Optimal. Leaf size=295 \[ \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4} \]
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Rubi [A]
time = 0.25, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {755, 837, 815,
649, 211, 266} \begin {gather*} \frac {6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac {8 a^3 e^5+c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {\sqrt {c} d \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 755
Rule 815
Rule 837
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}-\frac {\int \frac {-5 c d^2-6 a e^2-5 c d e x}{(d+e x) \left (a+c x^2\right )^3} \, dx}{6 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {\int \frac {3 c \left (5 c^2 d^4+11 a c d^2 e^2+8 a^2 e^4\right )+3 c^2 d e \left (5 c d^2+11 a e^2\right ) x}{(d+e x) \left (a+c x^2\right )^2} \, dx}{24 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac {\int \frac {-3 c^2 \left (5 c^3 d^6+16 a c^2 d^4 e^2+19 a^2 c d^2 e^4+16 a^3 e^6\right )-3 c^3 d e \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{(d+e x) \left (a+c x^2\right )} \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac {\int \left (-\frac {48 a^3 c^2 e^8}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {3 c^3 \left (5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {c \int \frac {5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {\left (c e^7\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (c d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right )\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 265, normalized size = 0.90 \begin {gather*} \frac {\frac {8 \left (c d^2+a e^2\right )^3 (a e+c d x)}{a \left (a+c x^2\right )^3}+\frac {2 \left (c d^2+a e^2\right )^2 \left (6 a^2 e^3+5 c^2 d^3 x+11 a c d e^2 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac {3 \left (c d^2+a e^2\right ) \left (8 a^3 e^5+5 c^3 d^5 x+16 a c^2 d^3 e^2 x+19 a^2 c d e^4 x\right )}{a^3 \left (a+c x^2\right )}+\frac {3 \sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{7/2}}+48 e^7 \log (d+e x)-24 e^7 \log \left (a+c x^2\right )}{48 \left (c d^2+a e^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 390, normalized size = 1.32
method | result | size |
default | \(\frac {e^{7} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}+\frac {c \left (\frac {\frac {c^{2} d \left (19 e^{6} a^{3}+35 e^{4} d^{2} a^{2} c +21 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}\right ) x^{5}}{16 a^{3}}+\left (\frac {1}{2} e^{7} a c +\frac {1}{2} e^{5} d^{2} c^{2}\right ) x^{4}+\frac {c d \left (17 e^{6} a^{3}+33 e^{4} d^{2} a^{2} c +21 d^{4} e^{2} c^{2} a +5 d^{6} c^{3}\right ) x^{3}}{6 a^{2}}+\left (\frac {5}{4} e^{7} a^{2}+\frac {3}{2} e^{5} d^{2} a c +\frac {1}{4} c^{2} d^{4} e^{3}\right ) x^{2}+\frac {d \left (29 e^{6} a^{3}+61 e^{4} d^{2} a^{2} c +43 d^{4} e^{2} c^{2} a +11 d^{6} c^{3}\right ) x}{16 a}+\frac {e \left (11 e^{6} a^{3}+18 e^{4} d^{2} a^{2} c +9 d^{4} e^{2} c^{2} a +2 d^{6} c^{3}\right )}{12 c}}{\left (c \,x^{2}+a \right )^{3}}+\frac {-\frac {8 e^{7} a^{3} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (35 a^{3} d \,e^{6}+35 a^{2} c \,d^{3} e^{4}+21 a \,c^{2} d^{5} e^{2}+5 c^{3} d^{7}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{16 a^{3}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}\) | \(390\) |
risch | \(\text {Expression too large to display}\) | \(3939\) |
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. 617 vs.
