Optimal. Leaf size=150 \[ \frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {\sqrt {a} e \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {d^3 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1266, 1661,
815, 649, 211, 266} \begin {gather*} \frac {\sqrt {a} e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (a e^2+3 c d^2\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a \left (d-e x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac {d^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 815
Rule 1266
Rule 1661
Rubi steps
\begin {align*} \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(d+e x) \left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {-\frac {a^2 d e}{c d^2+a e^2}-\frac {a \left (2 c d^2+a e^2\right ) x}{c d^2+a e^2}}{(d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )}{4 a c}\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\text {Subst}\left (\int \left (\frac {2 a c d^3 e}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {a \left (3 a c d^2 e+a^2 e^3+2 c^2 d^3 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a c}\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {\text {Subst}\left (\int \frac {3 a c d^2 e+a^2 e^3+2 c^2 d^3 x}{a+c x^2} \, dx,x,x^2\right )}{4 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {\left (c d^3\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {\left (a e \left (3 c d^2+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {\sqrt {a} e \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {d^3 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 142, normalized size = 0.95 \begin {gather*} \frac {\sqrt {a} e \left (3 c d^2+a e^2\right ) \left (a+c x^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )+\sqrt {c} \left (a \left (c d^2+a e^2\right ) \left (d-e x^2\right )-2 c d^3 \left (a+c x^4\right ) \log \left (d+e x^2\right )+c d^3 \left (a+c x^4\right ) \log \left (a+c x^4\right )\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 146, normalized size = 0.97
method | result | size |
default | \(-\frac {d^{3} \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {\frac {-\frac {a e \left (a \,e^{2}+c \,d^{2}\right ) x^{2}}{2 c}+\frac {a d \left (a \,e^{2}+c \,d^{2}\right )}{2 c}}{c \,x^{4}+a}+\frac {c \,d^{3} \ln \left (c \,x^{4}+a \right )+\frac {\left (a^{2} e^{3}+3 a \,d^{2} e c \right ) \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 c}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}\) | \(146\) |
risch | \(\frac {-\frac {a e \,x^{2}}{4 c \left (a \,e^{2}+c \,d^{2}\right )}+\frac {a d}{4 c \left (a \,e^{2}+c \,d^{2}\right )}}{c \,x^{4}+a}-\frac {d^{3} \ln \left (e \,x^{2}+d \right )}{2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a^{2} c^{3} e^{4}+2 a \,c^{4} d^{2} e^{2}+c^{5} d^{4}\right ) \textit {\_Z}^{2}-4 c^{3} d^{3} \textit {\_Z} +a \,e^{2}+4 c \,d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (2 a^{3} c^{3} e^{7}+2 a^{2} c^{4} d^{2} e^{5}-2 a \,c^{5} d^{4} e^{3}-2 c^{6} d^{6} e \right ) \textit {\_R}^{2}+\left (a^{2} c^{2} d \,e^{5}-6 a \,c^{3} d^{3} e^{3}-7 c^{4} d^{5} e \right ) \textit {\_R} +2 a^{2} e^{5}+8 a c \,d^{2} e^{3}+4 c^{2} d^{4} e \right ) x^{2}+\left (3 a^{3} c^{3} d \,e^{6}+5 a^{2} c^{4} d^{3} e^{4}+a \,c^{5} d^{5} e^{2}-c^{6} d^{7}\right ) \textit {\_R}^{2}+\left (2 a^{2} c^{2} d^{2} e^{4}-2 c^{4} d^{6}\right ) \textit {\_R} +2 a^{2} d \,e^{4}+8 a c \,d^{3} e^{2}+8 c^{2} d^{5}\right )\right )}{8}\) | \(381\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 191, normalized size = 1.27 \begin {gather*} \frac {d^{3} \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {d^{3} \log \left (x^{2} e + d\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt {a c}} - \frac {a x^{2} e - a d}{4 \, {\left (a c^{2} d^{2} + {\left (c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4} + a^{2} c e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.63, size = 460, normalized size = 3.07 \begin {gather*} \left [-\frac {2 \, a c d^{2} x^{2} e - 2 \, a c d^{3} + 2 \, a^{2} x^{2} e^{3} - 2 \, a^{2} d e^{2} - {\left ({\left (a c x^{4} + a^{2}\right )} e^{3} + 3 \, {\left (c^{2} d^{2} x^{4} + a c d^{2}\right )} e\right )} \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{4} + 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right ) - 2 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) + 4 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (x^{2} e + d\right )}{8 \, {\left (c^{4} d^{4} x^{4} + a c^{3} d^{4} + {\left (a^{2} c^{2} x^{4} + a^{3} c\right )} e^{4} + 2 \, {\left (a c^{3} d^{2} x^{4} + a^{2} c^{2} d^{2}\right )} e^{2}\right )}}, -\frac {a c d^{2} x^{2} e - a c d^{3} + a^{2} x^{2} e^{3} - a^{2} d e^{2} - {\left ({\left (a c x^{4} + a^{2}\right )} e^{3} + 3 \, {\left (c^{2} d^{2} x^{4} + a c d^{2}\right )} e\right )} \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right ) - {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) + 2 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (x^{2} e + d\right )}{4 \, {\left (c^{4} d^{4} x^{4} + a c^{3} d^{4} + {\left (a^{2} c^{2} x^{4} + a^{3} c\right )} e^{4} + 2 \, {\left (a c^{3} d^{2} x^{4} + a^{2} c^{2} d^{2}\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 = 4.38, size = 223, normalized size = 1.49 \begin {gather*} -\frac {d^{3} e \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac {d^{3} \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt {a c}} - \frac {c^{2} d^{3} x^{4} + a c d^{2} x^{2} e + a^{2} x^{2} e^{3} - a^{2} d e^{2}}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (c x^{4} + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.49, size = 647, normalized size = 4.31 \begin {gather*} \frac {\frac {a\,d}{4\,c\,\left (c\,d^2+a\,e^2\right )}-\frac {a\,e\,x^2}{4\,c\,\left (c\,d^2+a\,e^2\right )}}{c\,x^4+a}-\frac {d^3\,\ln \left (e\,x^2+d\right )}{2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {\ln \left (36\,c^8\,d^{10}\,x^2+36\,c^6\,d^{10}\,\sqrt {-a\,c^3}+a^5\,c\,e^{10}\,\sqrt {-a\,c^3}+a^5\,c^3\,e^{10}\,x^2-22\,a^2\,d^4\,e^6\,{\left (-a\,c^3\right )}^{3/2}-81\,c^2\,d^8\,e^2\,{\left (-a\,c^3\right )}^{3/2}+60\,a^2\,c^6\,d^6\,e^4\,x^2+22\,a^3\,c^5\,d^4\,e^6\,x^2+8\,a^4\,c^4\,d^2\,e^8\,x^2+8\,a^4\,c^2\,d^2\,e^8\,\sqrt {-a\,c^3}-60\,a\,c\,d^6\,e^4\,{\left (-a\,c^3\right )}^{3/2}+81\,a\,c^7\,d^8\,e^2\,x^2\right )\,\left (2\,c^3\,d^3+a\,e^3\,\sqrt {-a\,c^3}+3\,c\,d^2\,e\,\sqrt {-a\,c^3}\right )}{8\,\left (a^2\,c^3\,e^4+2\,a\,c^4\,d^2\,e^2+c^5\,d^4\right )}-\frac {\ln \left (36\,c^8\,d^{10}\,x^2-36\,c^6\,d^{10}\,\sqrt {-a\,c^3}-a^5\,c\,e^{10}\,\sqrt {-a\,c^3}+a^5\,c^3\,e^{10}\,x^2+22\,a^2\,d^4\,e^6\,{\left (-a\,c^3\right )}^{3/2}+81\,c^2\,d^8\,e^2\,{\left (-a\,c^3\right )}^{3/2}+60\,a^2\,c^6\,d^6\,e^4\,x^2+22\,a^3\,c^5\,d^4\,e^6\,x^2+8\,a^4\,c^4\,d^2\,e^8\,x^2-8\,a^4\,c^2\,d^2\,e^8\,\sqrt {-a\,c^3}+60\,a\,c\,d^6\,e^4\,{\left (-a\,c^3\right )}^{3/2}+81\,a\,c^7\,d^8\,e^2\,x^2\right )\,\left (a\,e^3\,\sqrt {-a\,c^3}-2\,c^3\,d^3+3\,c\,d^2\,e\,\sqrt {-a\,c^3}\right )}{8\,\left (a^2\,c^3\,e^4+2\,a\,c^4\,d^2\,e^2+c^5\,d^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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