Optimal. Leaf size=71 \[ -\frac {\left (c d^2-a e^2\right )^2}{2 e^3 (d+e x)^2}+\frac {2 c d \left (c d^2-a e^2\right )}{e^3 (d+e x)}+\frac {c^2 d^2 \log (d+e x)}{e^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45}
\begin {gather*} \frac {2 c d \left (c d^2-a e^2\right )}{e^3 (d+e x)}-\frac {\left (c d^2-a e^2\right )^2}{2 e^3 (d+e x)^2}+\frac {c^2 d^2 \log (d+e x)}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^5} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^3} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^3}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^2}+\frac {c^2 d^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^2}{2 e^3 (d+e x)^2}+\frac {2 c d \left (c d^2-a e^2\right )}{e^3 (d+e x)}+\frac {c^2 d^2 \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 59, normalized size = 0.83 \begin {gather*} \frac {\frac {\left (c d^2-a e^2\right ) \left (a e^2+c d (3 d+4 e x)\right )}{(d+e x)^2}+2 c^2 d^2 \log (d+e x)}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 81, normalized size = 1.14
method | result | size |
risch | \(\frac {-\frac {2 c d \left (e^{2} a -c \,d^{2}\right ) x}{e^{2}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}-3 c^{2} d^{4}}{2 e^{3}}}{\left (e x +d \right )^{2}}+\frac {c^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}\) | \(78\) |
default | \(-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{2 e^{3} \left (e x +d \right )^{2}}-\frac {2 c d \left (e^{2} a -c \,d^{2}\right )}{e^{3} \left (e x +d \right )}+\frac {c^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}\) | \(81\) |
norman | \(\frac {-\frac {d^{2} \left (a^{2} e^{5}+2 a c \,e^{3} d^{2}-3 c^{2} d^{4} e \right )}{2 e^{4}}-\frac {\left (a^{2} e^{5}+10 a c \,e^{3} d^{2}-11 c^{2} d^{4} e \right ) x^{2}}{2 e^{2}}-\frac {2 d \left (a c \,e^{3}-e \,c^{2} d^{2}\right ) x^{3}}{e}-\frac {d \left (a^{2} e^{5}+4 a c \,e^{3} d^{2}-5 c^{2} d^{4} e \right ) x}{e^{3}}}{\left (e x +d \right )^{4}}+\frac {c^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}\) | \(156\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 85, normalized size = 1.20 \begin {gather*} c^{2} d^{2} e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4} + 4 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} x}{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 = 2.05, size = 107, normalized size = 1.51 \begin {gather*} \frac {4 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} - 4 \, a c d x e^{3} - 2 \, a c d^{2} e^{2} - a^{2} e^{4} + 2 \, {\left (c^{2} d^{2} x^{2} e^{2} + 2 \, c^{2} d^{3} x e + c^{2} d^{4}\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 = 0.28, size = 90, normalized size = 1.27 \begin {gather*} \frac {c^{2} d^{2} \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{4} - 2 a c d^{2} e^{2} + 3 c^{2} d^{4} + x \left (- 4 a c d e^{3} + 4 c^{2} d^{3} e\right )}{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 = 0.85, size = 118, normalized size = 1.66 \begin {gather*} -c^{2} d^{2} e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{2} \, {\left (\frac {4 \, c^{2} d^{3} e^{3}}{x e + d} - \frac {c^{2} d^{4} e^{3}}{{\left (x e + d\right )}^{2}} - \frac {4 \, a c d e^{5}}{x e + d} + \frac {2 \, a c d^{2} e^{5}}{{\left (x e + d\right )}^{2}} - \frac {a^{2} e^{7}}{{\left (x e + d\right )}^{2}}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 89, normalized size = 1.25 \begin {gather*} \frac {c^2\,d^2\,\ln \left (d+e\,x\right )}{e^3}-\frac {\frac {a^2\,e^4+2\,a\,c\,d^2\,e^2-3\,c^2\,d^4}{2\,e^3}+\frac {2\,c\,d\,x\,\left (a\,e^2-c\,d^2\right )}{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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