Optimal. Leaf size=124 \[ -\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{4 e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{5 e^4}-\frac {c (2 c d-b e) (d+e x)^6}{2 e^4}+\frac {2 c^2 (d+e x)^7}{7 e^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785}
\begin {gather*} \frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac {c (d+e x)^6 (2 c d-b e)}{2 e^4}+\frac {2 c^2 (d+e x)^7}{7 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^3}{e^3}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^4}{e^3}-\frac {3 c (2 c d-b e) (d+e x)^5}{e^3}+\frac {2 c^2 (d+e x)^6}{e^3}\right ) \, dx\\ &=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{4 e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{5 e^4}-\frac {c (2 c d-b e) (d+e x)^6}{2 e^4}+\frac {2 c^2 (d+e x)^7}{7 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 175, normalized size = 1.41 \begin {gather*} a b d^3 x+\frac {1}{2} d^2 \left (b^2 d+2 a c d+3 a b e\right ) x^2+d \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^3+\frac {1}{4} \left (2 c^2 d^3+b e^2 (3 b d+a e)+3 c d e (3 b d+2 a e)\right ) x^4+\frac {1}{5} e \left (6 c^2 d^2+b^2 e^2+c e (9 b d+2 a e)\right ) x^5+\frac {1}{2} c e^2 (2 c d+b e) x^6+\frac {2}{7} c^2 e^3 x^7 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 221, normalized size = 1.78
method | result | size |
norman | \(\frac {2 c^{2} e^{3} x^{7}}{7}+\left (\frac {1}{2} e^{3} b c +c^{2} d \,e^{2}\right ) x^{6}+\left (\frac {2}{5} a c \,e^{3}+\frac {1}{5} b^{2} e^{3}+\frac {9}{5} d \,e^{2} b c +\frac {6}{5} d^{2} e \,c^{2}\right ) x^{5}+\left (\frac {1}{4} a b \,e^{3}+\frac {3}{2} a d \,e^{2} c +\frac {3}{4} b^{2} d \,e^{2}+\frac {9}{4} d^{2} e b c +\frac {1}{2} c^{2} d^{3}\right ) x^{4}+\left (d \,e^{2} a b +2 d^{2} e a c +b^{2} d^{2} e +b c \,d^{3}\right ) x^{3}+\left (\frac {3}{2} d^{2} e a b +d^{3} a c +\frac {1}{2} b^{2} d^{3}\right ) x^{2}+a b \,d^{3} x\) | \(183\) |
gosper | \(\frac {2}{7} c^{2} e^{3} x^{7}+\frac {1}{2} x^{6} e^{3} b c +x^{6} c^{2} d \,e^{2}+\frac {2}{5} x^{5} a c \,e^{3}+\frac {1}{5} x^{5} b^{2} e^{3}+\frac {9}{5} x^{5} d \,e^{2} b c +\frac {6}{5} x^{5} d^{2} e \,c^{2}+\frac {1}{4} x^{4} a b \,e^{3}+\frac {3}{2} x^{4} a d \,e^{2} c +\frac {3}{4} x^{4} b^{2} d \,e^{2}+\frac {9}{4} x^{4} d^{2} e b c +\frac {1}{2} x^{4} c^{2} d^{3}+a b d \,e^{2} x^{3}+2 a c \,d^{2} e \,x^{3}+b^{2} d^{2} e \,x^{3}+b c \,d^{3} x^{3}+\frac {3}{2} x^{2} d^{2} e a b +x^{2} d^{3} a c +\frac {1}{2} x^{2} b^{2} d^{3}+a b \,d^{3} x\) | \(212\) |
risch | \(\frac {2}{7} c^{2} e^{3} x^{7}+\frac {1}{2} x^{6} e^{3} b c +x^{6} c^{2} d \,e^{2}+\frac {2}{5} x^{5} a c \,e^{3}+\frac {1}{5} x^{5} b^{2} e^{3}+\frac {9}{5} x^{5} d \,e^{2} b c +\frac {6}{5} x^{5} d^{2} e \,c^{2}+\frac {1}{4} x^{4} a b \,e^{3}+\frac {3}{2} x^{4} a d \,e^{2} c +\frac {3}{4} x^{4} b^{2} d \,e^{2}+\frac {9}{4} x^{4} d^{2} e b c +\frac {1}{2} x^{4} c^{2} d^{3}+a b d \,e^{2} x^{3}+2 a c \,d^{2} e \,x^{3}+b^{2} d^{2} e \,x^{3}+b c \,d^{3} x^{3}+\frac {3}{2} x^{2} d^{2} e a b +x^{2} d^{3} a c +\frac {1}{2} x^{2} b^{2} d^{3}+a b \,d^{3} x\) | \(212\) |
