Optimal. Leaf size=247 \[ -\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} c^{3/4}} \]
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Rubi [A] time = 0.15, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1168, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} c^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {d+e x^2}{a+c x^4} \, dx &=\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 c}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 c}\\ &=\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}+\frac {\left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\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} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\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} a^{3/4} c^{3/4}}\\ &=-\frac {\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} a^{3/4} c^{3/4}}+\frac {\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} a^{3/4} c^{3/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \operatorname {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} a^{3/4} c^{3/4}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \operatorname {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} a^{3/4} c^{3/4}}\\ &=-\frac {\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} a^{3/4} c^{3/4}}+\frac {\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} a^{3/4} c^{3/4}}-\frac {\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} a^{3/4} c^{3/4}}+\frac {\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} a^{3/4} c^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 183, normalized size = 0.74 \begin {gather*} \frac {-\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )\right )-2 \left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {a} e+\sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} c^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^2}{a+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.98, size = 767, normalized size = 3.11 \begin {gather*} -\frac {1}{4} \, \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x + {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + a c^{2} d^{3} - a^{2} c d e^{2}\right )} \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}}\right ) + \frac {1}{4} \, \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x - {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + a c^{2} d^{3} - a^{2} c d e^{2}\right )} \sqrt {-\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} + 2 \, d e}{a c}}\right ) + \frac {1}{4} \, \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x + {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}}\right ) - \frac {1}{4} \, \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}} \log \left (-{\left (c^{2} d^{4} - a^{2} e^{4}\right )} x - {\left (a^{3} c^{2} e \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - a c^{2} d^{3} + a^{2} c d e^{2}\right )} \sqrt {\frac {a c \sqrt {-\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{3} c^{3}}} - 2 \, d e}{a c}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 245, normalized size = 0.99 \begin {gather*} \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d + \left (a c^{3}\right )^{\frac {3}{4}} e\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 )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d + \left (a c^{3}\right )^{\frac {3}{4}} e\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 )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 260, normalized size = 1.05 \begin {gather*} \frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 a}+\frac {\sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {\sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {\sqrt {2}\, e \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.53, size = 221, normalized size = 0.89 \begin {gather*} \frac {\sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\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 )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (\sqrt {c} d + \sqrt {a} e\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 )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} c^{\frac {3}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 599, normalized size = 2.43 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {8\,c^3\,d^2\,x\,\sqrt {\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}-\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}-\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}-\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}-\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}\right )\,\sqrt {-\frac {c\,d^2\,\sqrt {-a^3\,c^3}-a\,e^2\,\sqrt {-a^3\,c^3}+2\,a^2\,c^2\,d\,e}{16\,a^3\,c^3}}-2\,\mathrm {atanh}\left (\frac {8\,c^3\,d^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}-\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}+\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a^3\,c^3}}{16\,a^3\,c^2}-\frac {d\,e}{8\,a\,c}-\frac {e^2\,\sqrt {-a^3\,c^3}}{16\,a^2\,c^3}}}{2\,c^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,c\,d^3\,\sqrt {-a^3\,c^3}}{a^2}+\frac {2\,d\,e^2\,\sqrt {-a^3\,c^3}}{a}}\right )\,\sqrt {-\frac {a\,e^2\,\sqrt {-a^3\,c^3}-c\,d^2\,\sqrt {-a^3\,c^3}+2\,a^2\,c^2\,d\,e}{16\,a^3\,c^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 109, normalized size = 0.44 \begin {gather*} \operatorname {RootSum} {\left (256 t^{4} a^{3} c^{3} + 64 t^{2} a^{2} c^{2} d e + a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}, \left (t \mapsto t \log {\left (x + \frac {64 t^{3} a^{3} c^{2} e + 12 t a^{2} c d e^{2} - 4 t a c^{2} d^{3}}{a^{2} e^{4} - c^{2} d^{4}} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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