Optimal. Leaf size=212 \[ \frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {e^5 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {4 a^2 e^3+c d x \left (7 a e^2+3 c d^2\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}+\frac {\sqrt {c} d \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^3}+\frac {a e+c d x}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac {e^5 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \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 )^3} \, dx &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}-\frac {\int \frac {-3 c d^2-4 a e^2-3 c d e x}{(d+e x) \left (a+c x^2\right )^2} \, dx}{4 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\int \frac {c \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )+c^2 d e \left (3 c d^2+7 a e^2\right ) x}{(d+e x) \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\int \left (\frac {8 a^2 c e^6}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {c^2 \left (3 c^2 d^5+10 a c d^3 e^2+15 a^2 d e^4-8 a^2 e^5 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac {c \int \frac {3 c^2 d^5+10 a c d^3 e^2+15 a^2 d e^4-8 a^2 e^5 x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {\left (c e^5\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac {\left (c d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {4 a^2 e^3+c d \left (3 c d^2+7 a e^2\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (c d^2+a e^2\right )^3}-\frac {e^5 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 180, normalized size = 0.85 \begin {gather*} \frac {\frac {2 \left (c d^2+a e^2\right )^2 (a e+c d x)}{a \left (a+c x^2\right )^2}+\frac {\left (c d^2+a e^2\right ) \left (4 a^2 e^3+3 c^2 d^3 x+7 a c d e^2 x\right )}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+8 e^5 \log (d+e x)-4 e^5 \log \left (a+c x^2\right )}{8 \left (c d^2+a e^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 243, normalized size = 1.15
method | result | size |
default | \(\frac {e^{5} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {c \left (\frac {\frac {c d \left (7 a^{2} e^{4}+10 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) x^{3}}{8 a^{2}}+\left (\frac {1}{2} e^{5} a +\frac {1}{2} e^{3} d^{2} c \right ) x^{2}+\frac {d \left (9 a^{2} e^{4}+14 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x}{8 a}+\frac {e \left (3 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{4 c}}{\left (c \,x^{2}+a \right )^{2}}+\frac {-\frac {4 a^{2} e^{5} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (15 a^{2} d \,e^{4}+10 a c \,d^{3} e^{2}+3 c^{2} d^{5}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{3}}\) | \(243\) |
risch | \(\text {Expression too large to display}\) | \(2654\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 365, normalized size = 1.72 \begin {gather*} -\frac {e^{5} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac {e^{5} \log \left (x e + d\right )}{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}} + \frac {{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt {a c}} + \frac {4 \, a^{2} c x^{2} e^{3} + 2 \, a^{2} c d^{2} e + {\left (3 \, c^{3} d^{3} + 7 \, a c^{2} d e^{2}\right )} x^{3} + 6 \, a^{3} e^{3} + {\left (5 \, a c^{2} d^{3} + 9 \, a^{2} c d e^{2}\right )} x}{8 \, {\left (a^{4} c^{2} d^{4} + 2 \, a^{5} c d^{2} e^{2} + a^{6} e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4}\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. 475 vs.
\(2 (187) = 374\).
