Optimal. Leaf size=46 \[ \frac {a (d+e x)^4}{4 e}+\frac {b (d+e x)^6}{6 e}+\frac {c (d+e x)^8}{8 e} \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1142, 14} \[ \frac {a (d+e x)^4}{4 e}+\frac {b (d+e x)^6}{6 e}+\frac {c (d+e x)^8}{8 e} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1142
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \left (a+b x^2+c x^4\right ) \, dx,x,d+e x\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \left (a x^3+b x^5+c x^7\right ) \, dx,x,d+e x\right )}{e}\\ &=\frac {a (d+e x)^4}{4 e}+\frac {b (d+e x)^6}{6 e}+\frac {c (d+e x)^8}{8 e}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 150, normalized size = 3.26 \[ \frac {1}{4} e^3 x^4 \left (a+10 b d^2+35 c d^4\right )+\frac {1}{3} d e^2 x^3 \left (3 a+10 b d^2+21 c d^4\right )+\frac {1}{2} d^2 e x^2 \left (3 a+5 b d^2+7 c d^4\right )+d^3 x \left (a+b d^2+c d^4\right )+\frac {1}{6} e^5 x^6 \left (b+21 c d^2\right )+d e^4 x^5 \left (b+7 c d^2\right )+c d e^6 x^7+\frac {1}{8} c e^7 x^8 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 175, normalized size = 3.80 \[ \frac {1}{8} x^{8} e^{7} c + x^{7} e^{6} d c + \frac {7}{2} x^{6} e^{5} d^{2} c + 7 x^{5} e^{4} d^{3} c + \frac {35}{4} x^{4} e^{3} d^{4} c + \frac {1}{6} x^{6} e^{5} b + 7 x^{3} e^{2} d^{5} c + x^{5} e^{4} d b + \frac {7}{2} x^{2} e d^{6} c + \frac {5}{2} x^{4} e^{3} d^{2} b + x d^{7} c + \frac {10}{3} x^{3} e^{2} d^{3} b + \frac {5}{2} x^{2} e d^{4} b + \frac {1}{4} x^{4} e^{3} a + x d^{5} b + x^{3} e^{2} d a + \frac {3}{2} x^{2} e d^{2} a + x d^{3} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 169, normalized size = 3.67 \[ \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} c d^{6} + \frac {3}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} c d^{4} e + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{3} c d^{2} e^{2} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} b d^{4} + \frac {1}{8} \, {\left (x^{2} e + 2 \, d x\right )}^{4} c e^{3} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{2} b d^{2} e + \frac {1}{6} \, {\left (x^{2} e + 2 \, d x\right )}^{3} b e^{2} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} a d^{2} + \frac {1}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} a e \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 298, normalized size = 6.48 \[ \frac {c \,e^{7} x^{8}}{8}+c d \,e^{6} x^{7}+\frac {\left (15 c \,d^{2} e^{5}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) e^{3}\right ) x^{6}}{6}+\frac {\left (13 c \,d^{3} e^{4}+3 \left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) d \,e^{2}+\left (4 c \,d^{3} e +2 d e b \right ) e^{3}\right ) x^{5}}{5}+\left (c \,d^{4}+b \,d^{2}+a \right ) d^{3} x +\frac {\left (4 c \,d^{4} e^{3}+3 \left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) d^{2} e +3 \left (4 c \,d^{3} e +2 d e b \right ) d \,e^{2}+\left (c \,d^{4}+b \,d^{2}+a \right ) e^{3}\right ) x^{4}}{4}+\frac {\left (\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) d^{3}+3 \left (4 c \,d^{3} e +2 d e b \right ) d^{2} e +3 \left (c \,d^{4}+b \,d^{2}+a \right ) d \,e^{2}\right ) x^{3}}{3}+\frac {\left (\left (4 c \,d^{3} e +2 d e b \right ) d^{3}+3 \left (c \,d^{4}+b \,d^{2}+a \right ) d^{2} e \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.02, size = 142, normalized size = 3.09 \[ \frac {1}{8} \, c e^{7} x^{8} + c d e^{6} x^{7} + \frac {1}{6} \, {\left (21 \, c d^{2} + b\right )} e^{5} x^{6} + {\left (7 \, c d^{3} + b d\right )} e^{4} x^{5} + \frac {1}{4} \, {\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} x^{4} + \frac {1}{3} \, {\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} x^{3} + \frac {1}{2} \, {\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e x^{2} + {\left (c d^{7} + b d^{5} + a d^{3}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 141, normalized size = 3.07 \[ x\,\left (c\,d^7+b\,d^5+a\,d^3\right )+\frac {e^5\,x^6\,\left (21\,c\,d^2+b\right )}{6}+\frac {c\,e^7\,x^8}{8}+\frac {e^3\,x^4\,\left (35\,c\,d^4+10\,b\,d^2+a\right )}{4}+\frac {d^2\,e\,x^2\,\left (7\,c\,d^4+5\,b\,d^2+3\,a\right )}{2}+\frac {d\,e^2\,x^3\,\left (21\,c\,d^4+10\,b\,d^2+3\,a\right )}{3}+d\,e^4\,x^5\,\left (7\,c\,d^2+b\right )+c\,d\,e^6\,x^7 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 178, normalized size = 3.87 \[ c d e^{6} x^{7} + \frac {c e^{7} x^{8}}{8} + x^{6} \left (\frac {b e^{5}}{6} + \frac {7 c d^{2} e^{5}}{2}\right ) + x^{5} \left (b d e^{4} + 7 c d^{3} e^{4}\right ) + x^{4} \left (\frac {a e^{3}}{4} + \frac {5 b d^{2} e^{3}}{2} + \frac {35 c d^{4} e^{3}}{4}\right ) + x^{3} \left (a d e^{2} + \frac {10 b d^{3} e^{2}}{3} + 7 c d^{5} e^{2}\right ) + x^{2} \left (\frac {3 a d^{2} e}{2} + \frac {5 b d^{4} e}{2} + \frac {7 c d^{6} e}{2}\right ) + x \left (a d^{3} + b d^{5} + c d^{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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