Optimal. Leaf size=156 \[ -\frac {1}{4 a d x^4}+\frac {e}{2 a d^2 x^2}+\frac {c^{3/2} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )}-\frac {\left (c d^2-a e^2\right ) \log (x)}{a^2 d^3}-\frac {e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )}+\frac {c^2 d \log \left (a+c x^4\right )}{4 a^2 \left (c d^2+a e^2\right )} \]
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Rubi [A]
time = 0.12, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1266, 908, 649,
211, 266} \begin {gather*} \frac {c^{3/2} e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )}+\frac {c^2 d \log \left (a+c x^4\right )}{4 a^2 \left (a e^2+c d^2\right )}-\frac {\log (x) \left (c d^2-a e^2\right )}{a^2 d^3}-\frac {e^4 \log \left (d+e x^2\right )}{2 d^3 \left (a e^2+c d^2\right )}+\frac {e}{2 a d^2 x^2}-\frac {1}{4 a d x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 908
Rule 1266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a d x^3}-\frac {e}{a d^2 x^2}+\frac {-c d^2+a e^2}{a^2 d^3 x}-\frac {e^5}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {c^2 (a e+c d x)}{a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a d x^4}+\frac {e}{2 a d^2 x^2}-\frac {\left (c d^2-a e^2\right ) \log (x)}{a^2 d^3}-\frac {e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )}+\frac {c^2 \text {Subst}\left (\int \frac {a e+c d x}{a+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )}\\ &=-\frac {1}{4 a d x^4}+\frac {e}{2 a d^2 x^2}-\frac {\left (c d^2-a e^2\right ) \log (x)}{a^2 d^3}-\frac {e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )}+\frac {\left (c^3 d\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2+a e^2\right )}+\frac {\left (c^2 e\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}\\ &=-\frac {1}{4 a d x^4}+\frac {e}{2 a d^2 x^2}+\frac {c^{3/2} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )}-\frac {\left (c d^2-a e^2\right ) \log (x)}{a^2 d^3}-\frac {e^4 \log \left (d+e x^2\right )}{2 d^3 \left (c d^2+a e^2\right )}+\frac {c^2 d \log \left (a+c x^4\right )}{4 a^2 \left (c d^2+a e^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 209, normalized size = 1.34 \begin {gather*} -\frac {a c d^4+a^2 d^2 e^2-2 a c d^3 e x^2-2 a^2 d e^3 x^2+2 \sqrt {a} c^{3/2} d^3 e x^4 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {a} c^{3/2} d^3 e x^4 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+4 c^2 d^4 x^4 \log (x)-4 a^2 e^4 x^4 \log (x)+2 a^2 e^4 x^4 \log \left (d+e x^2\right )-c^2 d^4 x^4 \log \left (a+c x^4\right )}{4 a^2 d^3 \left (c d^2+a e^2\right ) x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 127, normalized size = 0.81
method | result | size |
default | \(-\frac {e^{4} \ln \left (e \,x^{2}+d \right )}{2 d^{3} \left (a \,e^{2}+c \,d^{2}\right )}+\frac {c^{2} \left (\frac {d \ln \left (c \,x^{4}+a \right )}{2}+\frac {a e \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) a^{2}}-\frac {1}{4 a d \,x^{4}}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (x \right )}{d^{3} a^{2}}+\frac {e}{2 a \,d^{2} x^{2}}\) | \(127\) |