time = 5.77, size = 975, normalized size = 4.60 \begin {gather*} \left [\frac {6 \, c^{4} d^{5} x^{3} + 10 \, a c^{3} d^{5} x + 4 \, a^{2} c^{2} d^{4} e - 8 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} e^{5} \log \left (c x^{2} + a\right ) + 16 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} e^{5} \log \left (x e + d\right ) + {\left (3 \, c^{4} d^{5} x^{4} + 6 \, a c^{3} d^{5} x^{2} + 3 \, a^{2} c^{2} d^{5} + 15 \, {\left (a^{2} c^{2} d x^{4} + 2 \, a^{3} c d x^{2} + a^{4} d\right )} e^{4} + 10 \, {\left (a c^{3} d^{3} x^{4} + 2 \, a^{2} c^{2} d^{3} x^{2} + a^{3} c d^{3}\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 ) + 4 \, {\left (2 \, a^{3} c x^{2} + 3 \, a^{4}\right )} e^{5} + 2 \, {\left (7 \, a^{2} c^{2} d x^{3} + 9 \, a^{3} c d x\right )} e^{4} + 8 \, {\left (a^{2} c^{2} d^{2} x^{2} + 2 \, a^{3} c d^{2}\right )} e^{3} + 4 \, {\left (5 \, a c^{3} d^{3} x^{3} + 7 \, a^{2} c^{2} d^{3} x\right )} e^{2}}{16 \, {\left (a^{2} c^{5} d^{6} x^{4} + 2 \, a^{3} c^{4} d^{6} x^{2} + a^{4} c^{3} d^{6} + {\left (a^{5} c^{2} x^{4} + 2 \, a^{6} c x^{2} + a^{7}\right )} e^{6} + 3 \, {\left (a^{4} c^{3} d^{2} x^{4} + 2 \, a^{5} c^{2} d^{2} x^{2} + a^{6} c d^{2}\right )} e^{4} + 3 \, {\left (a^{3} c^{4} d^{4} x^{4} + 2 \, a^{4} c^{3} d^{4} x^{2} + a^{5} c^{2} d^{4}\right )} e^{2}\right )}}, \frac {3 \, c^{4} d^{5} x^{3} + 5 \, a c^{3} d^{5} x + 2 \, a^{2} c^{2} d^{4} e - 4 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} e^{5} \log \left (c x^{2} + a\right ) + 8 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )} e^{5} \log \left (x e + d\right ) + {\left (3 \, c^{4} d^{5} x^{4} + 6 \, a c^{3} d^{5} x^{2} + 3 \, a^{2} c^{2} d^{5} + 15 \, {\left (a^{2} c^{2} d x^{4} + 2 \, a^{3} c d x^{2} + a^{4} d\right )} e^{4} + 10 \, {\left (a c^{3} d^{3} x^{4} + 2 \, a^{2} c^{2} d^{3} x^{2} + a^{3} c d^{3}\right )} e^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + 2 \, {\left (2 \, a^{3} c x^{2} + 3 \, a^{4}\right )} e^{5} + {\left (7 \, a^{2} c^{2} d x^{3} + 9 \, a^{3} c d x\right )} e^{4} + 4 \, {\left (a^{2} c^{2} d^{2} x^{2} + 2 \, a^{3} c d^{2}\right )} e^{3} + 2 \, {\left (5 \, a c^{3} d^{3} x^{3} + 7 \, a^{2} c^{2} d^{3} x\right )} e^{2}}{8 \, {\left (a^{2} c^{5} d^{6} x^{4} + 2 \, a^{3} c^{4} d^{6} x^{2} + a^{4} c^{3} d^{6} + {\left (a^{5} c^{2} x^{4} + 2 \, a^{6} c x^{2} + a^{7}\right )} e^{6} + 3 \, {\left (a^{4} c^{3} d^{2} x^{4} + 2 \, a^{5} c^{2} d^{2} x^{2} + a^{6} c d^{2}\right )} e^{4} + 3 \, {\left (a^{3} c^{4} d^{4} x^{4} + 2 \, a^{4} c^{3} d^{4} x^{2} + a^{5} c^{2} d^{4}\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 [A]
