Optimal. Leaf size=337 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-\sqrt{a} \sqrt{b} e-a g+b c\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-\sqrt{a} \sqrt{b} e-a g+b c\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e-a g+b c\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{a} \sqrt{b} e-a g+b c\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{(b d-a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{f \log \left (a+b x^4\right )}{4 b}+\frac{g x}{b}+\frac{h x^2}{2 b} \]
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Rubi [A] time = 0.399356, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.371, Rules used = {1885, 1887, 1168, 1162, 617, 204, 1165, 628, 1819, 1810, 635, 205, 260} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-\sqrt{a} \sqrt{b} e-a g+b c\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-\sqrt{a} \sqrt{b} e-a g+b c\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e-a g+b c\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{a} \sqrt{b} e-a g+b c\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{(b d-a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{f \log \left (a+b x^4\right )}{4 b}+\frac{g x}{b}+\frac{h x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 1885
Rule 1887
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 1819
Rule 1810
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{a+b x^4} \, dx &=\int \left (\frac{c+e x^2+g x^4}{a+b x^4}+\frac{x \left (d+f x^2+h x^4\right )}{a+b x^4}\right ) \, dx\\ &=\int \frac{c+e x^2+g x^4}{a+b x^4} \, dx+\int \frac{x \left (d+f x^2+h x^4\right )}{a+b x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+f x+h x^2}{a+b x^2} \, dx,x,x^2\right )+\int \left (\frac{g}{b}+\frac{b c-a g+b e x^2}{b \left (a+b x^4\right )}\right ) \, dx\\ &=\frac{g x}{b}+\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{h}{b}+\frac{b d-a h+b f x}{b \left (a+b x^2\right )}\right ) \, dx,x,x^2\right )+\frac{\int \frac{b c-a g+b e x^2}{a+b x^4} \, dx}{b}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{b d-a h+b f x}{a+b x^2} \, dx,x,x^2\right )}{2 b}+\frac{\left (b c-\sqrt{a} \sqrt{b} e-a g\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{2 \sqrt{a} b^{3/2}}+\frac{\left (b c+\sqrt{a} \sqrt{b} e-a g\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{2 \sqrt{a} b^{3/2}}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{1}{2} f \operatorname{Subst}\left (\int \frac{x}{a+b x^2} \, dx,x,x^2\right )-\frac{\left (b c-\sqrt{a} \sqrt{b} e-a g\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\left (b c-\sqrt{a} \sqrt{b} e-a g\right ) \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\left (b c+\sqrt{a} \sqrt{b} e-a g\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \sqrt{a} b^{3/2}}+\frac{\left (b c+\sqrt{a} \sqrt{b} e-a g\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 \sqrt{a} b^{3/2}}+\frac{(b d-a h) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{2 b}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{(b d-a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{\left (b c-\sqrt{a} \sqrt{b} e-a g\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\left (b c-\sqrt{a} \sqrt{b} e-a g\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{f \log \left (a+b x^4\right )}{4 b}+\frac{\left (b c+\sqrt{a} \sqrt{b} e-a g\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\left (b c+\sqrt{a} \sqrt{b} e-a g\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{(b d-a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{\left (b c+\sqrt{a} \sqrt{b} e-a g\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\left (b c+\sqrt{a} \sqrt{b} e-a g\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{5/4}}-\frac{\left (b c-\sqrt{a} \sqrt{b} e-a g\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{\left (b c-\sqrt{a} \sqrt{b} e-a g\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{5/4}}+\frac{f \log \left (a+b x^4\right )}{4 b}\\ \end{align*}
Mathematica [A] time = 0.324556, size = 342, normalized size = 1.01 \[ \frac{-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-2 a^{5/4} h+\sqrt{2} \sqrt{a} b^{3/4} e+2 \sqrt [4]{a} b d-\sqrt{2} a \sqrt [4]{b} g+\sqrt{2} b^{5/4} c\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (2 a^{5/4} h+\sqrt{2} \sqrt{a} b^{3/4} e-2 \sqrt [4]{a} b d-\sqrt{2} a \sqrt [4]{b} g+\sqrt{2} b^{5/4} c\right )+\sqrt [4]{b} \left (2 a^{3/4} \sqrt [4]{b} \left (f \log \left (a+b x^4\right )+2 x (2 g+h x)\right )+\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{a} \sqrt{b} e+a g-b c\right )+\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-\sqrt{a} \sqrt{b} e-a g+b c\right )\right )}{8 a^{3/4} b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 462, normalized size = 1.4 \begin{align*}{\frac{h{x}^{2}}{2\,b}}+{\frac{gx}{b}}-{\frac{\sqrt{2}g}{4\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{c\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{\sqrt{2}g}{4\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{c\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{\sqrt{2}g}{8\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c\sqrt{2}}{8\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{ah}{2\,b}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{d}{2}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e\sqrt{2}}{8\,b}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e\sqrt{2}}{4\,b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e\sqrt{2}}{4\,b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{f\ln \left ( b{x}^{4}+a \right ) }{4\,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 [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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Giac [A] time = 1.08285, size = 506, normalized size = 1.5 \begin{align*} \frac{f \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac{b h x^{2} + 2 \, b g x}{2 \, b^{2}} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{a b} b^{2} d + \sqrt{2} \sqrt{a b} a b h + \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{1}{4}} a b g + \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 + \sqrt{2} \sqrt{a b} a b h + \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{1}{4}} a b g + \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{1}{4}} a b g - \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{1}{4}} a b g - \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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