Optimal. Leaf size=87 \[ \frac {c \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1876, 212, 208, 205, 275} \[ \frac {c \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 275
Rule 1876
Rubi steps
\begin {align*} \int \frac {c+d x}{a-b x^4} \, dx &=\int \left (\frac {c}{a-b x^4}+\frac {d x}{a-b x^4}\right ) \, dx\\ &=c \int \frac {1}{a-b x^4} \, dx+d \int \frac {x}{a-b x^4} \, dx\\ &=\frac {c \int \frac {1}{\sqrt {a}-\sqrt {b} x^2} \, dx}{2 \sqrt {a}}+\frac {c \int \frac {1}{\sqrt {a}+\sqrt {b} x^2} \, dx}{2 \sqrt {a}}+\frac {1}{2} d \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )\\ &=\frac {c \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 134, normalized size = 1.54 \[ \frac {-\left (\sqrt [4]{a} d+\sqrt [4]{b} c\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\sqrt [4]{b} c \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+2 \sqrt [4]{b} c \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt [4]{a} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )-\sqrt [4]{a} d \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a^{3/4} \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 225, normalized size = 2.59 \[ \frac {\sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, a b} - \frac {\sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, a b} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {-a b} b d + \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{2}} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {-a b} b d + \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 101, normalized size = 1.16 \[ -\frac {d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{4 \sqrt {a b}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.88, size = 126, normalized size = 1.45 \[ \frac {c \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{4 \, \sqrt {a} \sqrt {b}} - \frac {d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{4 \, \sqrt {a} \sqrt {b}} - \frac {c \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.01, size = 182, normalized size = 2.09 \[ \left \{\begin {array}{cl} \frac {2\,c+3\,d\,x}{6\,b\,x^3} & \text {\ if\ \ }a=0\\ \frac {\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/4}\,x}{a^{1/4}}-1\right )\,\left (2\,a^{1/4}\,d+\sqrt {2}\,{\left (-b\right )}^{1/4}\,c\right )}{4\,a^{3/4}\,\sqrt {-b}}-\frac {\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/4}\,x}{a^{1/4}}+1\right )\,\left (4\,a^{1/4}\,d-2\,\sqrt {2}\,{\left (-b\right )}^{1/4}\,c\right )}{8\,a^{3/4}\,\sqrt {-b}}+\frac {\sqrt {2}\,c\,\ln \left (\frac {\sqrt {-b}\,x^2+\sqrt {a}+\sqrt {2}\,a^{1/4}\,{\left (-b\right )}^{1/4}\,x}{\sqrt {-b}\,x^2+\sqrt {a}-\sqrt {2}\,a^{1/4}\,{\left (-b\right )}^{1/4}\,x}\right )}{8\,a^{3/4}\,{\left (-b\right )}^{1/4}} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 126, normalized size = 1.45 \[ - \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{2} - 32 t^{2} a^{2} b d^{2} - 16 t a b c^{2} d + a d^{4} - b c^{4}, \left (t \mapsto t \log {\left (x + \frac {- 128 t^{3} a^{3} b d^{2} + 16 t^{2} a^{2} b c^{2} d + 8 t a^{2} d^{4} - 4 t a b c^{4} + 5 a c^{2} d^{3}}{4 a c d^{4} + b c^{5}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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