\(2 (263) = 526\).
time = 0.52, size = 617, normalized size = 2.09 \begin {gather*} -\frac {e^{7} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {e^{7} \log \left (x e + d\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt {a c}} + \frac {24 \, a^{3} c^{2} x^{4} e^{5} + 8 \, a^{3} c^{2} d^{4} e + 28 \, a^{4} c d^{2} e^{3} + 3 \, {\left (5 \, c^{5} d^{5} + 16 \, a c^{4} d^{3} e^{2} + 19 \, a^{2} c^{3} d e^{4}\right )} x^{5} + 44 \, a^{5} e^{5} + 8 \, {\left (5 \, a c^{4} d^{5} + 16 \, a^{2} c^{3} d^{3} e^{2} + 17 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 12 \, {\left (a^{3} c^{2} d^{2} e^{3} + 5 \, a^{4} c e^{5}\right )} x^{2} + 3 \, {\left (11 \, a^{2} c^{3} d^{5} + 32 \, a^{3} c^{2} d^{3} e^{2} + 29 \, a^{4} c d e^{4}\right )} x}{48 \, {\left (a^{6} c^{3} d^{6} + 3 \, a^{7} c^{2} d^{4} e^{2} + 3 \, a^{8} c d^{2} e^{4} + a^{9} e^{6} + {\left (a^{3} c^{6} d^{6} + 3 \, a^{4} c^{5} d^{4} e^{2} + 3 \, a^{5} c^{4} d^{2} e^{4} + a^{6} c^{3} e^{6}\right )} x^{6} + 3 \, {\left (a^{4} c^{5} d^{6} + 3 \, a^{5} c^{4} d^{4} e^{2} + 3 \, a^{6} c^{3} d^{2} e^{4} + a^{7} c^{2} e^{6}\right )} x^{4} + 3 \, {\left (a^{5} c^{4} d^{6} + 3 \, a^{6} c^{3} d^{4} e^{2} + 3 \, a^{7} c^{2} d^{2} e^{4} + a^{8} c e^{6}\right )} x^{2}\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. 842 vs.
\(2 (263) = 526\).
time = 19.70, size = 1709, normalized size = 5.79 \begin {gather*} \left [\frac {30 \, c^{6} d^{7} x^{5} + 80 \, a c^{5} d^{7} x^{3} + 66 \, a^{2} c^{4} d^{7} x + 16 \, a^{3} c^{3} d^{6} e - 48 \, {\left (a^{3} c^{3} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{5} c x^{2} + a^{6}\right )} e^{7} \log \left (c x^{2} + a\right ) + 96 \, {\left (a^{3} c^{3} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{5} c x^{2} + a^{6}\right )} e^{7} \log \left (x e + d\right ) + 3 \, {\left (5 \, c^{6} d^{7} x^{6} + 15 \, a c^{5} d^{7} x^{4} + 15 \, a^{2} c^{4} d^{7} x^{2} + 5 \, a^{3} c^{3} d^{7} + 35 \, {\left (a^{3} c^{3} d x^{6} + 3 \, a^{4} c^{2} d x^{4} + 3 \, a^{5} c d x^{2} + a^{6} d\right )} e^{6} + 35 \, {\left (a^{2} c^{4} d^{3} x^{6} + 3 \, a^{3} c^{3} d^{3} x^{4} + 3 \, a^{4} c^{2} d^{3} x^{2} + a^{5} c d^{3}\right )} e^{4} + 21 \, {\left (a c^{5} d^{5} x^{6} + 3 \, a^{2} c^{4} d^{5} x^{4} + 3 \, a^{3} c^{3} d^{5} x^{2} + a^{4} c^{2} d^{5}\right )} e^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} + 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + 8 \, {\left (6 \, a^{4} c^{2} x^{4} + 15 \, a^{5} c x^{2} + 11 \, a^{6}\right )} e^{7} + 2 \, {\left (57 \, a^{3} c^{3} d x^{5} + 136 \, a^{4} c^{2} d x^{3} + 87 \, a^{5} c d x\right )} e^{6} + 48 \, {\left (a^{3} c^{3} d^{2} x^{4} + 3 \, a^{4} c^{2} d^{2} x^{2} + 3 \, a^{5} c d^{2}\right )} e^{5} + 6 \, {\left (35 \, a^{2} c^{4} d^{3} x^{5} + 88 \, a^{3} c^{3} d^{3} x^{3} + 61 \, a^{4} c^{2} d^{3} x\right )} e^{4} + 24 \, {\left (a^{3} c^{3} d^{4} x^{2} + 3 \, a^{4} c^{2} d^{4}\right )} e^{3} + 6 \, {\left (21 \, a c^{5} d^{5} x^{5} + 