Optimal. Leaf size=35 \[ \frac {(a e+c d x)^3}{3 \left (c d^2-a e^2\right ) (d+e x)^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 37}
\begin {gather*} \frac {(a e+c d x)^3}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
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)^6} \, dx &=\int \frac {(a e+c d x)^2}{(d+e x)^4} \, dx\\ &=\frac {(a e+c d x)^3}{3 \left (c d^2-a e^2\right ) (d+e x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 59, normalized size = 1.69 \begin {gather*} -\frac {a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs.
\(2(33)=66\).
time = 0.67, size = 83, normalized size = 2.37
method | result | size |
gosper | \(-\frac {3 c^{2} d^{2} x^{2} e^{2}+3 a c d \,e^{3} x +3 c^{2} d^{3} e x +a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3} \left (e x +d \right )^{3}}\) | \(70\) |
risch | \(\frac {-\frac {c^{2} d^{2} x^{2}}{e}-\frac {c d \left (e^{2} a +c \,d^{2}\right ) x}{e^{2}}-\frac {a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3}}}{\left (e x +d \right )^{3}}\) | \(72\) |
default | \(-\frac {c d \left (e^{2} a -c \,d^{2}\right )}{e^{3} \left (e x +d \right )^{2}}-\frac {d^{2} c^{2}}{e^{3} \left (e x +d \right )}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3} \left (e x +d \right )^{3}}\) | \(83\) |
norman | \(\frac {-\frac {d^{2} \left (a^{2} e^{6}+a c \,d^{2} e^{4}+c^{2} d^{4} e^{2}\right )}{3 e^{5}}-\frac {\left (a^{2} e^{6}+7 a c \,d^{2} e^{4}+10 c^{2} d^{4} e^{2}\right ) x^{2}}{3 e^{3}}-e \,c^{2} d^{2} x^{4}-\frac {d \left (a c \,e^{4}+3 c^{2} d^{2} e^{2}\right ) x^{3}}{e^{2}}-\frac {d \left (2 a^{2} e^{6}+5 a c \,d^{2} e^{4}+5 c^{2} d^{4} e^{2}\right ) x}{3 e^{4}}}{\left (e x +d \right )^{5}}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (34) = 68\).
time = 0.27, size = 87, normalized size = 2.49 \begin {gather*} -\frac {3 \, c^{2} d^{2} x^{2} e^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (34) = 68\).
time = 3.39, size = 87, normalized size = 2.49 \begin {gather*} -\frac {3 \, c^{2} d^{3} x e + c^{2} d^{4} + 3 \, a c d x e^{3} + a^{2} e^{4} + {\left (3 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}}{3 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (27) = 54\).
time = 0.43, size = 99, normalized size = 2.83 \begin {gather*} \frac {- a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 3 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} - 3 c^{2} d^{3} e\right )}{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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Giac [A]
time = 0.89, size = 66, normalized size = 1.89 \begin {gather*} -\frac {{\left (3 \, c^{2} d^{2} x^{2} e^{2} + 3 \, c^{2} d^{3} x e + c^{2} d^{4} + 3 \, a c d x e^{3} + a c d^{2} e^{2} + a^{2} e^{4}\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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Mupad [B]
time = 0.57, size = 65, normalized size = 1.86 \begin {gather*} -\frac {\frac {a^2\,e}{3}-d\,\left (\frac {c^2\,x^3}{3}-a\,c\,x\right )+\frac {a\,c\,d^2}{3\,e}}{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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