Optimal. Leaf size=53 \[ \frac {-c d^2-a e^2}{2 e^3 (d+e x)^2}+\frac {2 c d}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 51, 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 = {711}
\begin {gather*} -\frac {a e^2+c d^2}{2 e^3 (d+e x)^2}+\frac {2 c d}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^3} \, dx &=\int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^3}-\frac {2 c d}{e^2 (d+e x)^2}+\frac {c}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {c d^2+a e^2}{2 e^3 (d+e x)^2}+\frac {2 c d}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 48, normalized size = 0.91 \begin {gather*} \frac {-a e^2+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 50, normalized size = 0.94
method | result | size |
norman | \(\frac {-\frac {e^{2} a -3 c \,d^{2}}{2 e^{3}}+\frac {2 c d x}{e^{2}}}{\left (e x +d \right )^{2}}+\frac {c \ln \left (e x +d \right )}{e^{3}}\) | \(47\) |
risch | \(\frac {-\frac {e^{2} a -3 c \,d^{2}}{2 e^{3}}+\frac {2 c d x}{e^{2}}}{\left (e x +d \right )^{2}}+\frac {c \ln \left (e x +d \right )}{e^{3}}\) | \(47\) |
default | \(\frac {2 c d}{e^{3} \left (e x +d \right )}+\frac {c \ln \left (e x +d \right )}{e^{3}}-\frac {e^{2} a +c \,d^{2}}{2 e^{3} \left (e x +d \right )^{2}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 54, normalized size = 1.02 \begin {gather*} c e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {4 \, c d x e + 3 \, c d^{2} - a e^{2}}{2 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.45, size = 71, normalized size = 1.34 \begin {gather*} \frac {4 \, c d x e + 3 \, c d^{2} - a e^{2} + 2 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )} \log \left (x e + d\right )}{2 \, {\left (x^{2} e^{5} + 2 \, d x e^{4} + d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.16, size = 56, normalized size = 1.06 \begin {gather*} \frac {c \log {\left (d + e x \right )}}{e^{3}} + \frac {- a e^{2} + 3 c d^{2} + 4 c d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.86, size = 46, normalized size = 0.87 \begin {gather*} c e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (4 \, c d x + {\left (3 \, c d^{2} - a e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 58, normalized size = 1.09 \begin {gather*} \frac {c\,\ln \left (d+e\,x\right )}{e^3}-\frac {\frac {a\,e^2-3\,c\,d^2}{2\,e^3}-\frac {2\,c\,d\,x}{e^2}}{d^2+2\,d\,e\,x+e^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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