Optimal. Leaf size=77 \[ -\frac {c d (d+e x)^4 \left (c d^2-a e^2\right )}{2 e^3}+\frac {(d+e x)^3 \left (c d^2-a e^2\right )^2}{3 e^3}+\frac {c^2 d^2 (d+e x)^5}{5 e^3} \]
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Rubi [A] time = 0.09, 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 = {610, 43} \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 43
Rule 610
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 = 0.03, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 105, normalized size = 1.36 \begin {gather*} \frac {1}{5} x^{5} e^{2} d^{2} c^{2} + \frac {1}{2} x^{4} e d^{3} c^{2} + \frac {1}{2} x^{4} e^{3} d c a + \frac {1}{3} x^{3} d^{4} c^{2} + \frac {4}{3} x^{3} e^{2} d^{2} c a + \frac {1}{3} x^{3} e^{4} a^{2} + x^{2} e d^{3} c a + x^{2} e^{3} d a^{2} + x e^{2} d^{2} a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, 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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maple [A] time = 0.04, size = 93, normalized size = 1.21 \begin {gather*} \frac {c^{2} d^{2} e^{2} x^{5}}{5}+a^{2} d^{2} e^{2} x +\frac {\left (a \,e^{2}+c \,d^{2}\right ) c d e \,x^{4}}{2}+\left (a \,e^{2}+c \,d^{2}\right ) a d e \,x^{2}+\frac {\left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 93, normalized size = 1.21 \begin {gather*} \frac {1}{5} \, c^{2} d^{2} e^{2} x^{5} + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} c d e x^{4} + a^{2} d^{2} e^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )}^{2} x^{3} + \frac {1}{3} \, {\left (2 \, c d e x^{3} + 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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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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sympy [A] time = 0.09, 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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