Optimal. Leaf size=54 \[ \frac {-a e^2-c d^2}{3 e^3 (d+e x)^3}-\frac {c}{e^3 (d+e x)}+\frac {c d}{e^3 (d+e x)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 52, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {697} \begin {gather*} -\frac {a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac {c}{e^3 (d+e x)}+\frac {c d}{e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^4} \, dx &=\int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^4}-\frac {2 c d}{e^2 (d+e x)^3}+\frac {c}{e^2 (d+e x)^2}\right ) \, dx\\ &=-\frac {c d^2+a e^2}{3 e^3 (d+e x)^3}+\frac {c d}{e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.72 \begin {gather*} -\frac {a e^2+c \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \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 {a+c x^2}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 63, normalized size = 1.17 \begin {gather*} -\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 37, normalized size = 0.69 \begin {gather*} -\frac {{\left (3 \, c x^{2} e^{2} + 3 \, c d x e + c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 0.94 \begin {gather*} \frac {c d}{\left (e x +d \right )^{2} e^{3}}-\frac {c}{\left (e x +d \right ) e^{3}}-\frac {a \,e^{2}+c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 63, normalized size = 1.17 \begin {gather*} -\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 63, normalized size = 1.17 \begin {gather*} -\frac {\frac {c\,d^2+a\,e^2}{3\,e^3}+\frac {c\,x^2}{e}+\frac {c\,d\,x}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 66, normalized size = 1.22 \begin {gather*} \frac {- a e^{2} - c d^{2} - 3 c d e x - 3 c e^{2} x^{2}}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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