Optimal. Leaf size=116 \[ \frac{\left (\sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} e+\sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.0954763, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1876, 275, 208, 1167, 205} \[ \frac{\left (\sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} e+\sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 1876
Rule 275
Rule 208
Rule 1167
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2}{a-b x^4} \, dx &=\int \left (\frac{d x}{a-b x^4}+\frac{c+e x^2}{a-b x^4}\right ) \, dx\\ &=d \int \frac{x}{a-b x^4} \, dx+\int \frac{c+e x^2}{a-b x^4} \, dx\\ &=\frac{1}{2} d \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )+\frac{1}{2} \left (-\frac{\sqrt{b} c}{\sqrt{a}}+e\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx+\frac{1}{2} \left (\frac{\sqrt{b} c}{\sqrt{a}}+e\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx\\ &=\frac{\left (\sqrt{b} c-\sqrt{a} e\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} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.050743, size = 187, normalized size = 1.61 \[ \frac{-\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (\sqrt [4]{a} \sqrt [4]{b} d+\sqrt{a} e+\sqrt{b} c\right )+2 \left (\sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt{b} c \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt [4]{a} \sqrt [4]{b} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )-\sqrt [4]{a} \sqrt [4]{b} d \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt{a} e \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a^{3/4} b^{3/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 161, normalized size = 1.4 \begin{align*}{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d}{4}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{e}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{4\,b}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.49401, size = 471, normalized size = 4.06 \begin{align*} - \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{3} + t^{2} \left (- 64 a^{2} b^{2} c e - 32 a^{2} b^{2} d^{2}\right ) + t \left (- 16 a^{2} b d e^{2} - 16 a b^{2} c^{2} d\right ) - a^{2} e^{4} + 2 a b c^{2} e^{2} - 4 a b c d^{2} e + a b d^{4} - b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{4} b^{2} e^{3} - 64 t^{3} a^{3} b^{3} c^{2} e + 128 t^{3} a^{3} b^{3} c d^{2} + 48 t^{2} a^{3} b^{2} c d e^{2} - 32 t^{2} a^{3} b^{2} d^{3} e - 16 t^{2} a^{2} b^{3} c^{3} d + 12 t a^{3} b c e^{4} + 12 t a^{3} b d^{2} e^{3} + 16 t a^{2} b^{2} c^{3} e^{2} - 36 t a^{2} b^{2} c^{2} d^{2} e - 8 t a^{2} b^{2} c d^{4} + 4 t a b^{3} c^{5} + 3 a^{3} d e^{5} - 5 a^{2} b c d^{3} e^{2} + 2 a^{2} b d^{5} e + 5 a b^{2} c^{4} d e - 5 a b^{2} c^{3} d^{3}}{a^{3} e^{6} + a^{2} b c^{2} e^{4} - 8 a^{2} b c d^{2} e^{3} + 4 a^{2} b d^{4} e^{2} - a b^{2} c^{4} e^{2} + 8 a b^{2} c^{3} d^{2} e - 4 a b^{2} c^{2} d^{4} - b^{3} c^{6}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08251, size = 396, normalized size = 3.41 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} d - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\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^{3}} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} d - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\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^{3}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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