Optimal. Leaf size=69 \[ \frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^3}-\frac {(2 c d-b e) (d+e x)^6}{6 e^3}+\frac {c (d+e x)^7}{7 e^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {712}
\begin {gather*} \frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^3}-\frac {(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac {c (d+e x)^7}{7 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^4}{e^2}+\frac {(-2 c d+b e) (d+e x)^5}{e^2}+\frac {c (d+e x)^6}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^3}-\frac {(2 c d-b e) (d+e x)^6}{6 e^3}+\frac {c (d+e x)^7}{7 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 135, normalized size = 1.96 \begin {gather*} a d^4 x+\frac {1}{2} d^3 (b d+4 a e) x^2+\frac {1}{3} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^3+\frac {1}{2} d e \left (2 c d^2+3 b d e+2 a e^2\right ) x^4+\frac {1}{5} e^2 \left (6 c d^2+4 b d e+a e^2\right ) x^5+\frac {1}{6} e^3 (4 c d+b e) x^6+\frac {1}{7} c e^4 x^7 \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs.
\(2(63)=126\).
time = 0.56, size = 136, normalized size = 1.97
method | result | size |
norman | \(\frac {e^{4} c \,x^{7}}{7}+\left (\frac {1}{6} e^{4} b +\frac {2}{3} d \,e^{3} c \right ) x^{6}+\left (\frac {1}{5} e^{4} a +\frac {4}{5} b d \,e^{3}+\frac {6}{5} d^{2} e^{2} c \right ) x^{5}+\left (a d \,e^{3}+\frac {3}{2} b \,d^{2} e^{2}+c \,d^{3} e \right ) x^{4}+\left (2 a \,d^{2} e^{2}+\frac {4}{3} b \,d^{3} e +\frac {1}{3} c \,d^{4}\right ) x^{3}+\left (2 d^{3} e a +\frac {1}{2} b \,d^{4}\right ) x^{2}+d^{4} a x\) | \(133\) |
default | \(\frac {e^{4} c \,x^{7}}{7}+\frac {\left (e^{4} b +4 d \,e^{3} c \right ) x^{6}}{6}+\frac {\left (e^{4} a +4 b d \,e^{3}+6 d^{2} e^{2} c \right ) x^{5}}{5}+\frac {\left (4 a d \,e^{3}+6 b \,d^{2} e^{2}+4 c \,d^{3} e \right ) x^{4}}{4}+\frac {\left (6 a \,d^{2} e^{2}+4 b \,d^{3} e +c \,d^{4}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e a +b \,d^{4}\right ) x^{2}}{2}+d^{4} a x\) | \(136\) |
gosper | \(\frac {1}{7} e^{4} c \,x^{7}+\frac {1}{6} x^{6} e^{4} b +\frac {2}{3} d \,e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{4} a +\frac {4}{5} x^{5} b d \,e^{3}+\frac {6}{5} x^{5} d^{2} e^{2} c +x^{4} a d \,e^{3}+\frac {3}{2} x^{4} b \,d^{2} e^{2}+x^{4} c \,d^{3} e +2 x^{3} a \,d^{2} e^{2}+\frac {4}{3} x^{3} b \,d^{3} e +\frac {1}{3} x^{3} c \,d^{4}+2 d^{3} e a \,x^{2}+\frac {1}{2} b \,d^{4} x^{2}+d^{4} a x\) | \(147\) |
risch | \(\frac {1}{7} e^{4} c \,x^{7}+\frac {1}{6} x^{6} e^{4} b +\frac {2}{3} d \,e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{4} a +\frac {4}{5} x^{5} b d \,e^{3}+\frac {6}{5} x^{5} d^{2} e^{2} c +x^{4} a d \,e^{3}+\frac {3}{2} x^{4} b \,d^{2} e^{2}+x^{4} c \,d^{3} e +2 x^{3} a \,d^{2} e^{2}+\frac {4}{3} x^{3} b \,d^{3} e +\frac {1}{3} x^{3} c \,d^{4}+2 d^{3} e a \,x^{2}+\frac {1}{2} b \,d^{4} x^{2}+d^{4} a x\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (64) = 128\).
time = 0.27, size = 129, normalized size = 1.87 \begin {gather*} \frac {1}{7} \, c x^{7} e^{4} + \frac {1}{6} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{6} + a d^{4} x + \frac {1}{5} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs.