56 \, a^{2} c^{4} d^{5} x^{3} + 43 \, a^{3} c^{3} d^{5} x\right )} e^{2}}{96 \, {\left (a^{3} c^{7} d^{8} x^{6} + 3 \, a^{4} c^{6} d^{8} x^{4} + 3 \, a^{5} c^{5} d^{8} x^{2} + a^{6} c^{4} d^{8} + {\left (a^{7} c^{3} x^{6} + 3 \, a^{8} c^{2} x^{4} + 3 \, a^{9} c x^{2} + a^{10}\right )} e^{8} + 4 \, {\left (a^{6} c^{4} d^{2} x^{6} + 3 \, a^{7} c^{3} d^{2} x^{4} + 3 \, a^{8} c^{2} d^{2} x^{2} + a^{9} c d^{2}\right )} e^{6} + 6 \, {\left (a^{5} c^{5} d^{4} x^{6} + 3 \, a^{6} c^{4} d^{4} x^{4} + 3 \, a^{7} c^{3} d^{4} x^{2} + a^{8} c^{2} d^{4}\right )} e^{4} + 4 \, {\left (a^{4} c^{6} d^{6} x^{6} + 3 \, a^{5} c^{5} d^{6} x^{4} + 3 \, a^{6} c^{4} d^{6} x^{2} + a^{7} c^{3} d^{6}\right )} e^{2}\right )}}, \frac {15 \, c^{6} d^{7} x^{5} + 40 \, a c^{5} d^{7} x^{3} + 33 \, a^{2} c^{4} d^{7} x + 8 \, a^{3} c^{3} d^{6} e - 24 \, {\left (a^{3} c^{3} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{5} c x^{2} + a^{6}\right )} e^{7} \log \left (c x^{2} + a\right ) + 48 \, {\left (a^{3} c^{3} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{5} c x^{2} + a^{6}\right )} e^{7} \log \left (x e + d\right ) + 3 \, {\left (5 \, c^{6} d^{7} x^{6} + 15 \, a c^{5} d^{7} x^{4} + 15 \, a^{2} c^{4} d^{7} x^{2} + 5 \, a^{3} c^{3} d^{7} + 35 \, {\left (a^{3} c^{3} d x^{6} + 3 \, a^{4} c^{2} d x^{4} + 3 \, a^{5} c d x^{2} + a^{6} d\right )} e^{6} + 35 \, {\left (a^{2} c^{4} d^{3} x^{6} + 3 \, a^{3} c^{3} d^{3} x^{4} + 3 \, a^{4} c^{2} d^{3} x^{2} + a^{5} c d^{3}\right )} e^{4} + 21 \, {\left (a c^{5} d^{5} x^{6} + 3 \, a^{2} c^{4} d^{5} x^{4} + 3 \, a^{3} c^{3} d^{5} x^{2} + a^{4} c^{2} d^{5}\right )} e^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + 4 \, {\left (6 \, a^{4} c^{2} x^{4} + 15 \, a^{5} c x^{2} + 11 \, a^{6}\right )} e^{7} + {\left (57 \, a^{3} c^{3} d x^{5} + 136 \, a^{4} c^{2} d x^{3} + 87 \, a^{5} c d x\right )} e^{6} + 24 \, {\left (a^{3} c^{3} d^{2} x^{4} + 3 \, a^{4} c^{2} d^{2} x^{2} + 3 \, a^{5} c d^{2}\right )} e^{5} + 3 \, {\left (35 \, a^{2} c^{4} d^{3} x^{5} + 88 \, a^{3} c^{3} d^{3} x^{3} + 61 \, a^{4} c^{2} d^{3} x\right )} e^{4} + 12 \, {\left (a^{3} c^{3} d^{4} x^{2} + 3 \, a^{4} c^{2} d^{4}\right )} e^{3} + 3 \, {\left (21 \, a c^{5} d^{5} x^{5} + 56 \, a^{2} c^{4} d^{5} x^{3} + 43 \, a^{3} c^{3} d^{5} x\right )} e^{2}}{48 \, {\left (a^{3} c^{7} d^{8} x^{6} + 3 \, a^{4} c^{6} d^{8} x^{4} + 3 \, a^{5} c^{5} d^{8} x^{2} + a^{6} c^{4} d^{8} + {\left (a^{7} c^{3} x^{6} + 3 \, a^{8} c^{2} x^{4} + 3 \, a^{9} c x^{2} + a^{10}\right )} e^{8} + 4 \, {\left (a^{6} c^{4} d^{2} x^{6} + 3 \, a^{7} c^{3} d^{2} x^{4} + 3 \, a^{8} c^{2} d^{2} x^{2} + a^{9} c d^{2}\right )} e^{6} + 6 \, {\left (a^{5} c^{5} d^{4} x^{6} + 3 \, a^{6} c^{4} d^{4} x^{4} + 3 \, a^{7} c^{3} d^{4} x^{2} + a^{8} c^{2} d^{4}\right )} e^{4} + 4 \, {\left (a^{4} c^{6} d^{6} x^{6} + 3 \, a^{5} c^{5} d^{6} x^{4} + 3 \, a^{6} c^{4} d^{6} x^{2} + a^{7} c^{3} d^{6}\right )} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 530 vs.