time = 1.31, size = 342, normalized size = 1.61 \begin {gather*} -\frac {e^{5} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac {e^{6} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac {{\left (3 \, c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt {a c}} + \frac {2 \, a^{2} c^{2} d^{4} e + 8 \, a^{3} c d^{2} e^{3} + 6 \, a^{4} e^{5} + {\left (3 \, c^{4} d^{5} + 10 \, a c^{3} d^{3} e^{2} + 7 \, a^{2} c^{2} d e^{4}\right )} x^{3} + 4 \, {\left (a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{2} + {\left (5 \, a c^{3} d^{5} + 14 \, a^{2} c^{2} d^{3} e^{2} + 9 \, a^{3} c d e^{4}\right )} x}{8 \, {\left (c d^{2} + a e^{2}\right )}^{3} {\left (c x^{2} + a\right )}^{2} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 987, normalized size = 4.66 \begin {gather*} \frac {\frac {c\,d^2\,e+3\,a\,e^3}{4\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {x^3\,\left (3\,c^3\,d^3+7\,a\,c^2\,d\,e^2\right )}{8\,a^2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {x\,\left (5\,c^2\,d^3+9\,a\,c\,d\,e^2\right )}{8\,a\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {c\,e^3\,x^2}{2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}}{a^2+2\,a\,c\,x^2+c^2\,x^4}+\frac {e^5\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^3}-\frac {\ln \left (9\,c^6\,d^{14}\,{\left (-a^5\,c\right )}^{3/2}-576\,a^{12}\,e^{14}\,\sqrt {-a^5\,c}-1326\,d^4\,e^{10}\,{\left (-a^5\,c\right )}^{5/2}+9\,a^7\,c^8\,d^{14}\,x+1377\,a^6\,d^2\,e^{12}\,{\left (-a^5\,c\right )}^{3/2}+576\,a^{14}\,c\,e^{14}\,x+319\,a^2\,c^4\,d^{10}\,e^4\,{\left (-a^5\,c\right )}^{3/2}+740\,a^3\,c^3\,d^8\,e^6\,{\left (-a^5\,c\right )}^{3/2}+1015\,a^4\,c^2\,d^6\,e^8\,{\left (-a^5\,c\right )}^{3/2}+78\,a^8\,c^7\,d^{12}\,e^2\,x+319\,a^9\,c^6\,d^{10}\,e^4\,x+740\,a^{10}\,c^5\,d^8\,e^6\,x+1015\,a^{11}\,c^4\,d^6\,e^8\,x+1326\,a^{12}\,c^3\,d^4\,e^{10}\,x+1377\,a^{13}\,c^2\,d^2\,e^{12}\,x+78\,a\,c^5\,d^{12}\,e^2\,{\left (-a^5\,c\right )}^{3/2}\right )\,\left (8\,a^5\,e^5+3\,c^2\,d^5\,\sqrt {-a^5\,c}+15\,a^2\,d\,e^4\,\sqrt {-a^5\,c}+10\,a\,c\,d^3\,e^2\,\sqrt {-a^5\,c}\right )}{16\,\left (a^8\,e^6+3\,a^7\,c\,d^2\,e^4+3\,a^6\,c^2\,d^4\,e^2+a^5\,c^3\,d^6\right )}+\frac {\ln \left (576\,a^{10}\,e^{14}\,\sqrt {-a^5\,c}+9\,a^5\,c^8\,d^{14}\,x+9\,a^3\,c^7\,d^{14}\,\sqrt {-a^5\,c}-1377\,a^4\,d^2\,e^{12}\,{\left (-a^5\,c\right )}^{3/2}-319\,c^4\,d^{10}\,e^4\,{\left (-a^5\,c\right )}^{3/2}+576\,a^{12}\,c\,e^{14}\,x-1015\,a^2\,c^2\,d^6\,e^8\,{\left (-a^5\,c\right )}^{3/2}+78\,a^4\,c^6\,d^{12}\,e^2\,\sqrt {-a^5\,c}+78\,a^6\,c^7\,d^{12}\,e^2\,x+319\,a^7\,c^6\,d^{10}\,e^4\,x+740\,a^8\,c^5\,d^8\,e^6\,x+1015\,a^9\,c^4\,d^6\,e^8\,x+1326\,a^{10}\,c^3\,d^4\,e^{10}\,x+1377\,a^{11}\,c^2\,d^2\,e^{12}\,x-740\,a\,c^3\,d^8\,e^6\,{\left (-a^5\,c\right )}^{3/2}-1326\,a^3\,c\,d^4\,e^{10}\,{\left (-a^5\,c\right )}^{3/2}\right )\,\left (3\,c^2\,d^5\,\sqrt {-a^5\,c}-8\,a^5\,e^5+15\,a^2\,d\,e^4\,\sqrt {-a^5\,c}+10\,a\,c\,d^3\,e^2\,\sqrt {-a^5\,c}\right )}{16\,\left (a^8\,e^6+3\,a^7\,c\,d^2\,e^4+3\,a^6\,c^2\,d^4\,e^2+a^5\,c^3\,d^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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