Optimal. Leaf size=26 \[ -\frac {e^2}{3 b d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {372, 261} \begin {gather*} -\frac {e^2}{3 b d \left (a+b (c+d x)^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 261
Rule 372
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b (c+d x)^3\right )^2} \, dx &=\frac {e^2 \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^2}{3 b d \left (a+b (c+d x)^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} -\frac {e^2}{3 b d \left (a+b (c+d x)^3\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 \frac {(c e+d e x)^2}{\left (a+b (c+d x)^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.20, size = 55, normalized size = 2.12 \begin {gather*} -\frac {e^{2}}{3 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + {\left (b^{2} c^{3} + a b\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 45, normalized size = 1.73 \begin {gather*} -\frac {e^{2}}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 47, normalized size = 1.81 \begin {gather*} -\frac {e^{2}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 55, normalized size = 2.12 \begin {gather*} -\frac {e^{2}}{3 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + {\left (b^{2} c^{3} + a b\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 46, normalized size = 1.77 \begin {gather*} -\frac {e^2}{3\,b\,d\,\left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.37, size = 60, normalized size = 2.31 \begin {gather*} - \frac {e^{2}}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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