Optimal. Leaf size=86 \[ \frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \]
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Rubi [A] time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1167, 205, 208} \begin {gather*} \frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 1167
Rubi steps
\begin {align*} \int \frac {c-d x^2}{a-b x^4} \, dx &=\frac {1}{2} \left (-\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx+\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx\\ &=\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 95, normalized size = 1.10 \begin {gather*} \frac {2 \left (\sqrt {a} d+\sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (\sqrt {b} c-\sqrt {a} d\right ) \left (\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )\right )}{4 a^{3/4} b^{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 {c-d x^2}{a-b x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.57, size = 755, normalized size = 8.78 \begin {gather*} \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x + {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x - {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x + {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) + \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x - {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 228, normalized size = 2.65 \begin {gather*} -\frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c + \sqrt {-a b} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c + \sqrt {-a b} b d\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c + \sqrt {-a b} b d\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 122, normalized size = 1.42 \begin {gather*} \frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 a}+\frac {d \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}-\frac {d \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.34, size = 109, normalized size = 1.27 \begin {gather*} \frac {{\left (\sqrt {b} c + \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (\sqrt {b} c - \sqrt {a} d\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.58, size = 579, normalized size = 6.73 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {8\,b^3\,c^2\,x\,\sqrt {-\frac {c\,d}{8\,a\,b}-\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3+\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}+\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}+\frac {8\,a\,b^2\,d^2\,x\,\sqrt {-\frac {c\,d}{8\,a\,b}-\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3+\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}+\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}\right )\,\sqrt {-\frac {a\,d^2\,\sqrt {a^3\,b^3}+b\,c^2\,\sqrt {a^3\,b^3}+2\,a^2\,b^2\,c\,d}{16\,a^3\,b^3}}-2\,\mathrm {atanh}\left (\frac {8\,b^3\,c^2\,x\,\sqrt {\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {c\,d}{8\,a\,b}+\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3-\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}-\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}+\frac {8\,a\,b^2\,d^2\,x\,\sqrt {\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {c\,d}{8\,a\,b}+\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3-\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}-\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}\right )\,\sqrt {\frac {a\,d^2\,\sqrt {a^3\,b^3}+b\,c^2\,\sqrt {a^3\,b^3}-2\,a^2\,b^2\,c\,d}{16\,a^3\,b^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 110, normalized size = 1.28 \begin {gather*} \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{3} + 64 t^{2} a^{2} b^{2} c d - a^{2} d^{4} + 2 a b c^{2} d^{2} - b^{2} c^{4}, \left (t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{3} b^{2} d - 12 t a^{2} b c d^{2} - 4 t a b^{2} c^{3}}{a^{2} d^{4} - b^{2} c^{4}} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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