Optimal. Leaf size=96 \[ \frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {c} \left (c d^2+a e^2\right )}-\frac {d \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )}+\frac {d \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )} \]
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Rubi [A]
time = 0.06, antiderivative size = 96, 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, 815, 649,
211, 266} \begin {gather*} \frac {\sqrt {a} e \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {c} \left (a e^2+c d^2\right )}+\frac {d \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )}-\frac {d \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 815
Rule 1266
Rubi steps
\begin {align*} \int \frac {x^3}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {d e}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {a e+c d x}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {d \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )}+\frac {\text {Subst}\left (\int \frac {a e+c d x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2+a e^2\right )}\\ &=-\frac {d \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )}+\frac {(c d) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2+a e^2\right )}+\frac {(a e) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2+a e^2\right )}\\ &=\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {c} \left (c d^2+a e^2\right )}-\frac {d \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )}+\frac {d \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 66, normalized size = 0.69 \begin {gather*} \frac {\frac {2 \sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {c}}-2 d \log \left (d+e x^2\right )+d \log \left (a+c x^4\right )}{4 c d^2+4 a e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 72, normalized size = 0.75
method | result | size |
default | \(-\frac {d \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )}+\frac {\frac {d \ln \left (c \,x^{4}+a \right )}{2}+\frac {a e \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a \,e^{2}+2 c \,d^{2}}\) | \(72\) |
risch | \(-\frac {d \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (1+\left (a c \,e^{2}+c^{2} d^{2}\right ) \textit {\_Z}^{2}-2 \textit {\_Z} c d \right )}{\sum }\textit {\_R} \ln \left (\left (\left (2 e^{3} a c -2 d^{2} e \,c^{2}\right ) \textit {\_R}^{2}-3 \textit {\_R} c d e +2 e \right ) x^{2}+\left (3 a c d \,e^{2}-c^{2} d^{3}\right ) \textit {\_R}^{2}-c \,d^{2} \textit {\_R} +2 d \right )\right )}{4}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 81, normalized size = 0.84 \begin {gather*} \frac {a \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right ) e}{2 \, {\left (c d^{2} + a e^{2}\right )} \sqrt {a c}} + \frac {d \log \left (c x^{4} + a\right )}{4 \, {\left (c d^{2} + a e^{2}\right )}} - \frac {d \log \left (x^{2} e + d\right )}{2 \, {\left (c d^{2} + a e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 147, normalized size = 1.53 \begin {gather*} \left [\frac {\sqrt {-\frac {a}{c}} e \log \left (\frac {c x^{4} + 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right ) + d \log \left (c x^{4} + a\right ) - 2 \, d \log \left (x^{2} e + d\right )}{4 \, {\left (c d^{2} + a e^{2}\right )}}, \frac {2 \, \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right ) e + d \log \left (c x^{4} + a\right ) - 2 \, d \log \left (x^{2} e + d\right )}{4 \, {\left (c d^{2} + a 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 = 4.58, size = 86, normalized size = 0.90 \begin {gather*} -\frac {d e \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{2} e + a e^{3}\right )}} + \frac {a \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right ) e}{2 \, {\left (c d^{2} + a e^{2}\right )} \sqrt {a c}} + \frac {d \log \left (c x^{4} + a\right )}{4 \, {\left (c d^{2} + a e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.94, size = 944, normalized size = 9.83 \begin {gather*} \frac {c\,d\,\ln \left (a^4\,e^6-9\,a\,c^3\,d^6-39\,a^3\,c\,d^2\,e^4+a^3\,e^6\,x^2\,\sqrt {-a\,c}-9\,c^3\,d^6\,x^2\,\sqrt {-a\,c}+79\,a^2\,c^2\,d^4\,e^2-42\,c\,d^5\,e\,{\left (-a\,c\right )}^{3/2}+76\,a\,d^3\,e^3\,{\left (-a\,c\right )}^{3/2}+10\,a^3\,d\,e^5\,\sqrt {-a\,c}+76\,a^2\,c^2\,d^3\,e^3\,x^2-42\,a\,c^3\,d^5\,e\,x^2-10\,a^3\,c\,d\,e^5\,x^2+39\,a\,d^2\,e^4\,x^2\,{\left (-a\,c\right )}^{3/2}-79\,c\,d^4\,e^2\,x^2\,{\left (-a\,c\right )}^{3/2}\right )}{4\,c^2\,d^2+4\,a\,c\,e^2}-\frac {d\,\ln \left (e\,x^2+d\right )}{2\,\left (c\,d^2+a\,e^2\right )}+\frac {c\,d\,\ln \left (9\,a\,c^3\,d^6-a^4\,e^6+39\,a^3\,c\,d^2\,e^4+a^3\,e^6\,x^2\,\sqrt {-a\,c}-9\,c^3\,d^6\,x^2\,\sqrt {-a\,c}-79\,a^2\,c^2\,d^4\,e^2+10\,a^3\,d\,e^5\,\sqrt {-a\,c}+42\,a\,c^2\,d^5\,e\,\sqrt {-a\,c}-76\,a^2\,c^2\,d^3\,e^3\,x^2+42\,a\,c^3\,d^5\,e\,x^2+10\,a^3\,c\,d\,e^5\,x^2-76\,a^2\,c\,d^3\,e^3\,\sqrt {-a\,c}+79\,a\,c^2\,d^4\,e^2\,x^2\,\sqrt {-a\,c}-39\,a^2\,c\,d^2\,e^4\,x^2\,\sqrt {-a\,c}\right )}{4\,c^2\,d^2+4\,a\,c\,e^2}-\frac {e\,\ln \left (a^4\,e^6-9\,a\,c^3\,d^6-39\,a^3\,c\,d^2\,e^4+a^3\,e^6\,x^2\,\sqrt {-a\,c}-9\,c^3\,d^6\,x^2\,\sqrt {-a\,c}+79\,a^2\,c^2\,d^4\,e^2-42\,c\,d^5\,e\,{\left (-a\,c\right )}^{3/2}+76\,a\,d^3\,e^3\,{\left (-a\,c\right )}^{3/2}+10\,a^3\,d\,e^5\,\sqrt {-a\,c}+76\,a^2\,c^2\,d^3\,e^3\,x^2-42\,a\,c^3\,d^5\,e\,x^2-10\,a^3\,c\,d\,e^5\,x^2+39\,a\,d^2\,e^4\,x^2\,{\left (-a\,c\right )}^{3/2}-79\,c\,d^4\,e^2\,x^2\,{\left (-a\,c\right )}^{3/2}\right )\,\sqrt {-a\,c}}{4\,c^2\,d^2+4\,a\,c\,e^2}+\frac {e\,\ln \left (9\,a\,c^3\,d^6-a^4\,e^6+39\,a^3\,c\,d^2\,e^4+a^3\,e^6\,x^2\,\sqrt {-a\,c}-9\,c^3\,d^6\,x^2\,\sqrt {-a\,c}-79\,a^2\,c^2\,d^4\,e^2+10\,a^3\,d\,e^5\,\sqrt {-a\,c}+42\,a\,c^2\,d^5\,e\,\sqrt {-a\,c}-76\,a^2\,c^2\,d^3\,e^3\,x^2+42\,a\,c^3\,d^5\,e\,x^2+10\,a^3\,c\,d\,e^5\,x^2-76\,a^2\,c\,d^3\,e^3\,\sqrt {-a\,c}+79\,a\,c^2\,d^4\,e^2\,x^2\,\sqrt {-a\,c}-39\,a^2\,c\,d^2\,e^4\,x^2\,\sqrt {-a\,c}\right )\,\sqrt {-a\,c}}{4\,c^2\,d^2+4\,a\,c\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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