Optimal. Leaf size=37 \[ \frac {a e^3 (c+d x)^4}{4 d}+\frac {b e^3 (c+d x)^7}{7 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {379, 14}
\begin {gather*} \frac {a e^3 (c+d x)^4}{4 d}+\frac {b e^3 (c+d x)^7}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 379
Rubi steps
\begin {align*} \int (c e+d e x)^3 \left (a+b (c+d x)^3\right ) \, dx &=\frac {e^3 \text {Subst}\left (\int x^3 \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int \left (a x^3+b x^6\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {a e^3 (c+d x)^4}{4 d}+\frac {b e^3 (c+d x)^7}{7 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(102\) vs. \(2(37)=74\).
time = 0.01, size = 102, normalized size = 2.76 \begin {gather*} e^3 \left (c^3 \left (a+b c^3\right ) x+\frac {3}{2} c^2 \left (a+2 b c^3\right ) d x^2+c \left (a+5 b c^3\right ) d^2 x^3+\frac {1}{4} \left (a+20 b c^3\right ) d^3 x^4+3 b c^2 d^4 x^5+b c d^5 x^6+\frac {1}{7} b d^6 x^7\right ) \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. \(153\) vs.
\(2(33)=66\).
time = 0.24, size = 154, normalized size = 4.16
method | result | size |
gosper | \(\frac {e^{3} x \left (4 d^{6} b \,x^{6}+28 c \,d^{5} b \,x^{5}+84 c^{2} d^{4} b \,x^{4}+140 x^{3} c^{3} b \,d^{3}+140 b \,c^{4} d^{2} x^{2}+84 x \,c^{5} b d +7 x^{3} a \,d^{3}+28 b \,c^{6}+28 x^{2} a c \,d^{2}+42 x a \,c^{2} d +28 c^{3} a \right )}{28}\) | \(110\) |
norman | \(\left (5 c^{3} e^{3} b \,d^{3}+\frac {1}{4} a \,d^{3} e^{3}\right ) x^{4}+\left (3 c^{5} e^{3} b d +\frac {3}{2} a \,c^{2} d \,e^{3}\right ) x^{2}+\left (b \,c^{6} e^{3}+a \,c^{3} e^{3}\right ) x +\left (5 c^{4} e^{3} b \,d^{2}+a c \,d^{2} e^{3}\right ) x^{3}+c \,e^{3} d^{5} b \,x^{6}+\frac {d^{6} e^{3} b \,x^{7}}{7}+3 c^{2} e^{3} d^{4} b \,x^{5}\) | \(137\) |
risch | \(\frac {1}{7} d^{6} e^{3} b \,x^{7}+c \,e^{3} d^{5} b \,x^{6}+3 c^{2} e^{3} d^{4} b \,x^{5}+5 e^{3} x^{4} c^{3} b \,d^{3}+\frac {1}{4} e^{3} d^{3} a \,x^{4}+5 e^{3} b \,c^{4} d^{2} x^{3}+e^{3} d^{2} a c \,x^{3}+3 e^{3} x^{2} c^{5} b d +\frac {3}{2} e^{3} d a \,c^{2} x^{2}+e^{3} b \,c^{6} x +a \,c^{3} e^{3} x\) | \(139\) |
default | \(\frac {d^{6} e^{3} b \,x^{7}}{7}+c \,e^{3} d^{5} b \,x^{6}+3 c^{2} e^{3} d^{4} b \,x^{5}+\frac {\left (19 c^{3} e^{3} b \,d^{3}+d^{3} e^{3} \left (b \,c^{3}+a \right )\right ) x^{4}}{4}+\frac {\left (12 c^{4} e^{3} b \,d^{2}+3 c \,e^{3} d^{2} \left (b \,c^{3}+a \right )\right ) x^{3}}{3}+\frac {\left (3 c^{5} e^{3} b d +3 c^{2} e^{3} d \left (b \,c^{3}+a \right )\right ) x^{2}}{2}+c^{3} e^{3} \left (b \,c^{3}+a \right ) x\) | \(154\) |
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. 118 vs.
\(2 (31) = 62\).
time = 0.30, size = 118, normalized size = 3.19 \begin {gather*} \frac {1}{7} \, b d^{6} x^{7} e^{3} + b c d^{5} x^{6} e^{3} + 3 \, b c^{2} d^{4} x^{5} e^{3} + \frac {1}{4} \, {\left (20 \, b c^{3} e^{3} + a e^{3}\right )} d^{3} x^{4} + {\left (5 \, b c^{4} e^{3} + a c e^{3}\right )} d^{2} x^{3} + \frac {3}{2} \, {\left (2 \, b c^{5} e^{3} + a c^{2} e^{3}\right )} d x^{2} + {\left (b c^{6} e^{3} + a c^{3} e^{3}\right )} x \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. 116 vs.
\(2 (33) = 66\).
time = 0.37, size = 116, normalized size = 3.14 \begin {gather*} \frac {1}{7} \, b d^{6} e^{3} x^{7} + b c d^{5} e^{3} x^{6} + 3 \, b c^{2} d^{4} e^{3} x^{5} + \frac {1}{4} \, {\left (20 \, b c^{3} + a\right )} d^{3} e^{3} x^{4} + {\left (5 \, b c^{4} + a c\right )} d^{2} e^{3} x^{3} + \frac {3}{2} \, {\left (2 \, b c^{5} + a c^{2}\right )} d e^{3} x^{2} + {\left (b c^{6} + a c^{3}\right )} e^{3} x \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. 144 vs.
\(2 (29) = 58\).
time = 0.02, size = 144, normalized size = 3.89 \begin {gather*} 3 b c^{2} d^{4} e^{3} x^{5} + b c d^{5} e^{3} x^{6} + \frac {b d^{6} e^{3} x^{7}}{7} + x^{4} \left (\frac {a d^{3} e^{3}}{4} + 5 b c^{3} d^{3} e^{3}\right ) + x^{3} \left (a c d^{2} e^{3} + 5 b c^{4} d^{2} e^{3}\right ) + x^{2} \cdot \left (\frac {3 a c^{2} d e^{3}}{2} + 3 b c^{5} d e^{3}\right ) + x \left (a c^{3} e^{3} + b c^{6} e^{3}\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. 127 vs.
\(2 (31) = 62\).
time = 1.36, size = 127, normalized size = 3.43 \begin {gather*} \frac {1}{7} \, b d^{6} x^{7} e^{3} + b c d^{5} x^{6} e^{3} + 3 \, b c^{2} d^{4} x^{5} e^{3} + 5 \, b c^{3} d^{3} x^{4} e^{3} + 5 \, b c^{4} d^{2} x^{3} e^{3} + 3 \, b c^{5} d x^{2} e^{3} + b c^{6} x e^{3} + \frac {1}{4} \, a d^{3} x^{4} e^{3} + a c d^{2} x^{3} e^{3} + \frac {3}{2} \, a c^{2} d x^{2} e^{3} + a c^{3} x e^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 113, normalized size = 3.05 \begin {gather*} \frac {d^3\,e^3\,x^4\,\left (20\,b\,c^3+a\right )}{4}+\frac {b\,d^6\,e^3\,x^7}{7}+c^3\,e^3\,x\,\left (b\,c^3+a\right )+\frac {3\,c^2\,d\,e^3\,x^2\,\left (2\,b\,c^3+a\right )}{2}+c\,d^2\,e^3\,x^3\,\left (5\,b\,c^3+a\right )+b\,c\,d^5\,e^3\,x^6+3\,b\,c^2\,d^4\,e^3\,x^5 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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