Optimal. Leaf size=39 \[ \frac {1}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^3+\frac {c d (d+e x)^4}{4 e^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {640, 45}
\begin {gather*} \frac {1}{3} (d+e x)^3 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^4}{4 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^2 \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^2}{e}+\frac {c d (d+e x)^3}{e}\right ) \, dx\\ &=\frac {1}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^3+\frac {c d (d+e x)^4}{4 e^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 51, normalized size = 1.31 \begin {gather*} \frac {1}{12} x \left (4 a e \left (3 d^2+3 d e x+e^2 x^2\right )+c d x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 69, normalized size = 1.77
method | result | size |
norman | \(\frac {c d \,e^{2} x^{4}}{4}+\left (\frac {1}{3} a \,e^{3}+\frac {2}{3} c \,d^{2} e \right ) x^{3}+\left (a d \,e^{2}+\frac {1}{2} c \,d^{3}\right ) x^{2}+a \,d^{2} e x\) | \(54\) |
gosper | \(\frac {x \left (3 c d \,e^{2} x^{3}+4 a \,e^{3} x^{2}+8 c \,d^{2} e \,x^{2}+12 a d \,e^{2} x +6 c \,d^{3} x +12 a \,d^{2} e \right )}{12}\) | \(56\) |
risch | \(\frac {1}{4} c d \,e^{2} x^{4}+\frac {1}{3} x^{3} a \,e^{3}+\frac {2}{3} x^{3} c \,d^{2} e +x^{2} a d \,e^{2}+\frac {1}{2} x^{2} c \,d^{3}+a \,d^{2} e x\) | \(56\) |
default | \(\frac {c d \,e^{2} x^{4}}{4}+\frac {\left (c \,d^{2} e +e \left (e^{2} a +c \,d^{2}\right )\right ) x^{3}}{3}+\frac {\left (d \left (e^{2} a +c \,d^{2}\right )+a d \,e^{2}\right ) x^{2}}{2}+a \,d^{2} e x\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 53, normalized size = 1.36 \begin {gather*} \frac {1}{4} \, c d x^{4} e^{2} + a d^{2} x e + \frac {1}{3} \, {\left (2 \, c d^{2} e + a e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{3} + 2 \, a d e^{2}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.82, size = 57, normalized size = 1.46 \begin {gather*} \frac {1}{2} \, c d^{3} x^{2} + \frac {1}{3} \, a x^{3} e^{3} + \frac {1}{4} \, {\left (c d x^{4} + 4 \, a d x^{2}\right )} e^{2} + \frac {1}{3} \, {\left (2 \, c d^{2} x^{3} + 3 \, a d^{2} x\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 56, normalized size = 1.44 \begin {gather*} a d^{2} e x + \frac {c d e^{2} x^{4}}{4} + x^{3} \left (\frac {a e^{3}}{3} + \frac {2 c d^{2} e}{3}\right ) + x^{2} \left (a d e^{2} + \frac {c d^{3}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 54, normalized size = 1.38 \begin {gather*} \frac {1}{4} \, c d x^{4} e^{2} + \frac {2}{3} \, c d^{2} x^{3} e + \frac {1}{2} \, c d^{3} x^{2} + \frac {1}{3} \, a x^{3} e^{3} + a d x^{2} e^{2} + a d^{2} x e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 53, normalized size = 1.36 \begin {gather*} x^2\,\left (\frac {c\,d^3}{2}+a\,d\,e^2\right )+x^3\,\left (\frac {2\,c\,d^2\,e}{3}+\frac {a\,e^3}{3}\right )+a\,d^2\,e\,x+\frac {c\,d\,e^2\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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