Optimal. Leaf size=239 \[ \frac {\sqrt {\sqrt {4 a c+b^2}-b} \tanh ^{-1}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}-b} \left (\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}-\frac {\sqrt {\sqrt {4 a c+b^2}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}+b} \left (-\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d} \]
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Rubi [A] time = 0.18, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2072} \[ \frac {\sqrt {\sqrt {4 a c+b^2}-b} \tanh ^{-1}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}-b} \left (\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}-\frac {\sqrt {\sqrt {4 a c+b^2}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}+b} \left (-\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d} \]
Antiderivative was successfully verified.
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Rule 2072
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx &=-\frac {\sqrt {b+\sqrt {b^2+4 a c}} \tan ^{-1}\left (\frac {\sqrt {b+\sqrt {b^2+4 a c}} x \left (b-\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}+\frac {\sqrt {-b+\sqrt {b^2+4 a c}} \tanh ^{-1}\left (\frac {\sqrt {-b+\sqrt {b^2+4 a c}} x \left (b+\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}\\ \end {align*}
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Mathematica [C] time = 0.68, size = 432, normalized size = 1.81 \[ \frac {\sqrt {\frac {4 c x^2}{\sqrt {4 a c+b^2}-b}+2} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (2 i \sqrt {a} \sqrt {c} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+\left (b-2 i \sqrt {a} \sqrt {c}\right ) \Pi \left (-\frac {i \left (b+\sqrt {b^2+4 a c}\right )}{2 \sqrt {a} \sqrt {c}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )-\left (b+2 i \sqrt {a} \sqrt {c}\right ) \Pi \left (\frac {i \left (b+\sqrt {b^2+4 a c}\right )}{2 \sqrt {a} \sqrt {c}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )\right )}{4 \sqrt {a} \sqrt {c} d \sqrt {-\frac {c}{\sqrt {4 a c+b^2}+b}} \sqrt {a+b x^2-c x^4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 6.82, size = 669, normalized size = 2.80 \[ -\frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} + \sqrt {-c x^{4} + b x^{2} + a} x^{2} + {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} + \sqrt {-c x^{4} + b x^{2} + a} x^{2} - {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) - \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} - \sqrt {-c x^{4} + b x^{2} + a} x^{2} + {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} - \sqrt {-c x^{4} + b x^{2} + a} x^{2} - {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c x^{4} + b x^{2} + a}}{c d x^{4} + a d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 568, normalized size = 2.38 \[ \frac {\sqrt {2}\, \arctan \left (\frac {-\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{4 \sqrt {-b +\sqrt {4 a c +b^{2}}}\, d}-\frac {\sqrt {2}\, \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{4 \sqrt {-b +\sqrt {4 a c +b^{2}}}\, d}-\frac {\sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\sqrt {4 a c +b^{2}}\right )}{32 a c d}+\frac {\sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}-\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\sqrt {4 a c +b^{2}}\right )}{32 a c d}+\frac {\sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\sqrt {4 a c +b^{2}}\right )}{32 a c d}-\frac {\sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}-\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\sqrt {4 a c +b^{2}}\right )}{32 a c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c x^{4} + b x^{2} + a}}{c d x^{4} + a d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {-c\,x^4+b\,x^2+a}}{c\,d\,x^4+a\,d} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {a + b x^{2} - c x^{4}}}{a + c x^{4}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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