Optimal. Leaf size=77 \[ \frac {\left (c d^2-a e^2\right )^2 (d+e x)^3}{3 e^3}-\frac {c d \left (c d^2-a e^2\right ) (d+e x)^4}{2 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {624, 45}
\begin {gather*} -\frac {c d (d+e x)^4 \left (c d^2-a e^2\right )}{2 e^3}+\frac {(d+e x)^3 \left (c d^2-a e^2\right )^2}{3 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 624
Rubi steps
\begin {align*} \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx &=\frac {\int \left (c d^2+c d e x\right )^2 \left (a e^2+c d e x\right )^2 \, dx}{c^2 d^2 e^2}\\ &=\frac {\int \left (\left (c d^2-a e^2\right )^2 \left (c d^2+c d e x\right )^2-2 \left (c d^2-a e^2\right ) \left (c d^2+c d e x\right )^3+\left (c d^2+c d e x\right )^4\right ) \, dx}{c^2 d^2 e^2}\\ &=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^3}{3 e^3}-\frac {c d \left (c d^2-a e^2\right ) (d+e x)^4}{2 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 87, normalized size = 1.13 \begin {gather*} \frac {1}{30} x \left (10 a^2 e^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a c d e x \left (6 d^2+8 d e x+3 e^2 x^2\right )+c^2 d^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 93, normalized size = 1.21
method | result | size |
default | \(\frac {c^{2} d^{2} e^{2} x^{5}}{5}+\frac {\left (e^{2} a +c \,d^{2}\right ) c d e \,x^{4}}{2}+\frac {\left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right ) x^{3}}{3}+a d e \left (e^{2} a +c \,d^{2}\right ) x^{2}+a^{2} e^{2} d^{2} x\) | \(93\) |
norman | \(\frac {c^{2} d^{2} e^{2} x^{5}}{5}+\left (\frac {1}{2} a c d \,e^{3}+\frac {1}{2} c^{2} d^{3} e \right ) x^{4}+\left (\frac {1}{3} a^{2} e^{4}+\frac {4}{3} a c \,d^{2} e^{2}+\frac {1}{3} c^{2} d^{4}\right ) x^{3}+\left (a^{2} e^{3} d +a c \,d^{3} e \right ) x^{2}+a^{2} e^{2} d^{2} x\) | \(100\) |
risch | \(\frac {1}{5} c^{2} d^{2} e^{2} x^{5}+\frac {1}{2} a c d \,e^{3} x^{4}+\frac {1}{2} c^{2} d^{3} e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{4}+\frac {4}{3} x^{3} a c \,d^{2} e^{2}+\frac {1}{3} x^{3} c^{2} d^{4}+a^{2} d \,e^{3} x^{2}+a c \,d^{3} e \,x^{2}+a^{2} e^{2} d^{2} x\) | \(106\) |
gosper | \(\frac {x \left (6 c^{2} d^{2} e^{2} x^{4}+15 x^{3} a c d \,e^{3}+15 x^{3} c^{2} d^{3} e +10 x^{2} a^{2} e^{4}+40 x^{2} a c \,d^{2} e^{2}+10 x^{2} c^{2} d^{4}+30 a^{2} d \,e^{3} x +30 a c \,d^{3} e x +30 a^{2} e^{2} d^{2}\right )}{30}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 91, normalized size = 1.18 \begin {gather*} \frac {1}{5} \, c^{2} d^{2} x^{5} e^{2} + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} c d x^{4} e + a^{2} d^{2} x e^{2} + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )}^{2} x^{3} + \frac {1}{3} \, {\left (2 \, c d x^{3} e + 3 \, {\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a d e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.16, size = 103, normalized size = 1.34 \begin {gather*} \frac {1}{3} \, c^{2} d^{4} x^{3} + \frac {1}{3} \, a^{2} x^{3} e^{4} + \frac {1}{2} \, {\left (a c d x^{4} + 2 \, a^{2} d x^{2}\right )} e^{3} + \frac {1}{15} \, {\left (3 \, c^{2} d^{2} x^{5} + 20 \, a c d^{2} x^{3} + 15 \, a^{2} d^{2} x\right )} e^{2} + \frac {1}{2} \, {\left (c^{2} d^{3} x^{4} + 2 \, a c d^{3} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 104, normalized size = 1.35 \begin {gather*} a^{2} d^{2} e^{2} x + \frac {c^{2} d^{2} e^{2} x^{5}}{5} + x^{4} \left (\frac {a c d e^{3}}{2} + \frac {c^{2} d^{3} e}{2}\right ) + x^{3} \left (\frac {a^{2} e^{4}}{3} + \frac {4 a c d^{2} e^{2}}{3} + \frac {c^{2} d^{4}}{3}\right ) + x^{2} \left (a^{2} d e^{3} + a c d^{3} e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 101, normalized size = 1.31 \begin {gather*} \frac {1}{5} \, c^{2} d^{2} x^{5} e^{2} + \frac {1}{2} \, c^{2} d^{3} x^{4} e + \frac {1}{3} \, c^{2} d^{4} x^{3} + \frac {1}{2} \, a c d x^{4} e^{3} + \frac {4}{3} \, a c d^{2} x^{3} e^{2} + a c d^{3} x^{2} e + \frac {1}{3} \, a^{2} x^{3} e^{4} + a^{2} d x^{2} e^{3} + a^{2} d^{2} x e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.57, size = 99, normalized size = 1.29 \begin {gather*} x^3\,\left (\frac {a^2\,e^4}{3}+\frac {4\,a\,c\,d^2\,e^2}{3}+\frac {c^2\,d^4}{3}\right )+x^2\,\left (a^2\,d\,e^3+c\,a\,d^3\,e\right )+x^4\,\left (\frac {c^2\,d^3\,e}{2}+\frac {a\,c\,d\,e^3}{2}\right )+a^2\,d^2\,e^2\,x+\frac {c^2\,d^2\,e^2\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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