\(2 (64) = 128\).
time = 2.94, size = 138, normalized size = 2.00 \begin {gather*} \frac {1}{3} \, c d^{4} x^{3} + \frac {1}{2} \, b d^{4} x^{2} + a d^{4} x + \frac {1}{210} \, {\left (30 \, c x^{7} + 35 \, b x^{6} + 42 \, a x^{5}\right )} e^{4} + \frac {1}{15} \, {\left (10 \, c d x^{6} + 12 \, b d x^{5} + 15 \, a d x^{4}\right )} e^{3} + \frac {1}{10} \, {\left (12 \, c d^{2} x^{5} + 15 \, b d^{2} x^{4} + 20 \, a d^{2} x^{3}\right )} e^{2} + \frac {1}{3} \, {\left (3 \, c d^{3} x^{4} + 4 \, b d^{3} x^{3} + 6 \, a d^{3} x^{2}\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. 146 vs.
\(2 (63) = 126\).
time = 0.02, size = 146, normalized size = 2.12 \begin {gather*} a d^{4} x + \frac {c e^{4} x^{7}}{7} + x^{6} \left (\frac {b e^{4}}{6} + \frac {2 c d e^{3}}{3}\right ) + x^{5} \left (\frac {a e^{4}}{5} + \frac {4 b d e^{3}}{5} + \frac {6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (a d e^{3} + \frac {3 b d^{2} e^{2}}{2} + c d^{3} e\right ) + x^{3} \cdot \left (2 a d^{2} e^{2} + \frac {4 b d^{3} e}{3} + \frac {c d^{4}}{3}\right ) + x^{2} \cdot \left (2 a d^{3} e + \frac {b d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 140 vs.
\(2 (64) = 128\).
time = 0.87, size = 140, normalized size = 2.03 \begin {gather*} \frac {1}{7} \, c x^{7} e^{4} + \frac {2}{3} \, c d x^{6} e^{3} + \frac {6}{5} \, c d^{2} x^{5} e^{2} + c d^{3} x^{4} e + \frac {1}{3} \, c d^{4} x^{3} + \frac {1}{6} \, b x^{6} e^{4} + \frac {4}{5} \, b d x^{5} e^{3} + \frac {3}{2} \, b d^{2} x^{4} e^{2} + \frac {4}{3} \, b d^{3} x^{3} e + \frac {1}{2} \, b d^{4} x^{2} + \frac {1}{5} \, a x^{5} e^{4} + a d x^{4} e^{3} + 2 \, a d^{2} x^{3} e^{2} + 2 \, a d^{3} x^{2} e + a d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.65, size = 131, normalized size = 1.90 \begin {gather*} x^2\,\left (\frac {b\,d^4}{2}+2\,a\,e\,d^3\right )+x^6\,\left (\frac {b\,e^4}{6}+\frac {2\,c\,d\,e^3}{3}\right )+x^3\,\left (\frac {c\,d^4}{3}+\frac {4\,b\,d^3\,e}{3}+2\,a\,d^2\,e^2\right )+x^5\,\left (\frac {6\,c\,d^2\,e^2}{5}+\frac {4\,b\,d\,e^3}{5}+\frac {a\,e^4}{5}\right )+\frac {c\,e^4\,x^7}{7}+a\,d^4\,x+\frac {d\,e\,x^4\,\left (2\,c\,d^2+3\,b\,d\,e+2\,a\,e^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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