risch | \(\frac {\frac {e \,x^{2}}{2 d^{2} a}-\frac {1}{4 d a}}{x^{4}}+\frac {\ln \left (x \right ) e^{2}}{d^{3} a}-\frac {\ln \left (x \right ) c}{d \,a^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{5} e^{2}+d^{2} a^{4} c \right ) \textit {\_Z}^{2}-2 a^{2} c^{2} d \textit {\_Z} +c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-6 a^{6} d^{4} e^{4}-7 a^{5} c \,d^{6} e^{2}-5 a^{4} c^{2} d^{8}\right ) \textit {\_R}^{3}+\left (-18 a^{4} c \,d^{3} e^{4}+16 a^{3} c^{2} d^{5} e^{2}+5 a^{2} c^{3} d^{7}\right ) \textit {\_R}^{2}+\left (-8 a^{3} c \,e^{6}+24 a^{2} c^{2} d^{2} e^{4}-8 a \,c^{3} d^{4} e^{2}\right ) \textit {\_R} -8 c^{3} d \,e^{4}\right ) x^{2}+\left (-2 a^{6} d^{5} e^{3}+2 a^{5} c \,d^{7} e \right ) \textit {\_R}^{3}+\left (8 a^{5} d^{2} e^{5}-12 a^{4} c \,d^{4} e^{3}+7 a^{3} c^{2} d^{6} e \right ) \textit {\_R}^{2}+\left (-16 a^{3} c d \,e^{5}+18 a^{2} c^{2} d^{3} e^{3}-4 a \,c^{3} d^{5} e \right ) \textit {\_R} +8 a \,c^{2} e^{5}-8 c^{3} d^{2} e^{3}\right )\right )}{4}-\frac {e^{4} \ln \left (e \,x^{2}+d \right )}{2 d^{3} \left (a \,e^{2}+c \,d^{2}\right )}\) | \(379\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 143, normalized size = 0.92 \begin {gather*} \frac {c^{2} d \log \left (c x^{4} + a\right )}{4 \, {\left (a^{2} c d^{2} + a^{3} e^{2}\right )}} + \frac {c^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right ) e}{2 \, {\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {a c}} - \frac {e^{4} \log \left (x^{2} e + d\right )}{2 \, {\left (c d^{5} + a d^{3} e^{2}\right )}} - \frac {{\left (c d^{2} - a e^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{2} d^{3}} + \frac {2 \, x^{2} e - d}{4 \, a d^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 54.53, size = 350, normalized size = 2.24 \begin {gather*} \left [\frac {a c d^{3} x^{4} \sqrt {-\frac {c}{a}} e \log \left (\frac {c x^{4} + 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) + c^{2} d^{4} x^{4} \log \left (c x^{4} + a\right ) + 2 \, a c d^{3} x^{2} e - 2 \, a^{2} x^{4} e^{4} \log \left (x^{2} e + d\right ) - a c d^{4} + 2 \, a^{2} d x^{2} e^{3} - a^{2} d^{2} e^{2} - 4 \, {\left (c^{2} d^{4} x^{4} - a^{2} x^{4} e^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{2} c d^{5} x^{4} + a^{3} d^{3} x^{4} e^{2}\right )}}, -\frac {2 \, a c d^{3} x^{4} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) e - c^{2} d^{4} x^{4} \log \left (c x^{4} + a\right ) - 2 \, a c d^{3} x^{2} e + 2 \, a^{2} x^{4} e^{4} \log \left (x^{2} e + d\right ) + a c d^{4} - 2 \, a^{2} d x^{2} e^{3} + a^{2} d^{2} e^{2} + 4 \, {\left (c^{2} d^{4} x^{4} - a^{2} x^{4} e^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{2} c d^{5} x^{4} + a^{3} d^{3} x^{4} 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 = 5.42, size = 168, normalized size = 1.08 \begin {gather*} \frac {c^{2} d \log \left (c x^{4} + a\right )}{4 \, {\left (a^{2} c d^{2} + a^{3} e^{2}\right )}} + \frac {c^{2} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right ) e}{2 \, {\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {a c}} - \frac {e^{5} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{5} e + a d^{3} e^{3}\right )}} - \frac {{\left (c d^{2} - a e^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{2} d^{3}} + \frac {3 \, c d^{2} x^{4} - 3 \, a x^{4} e^{2} + 2 \, a d x^{2} e - a d^{2}}{4 \, a^{2} d^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.87, size = 1017, normalized size = 6.52 \begin {gather*} \frac {\ln \left (25\,a^2\,c^9\,d^{20}\,{\left (-a^5\,c^3\right )}^{3/2}-64\,a^{19}\,c^4\,e^{20}\,x^2-25\,a^9\,c^{14}\,d^{20}\,x^2-64\,a^{17}\,c^2\,e^{20}\,\sqrt {-a^5\,c^3}+100\,a^3\,d^8\,e^{12}\,{\left (-a^5\,c^3\right )}^{5/2}+128\,a^{11}\,d^2\,e^{18}\,{\left (-a^5\,c^3\right )}^{3/2}-112\,c^3\,d^{14}\,e^6\,{\left (-a^5\,c^3\right )}^{5/2}-76\,a^{10}\,c^{13}\,d^{18}\,e^2\,x^2-138\,a^{11}\,c^{12}\,d^{16}\,e^4\,x^2-112\,a^{12}\,c^{11}\,d^{14}\,e^6\,x^2+55\,a^{13}\,c^{10}\,d^{12}\,e^8\,x^2+104\,a^{14}\,c^9\,d^{10}\,e^{10}\,x^2+100\,a^{15}\,c^8\,d^8\,e^{12}\,x^2+172\,a^{16}\,c^7\,d^6\,e^{14}\,x^2+32\,a^{17}\,c^6\,d^4\,e^{16}\,x^2-128\,a^{18}\,c^5\,d^2\,e^{18}\,x^2+55\,a\,c^2\,d^{12}\,e^8\,{\left (-a^5\,c^3\right )}^{5/2}+104\,a^2\,c\,d^{10}\,e^{10}\,{\left (-a^5\,c^3\right )}^{5/2}-32\,a^{10}\,c\,d^4\,e^{16}\,{\left (-a^5\,c^3\right )}^{3/2}+76\,a^3\,c^8\,d^{18}\,e^2\,{\left (-a^5\,c^3\right )}^{3/2}+138\,a^4\,c^7\,d^{16}\,e^4\,{\left (-a^5\,c^3\right )}^{3/2}-172\,a^9\,c^2\,d^6\,e^{14}\,{\left (-a^5\,c^3\right )}^{3/2}\right )\,\left (e\,\sqrt {-a^5\,c^3}+a^2\,c^2\,d\right )}{4\,a^5\,e^2+4\,c\,a^4\,d^2}-\frac {e^4\,\ln \left (e\,x^2+d\right )}{2\,\left (c\,d^5+a\,d^3\,e^2\right )}-\frac {\ln \left (25\,a^9\,c^{14}\,d^{20}\,x^2+64\,a^{19}\,c^4\,e^{20}\,x^2+25\,a^2\,c^9\,d^{20}\,{\left (-a^5\,c^3\right )}^{3/2}-64\,a^{17}\,c^2\,e^{20}\,\sqrt {-a^5\,c^3}+100\,a^3\,d^8\,e^{12}\,{\left (-a^5\,c^3\right )}^{5/2}+128\,a^{11}\,d^2\,e^{18}\,{\left (-a^5\,c^3\right )}^{3/2}-112\,c^3\,d^{14}\,e^6\,{\left (-a^5\,c^3\right )}^{5/2}+76\,a^{10}\,c^{13}\,d^{18}\,e^2\,x^2+138\,a^{11}\,c^{12}\,d^{16}\,e^4\,x^2+112\,a^{12}\,c^{11}\,d^{14}\,e^6\,x^2-55\,a^{13}\,c^{10}\,d^{12}\,e^8\,x^2-104\,a^{14}\,c^9\,d^{10}\,e^{10}\,x^2-100\,a^{15}\,c^8\,d^8\,e^{12}\,x^2-172\,a^{16}\,c^7\,d^6\,e^{14}\,x^2-32\,a^{17}\,c^6\,d^4\,e^{16}\,x^2+128\,a^{18}\,c^5\,d^2\,e^{18}\,x^2+55\,a\,c^2\,d^{12}\,e^8\,{\left (-a^5\,c^3\right )}^{5/2}+104\,a^2\,c\,d^{10}\,e^{10}\,{\left (-a^5\,c^3\right )}^{5/2}-32\,a^{10}\,c\,d^4\,e^{16}\,{\left (-a^5\,c^3\right )}^{3/2}+76\,a^3\,c^8\,d^{18}\,e^2\,{\left (-a^5\,c^3\right )}^{3/2}+138\,a^4\,c^7\,d^{16}\,e^4\,{\left (-a^5\,c^3\right )}^{3/2}-172\,a^9\,c^2\,d^6\,e^{14}\,{\left (-a^5\,c^3\right )}^{3/2}\right )\,\left (e\,\sqrt {-a^5\,c^3}-a^2\,c^2\,d\right )}{4\,\left (a^5\,e^2+c\,a^4\,d^2\right )}-\frac {\frac {1}{4\,a\,d}-\frac {e\,x^2}{2\,a\,d^2}}{x^4}+\frac {\ln \left (x\right )\,\left (a\,e^2-c\,d^2\right )}{a^2\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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