\(2 (263) = 526\).
time = 1.38, size = 530, normalized size = 1.80 \begin {gather*} -\frac {e^{7} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {e^{8} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{8} e + 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} + 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} + \frac {{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt {a c}} + \frac {8 \, a^{3} c^{3} d^{6} e + 36 \, a^{4} c^{2} d^{4} e^{3} + 72 \, a^{5} c d^{2} e^{5} + 44 \, a^{6} e^{7} + 3 \, {\left (5 \, c^{6} d^{7} + 21 \, a c^{5} d^{5} e^{2} + 35 \, a^{2} c^{4} d^{3} e^{4} + 19 \, a^{3} c^{3} d e^{6}\right )} x^{5} + 24 \, {\left (a^{3} c^{3} d^{2} e^{5} + a^{4} c^{2} e^{7}\right )} x^{4} + 8 \, {\left (5 \, a c^{5} d^{7} + 21 \, a^{2} c^{4} d^{5} e^{2} + 33 \, a^{3} c^{3} d^{3} e^{4} + 17 \, a^{4} c^{2} d e^{6}\right )} x^{3} + 12 \, {\left (a^{3} c^{3} d^{4} e^{3} + 6 \, a^{4} c^{2} d^{2} e^{5} + 5 \, a^{5} c e^{7}\right )} x^{2} + 3 \, {\left (11 \, a^{2} c^{4} d^{7} + 43 \, a^{3} c^{3} d^{5} e^{2} + 61 \, a^{4} c^{2} d^{3} e^{4} + 29 \, a^{5} c d e^{6}\right )} x}{48 \, {\left (c d^{2} + a e^{2}\right )}^{4} {\left (c x^{2} + a\right )}^{3} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.82, size = 1470, normalized size = 4.98 \begin {gather*} \frac {\frac {11\,a^2\,e^5+7\,a\,c\,d^2\,e^3+2\,c^2\,d^4\,e}{12\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x^2\,\left (c^2\,d^2\,e^3+5\,a\,c\,e^5\right )}{4\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x\,\left (29\,a^2\,c\,d\,e^4+32\,a\,c^2\,d^3\,e^2+11\,c^3\,d^5\right )}{16\,a\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {c^2\,e^5\,x^4}{2\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x^3\,\left (17\,a^2\,c^2\,d\,e^4+16\,a\,c^3\,d^3\,e^2+5\,c^4\,d^5\right )}{6\,a^2\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}+\frac {x^5\,\left (19\,a^2\,c^3\,d\,e^4+16\,a\,c^4\,d^3\,e^2+5\,c^5\,d^5\right )}{16\,a^3\,\left (a^3\,e^6+3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2+c^3\,d^6\right )}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6}+\frac {e^7\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^4}-\frac {\ln \left (25\,a^7\,c^{10}\,d^{18}\,x-2304\,a^{13}\,e^{18}\,\sqrt {-a^7\,c}-25\,a^4\,c^9\,d^{18}\,\sqrt {-a^7\,c}+5833\,a^5\,d^2\,e^{16}\,{\left (-a^7\,c\right )}^{3/2}+3612\,c^5\,d^{12}\,e^6\,{\left (-a^7\,c\right )}^{3/2}+2304\,a^{16}\,c\,e^{18}\,x+9660\,a^2\,c^3\,d^8\,e^{10}\,{\left (-a^7\,c\right )}^{3/2}+8820\,a^3\,c^2\,d^6\,e^{12}\,{\left (-a^7\,c\right )}^{3/2}-260\,a^5\,c^8\,d^{16}\,e^2\,\sqrt {-a^7\,c}-1236\,a^6\,c^7\,d^{14}\,e^4\,\sqrt {-a^7\,c}+260\,a^8\,c^9\,d^{16}\,e^2\,x+1236\,a^9\,c^8\,d^{14}\,e^4\,x+3612\,a^{10}\,c^7\,d^{12}\,e^6\,x+7126\,a^{11}\,c^6\,d^{10}\,e^8\,x+9660\,a^{12}\,c^5\,d^8\,e^{10}\,x+8820\,a^{13}\,c^4\,d^6\,e^{12}\,x+7204\,a^{14}\,c^3\,d^4\,e^{14}\,x+5833\,a^{15}\,c^2\,d^2\,e^{16}\,x+7126\,a\,c^4\,d^{10}\,e^8\,{\left (-a^7\,c\right )}^{3/2}+7204\,a^4\,c\,d^4\,e^{14}\,{\left (-a^7\,c\right )}^{3/2}\right )\,\left (16\,a^7\,e^7+5\,c^3\,d^7\,\sqrt {-a^7\,c}+35\,a^3\,d\,e^6\,\sqrt {-a^7\,c}+21\,a\,c^2\,d^5\,e^2\,\sqrt {-a^7\,c}+35\,a^2\,c\,d^3\,e^4\,\sqrt {-a^7\,c}\right )}{32\,\left (a^{11}\,e^8+4\,a^{10}\,c\,d^2\,e^6+6\,a^9\,c^2\,d^4\,e^4+4\,a^8\,c^3\,d^6\,e^2+a^7\,c^4\,d^8\right )}+\frac {\ln \left (2304\,a^{13}\,e^{18}\,\sqrt {-a^7\,c}+25\,a^7\,c^{10}\,d^{18}\,x+25\,a^4\,c^9\,d^{18}\,\sqrt {-a^7\,c}-5833\,a^5\,d^2\,e^{16}\,{\left (-a^7\,c\right )}^{3/2}-3612\,c^5\,d^{12}\,e^6\,{\left (-a^7\,c\right )}^{3/2}+2304\,a^{16}\,c\,e^{18}\,x-9660\,a^2\,c^3\,d^8\,e^{10}\,{\left (-a^7\,c\right )}^{3/2}-8820\,a^3\,c^2\,d^6\,e^{12}\,{\left (-a^7\,c\right )}^{3/2}+260\,a^5\,c^8\,d^{16}\,e^2\,\sqrt {-a^7\,c}+1236\,a^6\,c^7\,d^{14}\,e^4\,\sqrt {-a^7\,c}+260\,a^8\,c^9\,d^{16}\,e^2\,x+1236\,a^9\,c^8\,d^{14}\,e^4\,x+3612\,a^{10}\,c^7\,d^{12}\,e^6\,x+7126\,a^{11}\,c^6\,d^{10}\,e^8\,x+9660\,a^{12}\,c^5\,d^8\,e^{10}\,x+8820\,a^{13}\,c^4\,d^6\,e^{12}\,x+7204\,a^{14}\,c^3\,d^4\,e^{14}\,x+5833\,a^{15}\,c^2\,d^2\,e^{16}\,x-7126\,a\,c^4\,d^{10}\,e^8\,{\left (-a^7\,c\right )}^{3/2}-7204\,a^4\,c\,d^4\,e^{14}\,{\left (-a^7\,c\right )}^{3/2}\right )\,\left (5\,c^3\,d^7\,\sqrt {-a^7\,c}-16\,a^7\,e^7+35\,a^3\,d\,e^6\,\sqrt {-a^7\,c}+21\,a\,c^2\,d^5\,e^2\,\sqrt {-a^7\,c}+35\,a^2\,c\,d^3\,e^4\,\sqrt {-a^7\,c}\right )}{32\,\left (a^{11}\,e^8+4\,a^{10}\,c\,d^2\,e^6+6\,a^9\,c^2\,d^4\,e^4+4\,a^8\,c^3\,d^6\,e^2+a^7\,c^4\,d^8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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