Optimal. Leaf size=712 \[ \frac {d x}{4 c \left (c d^2+a e^2\right )}-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} \left (\sqrt {c} d-3 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{a} \left (\sqrt {c} d-3 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )} \]
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Rubi [A]
time = 0.44, antiderivative size = 712, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1328, 1290,
1294, 1182, 1176, 631, 210, 1179, 642, 1302, 211} \begin {gather*} \frac {\sqrt [4]{a} d^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} c^{3/4} \left (a e^2+c d^2\right )^2}-\frac {\sqrt [4]{a} d^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {c} d-\sqrt {a} e\right )}{2 \sqrt {2} c^{3/4} \left (a e^2+c d^2\right )^2}+\frac {\sqrt [4]{a} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {c} d-3 \sqrt {a} e\right )}{8 \sqrt {2} c^{7/4} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {c} d-3 \sqrt {a} e\right )}{8 \sqrt {2} c^{7/4} \left (a e^2+c d^2\right )}+\frac {d^{7/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} \left (a e^2+c d^2\right )^2}+\frac {\sqrt [4]{a} d^2 \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (a e^2+c d^2\right )^2}-\frac {\sqrt [4]{a} d^2 \left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (a e^2+c d^2\right )^2}+\frac {\sqrt [4]{a} \left (3 \sqrt {a} e+\sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{a} \left (3 \sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (a e^2+c d^2\right )}-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac {d x}{4 c \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1290
Rule 1294
Rule 1302
Rule 1328
Rubi steps
\begin {align*} \int \frac {x^8}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=-\frac {a \int \frac {x^4 \left (d-e x^2\right )}{\left (a+c x^4\right )^2} \, dx}{c d^2+a e^2}+\frac {d^2 \int \frac {x^4}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx}{c d^2+a e^2}\\ &=-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\int \frac {x^2 \left (-3 a e-c d x^2\right )}{a+c x^4} \, dx}{4 c \left (c d^2+a e^2\right )}+\frac {d^2 \int \left (\frac {d^2}{\left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac {a \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (a+c x^4\right )}\right ) \, dx}{c d^2+a e^2}\\ &=\frac {d x}{4 c \left (c d^2+a e^2\right )}-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac {\left (a d^2\right ) \int \frac {d-e x^2}{a+c x^4} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {d^4 \int \frac {1}{d+e x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {\int \frac {-a c d+3 a c e x^2}{a+c x^4} \, dx}{4 c^2 \left (c d^2+a e^2\right )}\\ &=\frac {d x}{4 c \left (c d^2+a e^2\right )}-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} \left (c d^2+a e^2\right )^2}-\frac {\left (a d^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 c \left (c d^2+a e^2\right )^2}-\frac {\left (a d^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 c \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {a} \left (\sqrt {c} d-3 \sqrt {a} e\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 c^2 \left (c d^2+a e^2\right )}-\frac {\left (\sqrt {a} \left (\sqrt {c} d+3 \sqrt {a} e\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 c^2 \left (c d^2+a e^2\right )}\\ &=\frac {d x}{4 c \left (c d^2+a e^2\right )}-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} \left (c d^2+a e^2\right )^2}-\frac {\left (a d^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c \left (c d^2+a e^2\right )^2}-\frac {\left (a d^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {a} \left (\sqrt {c} d-3 \sqrt {a} e\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 c^2 \left (c d^2+a e^2\right )}-\frac {\left (\sqrt {a} \left (\sqrt {c} d-3 \sqrt {a} e\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 c^2 \left (c d^2+a e^2\right )}+\frac {\left (\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {\left (\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}\\ &=\frac {d x}{4 c \left (c d^2+a e^2\right )}-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\left (a^{3/4} d^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\left (a^{3/4} d^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt [4]{a} \left (\sqrt {c} d-3 \sqrt {a} e\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {\left (\sqrt [4]{a} \left (\sqrt {c} d-3 \sqrt {a} e\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}\\ &=\frac {d x}{4 c \left (c d^2+a e^2\right )}-\frac {x^3 \left (a e+c d x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} \left (c d^2+a e^2\right )^2}+\frac {a^{3/4} d^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} \left (\sqrt {c} d-3 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {a^{3/4} d^2 \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{a} \left (\sqrt {c} d-3 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}+\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}+\frac {\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}-\frac {\sqrt [4]{a} d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} c^{3/4} \left (c d^2+a e^2\right )^2}-\frac {\sqrt [4]{a} \left (\sqrt {c} d+3 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} c^{7/4} \left (c d^2+a e^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 431, normalized size = 0.61 \begin {gather*} \frac {\frac {8 a \left (c d^2+a e^2\right ) x \left (d-e x^2\right )}{c \left (a+c x^4\right )}+\frac {32 d^{7/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {2 \sqrt {2} \sqrt [4]{a} \left (-5 c^{3/2} d^3+7 \sqrt {a} c d^2 e-a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{7/4}}+\frac {2 \sqrt {2} \sqrt [4]{a} \left (-5 c^{3/2} d^3+7 \sqrt {a} c d^2 e-a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{7/4}}+\frac {\sqrt {2} \sqrt [4]{a} \left (5 c^{3/2} d^3+7 \sqrt {a} c d^2 e+a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{7/4}}-\frac {\sqrt {2} \sqrt [4]{a} \left (5 c^{3/2} d^3+7 \sqrt {a} c d^2 e+a \sqrt {c} d e^2+3 a^{3/2} e^3\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{7/4}}}{32 \left (c d^2+a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 334, normalized size = 0.47
method | result | size |
default | \(\frac {d^{4} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {d e}}-\frac {a \left (\frac {\frac {e \left (a \,e^{2}+c \,d^{2}\right ) x^{3}}{4 c}-\frac {d \left (a \,e^{2}+c \,d^{2}\right ) x}{4 c}}{c \,x^{4}+a}+\frac {\frac {\left (d \,e^{2} a +5 c \,d^{3}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-3 a \,e^{3}-7 c \,d^{2} e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{4 c}\right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}\) | \(334\) |
risch | \(\text {Expression too large to display}\) | \(1581\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 487, normalized size = 0.68 \begin {gather*} \frac {d^{\frac {7}{2}} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {a {\left (\frac {2 \, \sqrt {2} {\left (5 \, c^{\frac {3}{2}} d^{3} - 7 \, \sqrt {a} c d^{2} e + a \sqrt {c} d e^{2} - 3 \, a^{\frac {3}{2}} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (5 \, c^{\frac {3}{2}} d^{3} - 7 \, \sqrt {a} c d^{2} e + a \sqrt {c} d e^{2} - 3 \, a^{\frac {3}{2}} e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (5 \, c^{\frac {3}{2}} d^{3} + 7 \, \sqrt {a} c d^{2} e + a \sqrt {c} d e^{2} + 3 \, a^{\frac {3}{2}} e^{3}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (5 \, c^{\frac {3}{2}} d^{3} + 7 \, \sqrt {a} c d^{2} e + a \sqrt {c} d e^{2} + 3 \, a^{\frac {3}{2}} e^{3}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}\right )}}{32 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}} - \frac {{\left (a c d^{2} e + a^{2} e^{3}\right )} x^{3} - {\left (a c d^{3} + a^{2} d e^{2}\right )} x}{4 \, {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + {\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4} + a^{3} c e^{4}\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. 4622 vs.
\(2 (534) = 1068\).
time = 10.10, size = 9273, normalized size = 13.02 \begin {gather*} \text {Too large to display} \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 = 3.89, size = 581, normalized size = 0.82 \begin {gather*} \frac {d^{\frac {7}{2}} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 7 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} c^{6} d^{4} + 2 \, \sqrt {2} a c^{5} d^{2} e^{2} + \sqrt {2} a^{2} c^{4} e^{4}\right )}} - \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} - 7 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} c^{6} d^{4} + 2 \, \sqrt {2} a c^{5} d^{2} e^{2} + \sqrt {2} a^{2} c^{4} e^{4}\right )}} - \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 7 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{16 \, {\left (\sqrt {2} c^{6} d^{4} + 2 \, \sqrt {2} a c^{5} d^{2} e^{2} + \sqrt {2} a^{2} c^{4} e^{4}\right )}} + \frac {{\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{3} + \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{2} + 7 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} a e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{16 \, {\left (\sqrt {2} c^{6} d^{4} + 2 \, \sqrt {2} a c^{5} d^{2} e^{2} + \sqrt {2} a^{2} c^{4} e^{4}\right )}} - \frac {a x^{3} e - a d x}{4 \, {\left (c x^{4} + a\right )} {\left (c^{2} d^{2} + a c e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.86, size = 2500, normalized size = 3.51 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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