default | \(\frac {2 c^{2} e^{3} x^{7}}{7}+\frac {\left (\left (b \,e^{3}+6 c d \,e^{2}\right ) c +2 e^{3} b c \right ) x^{6}}{6}+\frac {\left (\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) c +\left (b \,e^{3}+6 c d \,e^{2}\right ) b +2 a c \,e^{3}\right ) x^{5}}{5}+\frac {\left (\left (3 b \,d^{2} e +2 c \,d^{3}\right ) c +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) b +\left (b \,e^{3}+6 c d \,e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (b c \,d^{3}+\left (3 b \,d^{2} e +2 c \,d^{3}\right ) b +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) a \right ) x^{3}}{3}+\frac {\left (b^{2} d^{3}+\left (3 b \,d^{2} e +2 c \,d^{3}\right ) a \right ) x^{2}}{2}+a b \,d^{3} x\) | \(221\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 177, normalized size = 1.43 \begin {gather*} \frac {2}{7} \, c^{2} x^{7} e^{3} + \frac {1}{2} \, {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{6} + a b d^{3} x + \frac {1}{5} \, {\left (6 \, c^{2} d^{2} e + 9 \, b c d e^{2} + b^{2} e^{3} + 2 \, a c e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, c^{2} d^{3} + 9 \, b c d^{2} e + a b e^{3} + 3 \, {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d\right )} x^{4} + {\left (b c d^{3} + a b d e^{2} + {\left (b^{2} e + 2 \, a c e\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a b d^{2} e + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.64, size = 180, normalized size = 1.45 \begin {gather*} \frac {1}{2} \, c^{2} d^{3} x^{4} + b c d^{3} x^{3} + a b d^{3} x + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} d^{3} x^{2} + \frac {1}{140} \, {\left (40 \, c^{2} x^{7} + 70 \, b c x^{6} + 35 \, a b x^{4} + 28 \, {\left (b^{2} + 2 \, a c\right )} x^{5}\right )} e^{3} + \frac {1}{20} \, {\left (20 \, c^{2} d x^{6} + 36 \, b c d x^{5} + 20 \, a b d x^{3} + 15 \, {\left (b^{2} + 2 \, a c\right )} d x^{4}\right )} e^{2} + \frac {1}{20} \, {\left (24 \, c^{2} d^{2} x^{5} + 45 \, b c d^{2} x^{4} + 30 \, a b d^{2} x^{2} + 20 \, {\left (b^{2} + 2 \, a c\right )} d^{2} x^{3}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 211, normalized size = 1.70 \begin {gather*} a b d^{3} x + \frac {2 c^{2} e^{3} x^{7}}{7} + x^{6} \left (\frac {b c e^{3}}{2} + c^{2} d e^{2}\right ) + x^{5} \cdot \left (\frac {2 a c e^{3}}{5} + \frac {b^{2} e^{3}}{5} + \frac {9 b c d e^{2}}{5} + \frac {6 c^{2} d^{2} e}{5}\right ) + x^{4} \left (\frac {a b e^{3}}{4} + \frac {3 a c d e^{2}}{2} + \frac {3 b^{2} d e^{2}}{4} + \frac {9 b c d^{2} e}{4} + \frac {c^{2} d^{3}}{2}\right ) + x^{3} \left (a b d e^{2} + 2 a c d^{2} e + b^{2} d^{2} e + b c d^{3}\right ) + x^{2} \cdot \left (\frac {3 a b d^{2} e}{2} + a c d^{3} + \frac {b^{2} d^{3}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 206, normalized size = 1.66 \begin {gather*} \frac {2}{7} \, c^{2} x^{7} e^{3} + c^{2} d x^{6} e^{2} + \frac {6}{5} \, c^{2} d^{2} x^{5} e + \frac {1}{2} \, c^{2} d^{3} x^{4} + \frac {1}{2} \, b c x^{6} e^{3} + \frac {9}{5} \, b c d x^{5} e^{2} + \frac {9}{4} \, b c d^{2} x^{4} e + b c d^{3} x^{3} + \frac {1}{5} \, b^{2} x^{5} e^{3} + \frac {2}{5} \, a c x^{5} e^{3} + \frac {3}{4} \, b^{2} d x^{4} e^{2} + \frac {3}{2} \, a c d x^{4} e^{2} + b^{2} d^{2} x^{3} e + 2 \, a c d^{2} x^{3} e + \frac {1}{2} \, b^{2} d^{3} x^{2} + a c d^{3} x^{2} + \frac {1}{4} \, a b x^{4} e^{3} + a b d x^{3} e^{2} + \frac {3}{2} \, a b d^{2} x^{2} e + a b d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 179, normalized size = 1.44 \begin {gather*} x^4\,\left (\frac {3\,b^2\,d\,e^2}{4}+\frac {9\,b\,c\,d^2\,e}{4}+\frac {a\,b\,e^3}{4}+\frac {c^2\,d^3}{2}+\frac {3\,a\,c\,d\,e^2}{2}\right )+x^3\,\left (b^2\,d^2\,e+c\,b\,d^3+a\,b\,d\,e^2+2\,a\,c\,d^2\,e\right )+x^2\,\left (\frac {b^2\,d^3}{2}+\frac {3\,a\,e\,b\,d^2}{2}+a\,c\,d^3\right )+x^5\,\left (\frac {b^2\,e^3}{5}+\frac {9\,b\,c\,d\,e^2}{5}+\frac {6\,c^2\,d^2\,e}{5}+\frac {2\,a\,c\,e^3}{5}\right )+\frac {2\,c^2\,e^3\,x^7}{7}+\frac {c\,e^2\,x^6\,\left (b\,e+2\,c\,d\right )}{2}+a\,b\,d^3\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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