} e + a c^{3} d x^{8} e + \frac {6}{7} \, b^{2} c^{2} d^{2} x^{7} + \frac {4}{7} \, a c^{3} d^{2} x^{7} + \frac {1}{2} \, b^{3} c x^{8} e^{2} + \frac {3}{2} \, a b c^{2} x^{8} e^{2} + \frac {8}{7} \, b^{3} c d x^{7} e + \frac {24}{7} \, a b c^{2} d x^{7} e + \frac {2}{3} \, b^{3} c d^{2} x^{6} + 2 \, a b c^{2} d^{2} x^{6} + \frac {1}{7} \, b^{4} x^{7} e^{2} + \frac {12}{7} \, a b^{2} c x^{7} e^{2} + \frac {6}{7} \, a^{2} c^{2} x^{7} e^{2} + \frac {1}{3} \, b^{4} d x^{6} e + 4 \, a b^{2} c d x^{6} e + 2 \, a^{2} c^{2} d x^{6} e + \frac {1}{5} \, b^{4} d^{2} x^{5} + \frac {12}{5} \, a b^{2} c d^{2} x^{5} + \frac {6}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {2}{3} \, a b^{3} x^{6} e^{2} + 2 \, a^{2} b c x^{6} e^{2} + \frac {8}{5} \, a b^{3} d x^{5} e + \frac {24}{5} \, a^{2} b c d x^{5} e + a b^{3} d^{2} x^{4} + 3 \, a^{2} b c d^{2} x^{4} + \frac {6}{5} \, a^{2} b^{2} x^{5} e^{2} + \frac {4}{5} \, a^{3} c x^{5} e^{2} + 3 \, a^{2} b^{2} d x^{4} e + 2 \, a^{3} c d x^{4} e + 2 \, a^{2} b^{2} d^{2} x^{3} + \frac {4}{3} \, a^{3} c d^{2} x^{3} + a^{3} b x^{4} e^{2} + \frac {8}{3} \, a^{3} b d x^{3} e + 2 \, a^{3} b d^{2} x^{2} + \frac {1}{3} \, a^{4} x^{3} e^{2} + a^{4} d x^{2} e + a^{4} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 445, normalized size = 1.01 \begin {gather*} x^5\,\left (\frac {4\,a^3\,c\,e^2}{5}+\frac {6\,a^2\,b^2\,e^2}{5}+\frac {24\,a^2\,b\,c\,d\,e}{5}+\frac {6\,a^2\,c^2\,d^2}{5}+\frac {8\,a\,b^3\,d\,e}{5}+\frac {12\,a\,b^2\,c\,d^2}{5}+\frac {b^4\,d^2}{5}\right )+x^3\,\left (\frac {a^4\,e^2}{3}+\frac {8\,a^3\,b\,d\,e}{3}+\frac {4\,c\,a^3\,d^2}{3}+2\,a^2\,b^2\,d^2\right )+x^7\,\left (\frac {6\,a^2\,c^2\,e^2}{7}+\frac {12\,a\,b^2\,c\,e^2}{7}+\frac {24\,a\,b\,c^2\,d\,e}{7}+\frac {4\,a\,c^3\,d^2}{7}+\frac {b^4\,e^2}{7}+\frac {8\,b^3\,c\,d\,e}{7}+\frac {6\,b^2\,c^2\,d^2}{7}\right )+x^9\,\left (\frac {2\,b^2\,c^2\,e^2}{3}+\frac {8\,b\,c^3\,d\,e}{9}+\frac {c^4\,d^2}{9}+\frac {4\,a\,c^3\,e^2}{9}\right )+x^6\,\left (2\,a^2\,b\,c\,e^2+2\,a^2\,c^2\,d\,e+\frac {2\,a\,b^3\,e^2}{3}+4\,a\,b^2\,c\,d\,e+2\,a\,b\,c^2\,d^2+\frac {b^4\,d\,e}{3}+\frac {2\,b^3\,c\,d^2}{3}\right )+x^4\,\left (a^3\,b\,e^2+2\,c\,a^3\,d\,e+3\,a^2\,b^2\,d\,e+3\,c\,a^2\,b\,d^2+a\,b^3\,d^2\right )+x^8\,\left (\frac {b^3\,c\,e^2}{2}+\frac {3\,b^2\,c^2\,d\,e}{2}+\frac {b\,c^3\,d^2}{2}+\frac {3\,a\,b\,c^2\,e^2}{2}+a\,c^3\,d\,e\right )+a^4\,d^2\,x+\frac {c^4\,e^2\,x^{11}}{11}+a^3\,d\,x^2\,\left (a\,e+2\,b\,d\right )+\frac {c^3\,e\,x^{10}\,\left (2\,b\,e+c\,d\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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