Optimal. Leaf size=62 \[ \frac {d (c d-b e) (d+e x)^4}{4 e^3}-\frac {(2 c d-b e) (d+e x)^5}{5 e^3}+\frac {c (d+e x)^6}{6 e^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {712}
\begin {gather*} -\frac {(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac {d (d+e x)^4 (c d-b e)}{4 e^3}+\frac {c (d+e x)^6}{6 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^3 \left (b x+c x^2\right ) \, dx &=\int \left (\frac {d (c d-b e) (d+e x)^3}{e^2}+\frac {(-2 c d+b e) (d+e x)^4}{e^2}+\frac {c (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac {d (c d-b e) (d+e x)^4}{4 e^3}-\frac {(2 c d-b e) (d+e x)^5}{5 e^3}+\frac {c (d+e x)^6}{6 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 67, normalized size = 1.08 \begin {gather*} \frac {1}{60} x^2 \left (30 b d^3+20 d^2 (c d+3 b e) x+45 d e (c d+b e) x^2+12 e^2 (3 c d+b e) x^3+10 c e^3 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 76, normalized size = 1.23
method | result | size |
norman | \(\frac {e^{3} c \,x^{6}}{6}+\left (\frac {1}{5} e^{3} b +\frac {3}{5} d \,e^{2} c \right ) x^{5}+\left (\frac {3}{4} b d \,e^{2}+\frac {3}{4} d^{2} e c \right ) x^{4}+\left (b \,d^{2} e +\frac {1}{3} d^{3} c \right ) x^{3}+\frac {b \,d^{3} x^{2}}{2}\) | \(74\) |
gosper | \(\frac {x^{2} \left (10 e^{3} c \,x^{4}+12 x^{3} e^{3} b +36 x^{3} d \,e^{2} c +45 x^{2} b d \,e^{2}+45 x^{2} d^{2} e c +60 x b \,d^{2} e +20 x \,d^{3} c +30 b \,d^{3}\right )}{60}\) | \(76\) |
default | \(\frac {e^{3} c \,x^{6}}{6}+\frac {\left (e^{3} b +3 d \,e^{2} c \right ) x^{5}}{5}+\frac {\left (3 b d \,e^{2}+3 d^{2} e c \right ) x^{4}}{4}+\frac {\left (3 b \,d^{2} e +d^{3} c \right ) x^{3}}{3}+\frac {b \,d^{3} x^{2}}{2}\) | \(76\) |
risch | \(\frac {1}{6} e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{3} b +\frac {3}{5} x^{5} d \,e^{2} c +\frac {3}{4} x^{4} b d \,e^{2}+\frac {3}{4} x^{4} d^{2} e c +x^{3} b \,d^{2} e +\frac {1}{3} x^{3} d^{3} c +\frac {1}{2} b \,d^{3} x^{2}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 71, normalized size = 1.15 \begin {gather*} \frac {1}{6} \, c x^{6} e^{3} + \frac {1}{2} \, b d^{3} x^{2} + \frac {1}{5} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + \frac {3}{4} \, {\left (c d^{2} e + b d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, b d^{2} e\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 78, normalized size = 1.26 \begin {gather*} \frac {1}{3} \, c d^{3} x^{3} + \frac {1}{2} \, b d^{3} x^{2} + \frac {1}{30} \, {\left (5 \, c x^{6} + 6 \, b x^{5}\right )} e^{3} + \frac {3}{20} \, {\left (4 \, c d x^{5} + 5 \, b d x^{4}\right )} e^{2} + \frac {1}{4} \, {\left (3 \, c d^{2} x^{4} + 4 \, b 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 = 0.01, size = 80, normalized size = 1.29 \begin {gather*} \frac {b d^{3} x^{2}}{2} + \frac {c e^{3} x^{6}}{6} + x^{5} \left (\frac {b e^{3}}{5} + \frac {3 c d e^{2}}{5}\right ) + x^{4} \cdot \left (\frac {3 b d e^{2}}{4} + \frac {3 c d^{2} e}{4}\right ) + x^{3} \left (b d^{2} e + \frac {c d^{3}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 74, normalized size = 1.19 \begin {gather*} \frac {1}{6} \, c x^{6} e^{3} + \frac {3}{5} \, c d x^{5} e^{2} + \frac {3}{4} \, c d^{2} x^{4} e + \frac {1}{3} \, c d^{3} x^{3} + \frac {1}{5} \, b x^{5} e^{3} + \frac {3}{4} \, b d x^{4} e^{2} + b d^{2} x^{3} e + \frac {1}{2} \, b d^{3} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 68, normalized size = 1.10 \begin {gather*} x^3\,\left (\frac {c\,d^3}{3}+b\,e\,d^2\right )+x^5\,\left (\frac {b\,e^3}{5}+\frac {3\,c\,d\,e^2}{5}\right )+\frac {b\,d^3\,x^2}{2}+\frac {c\,e^3\,x^6}{6}+\frac {3\,d\,e\,x^4\,\left (b\,e+c\,d\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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