Optimal. Leaf size=413 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{(a h+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{4 a f-x \left (2 x (a h+3 b d)+a g+7 b c+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{x \left (x (b d-a h)-a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]
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Rubi [A] time = 0.48642, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {1858, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{(a h+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{4 a f-x \left (2 x (a h+3 b d)+a g+7 b c+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{x \left (x (b d-a h)-a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1858
Rule 1854
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{\left (a+b x^4\right )^3} \, dx &=\frac{x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac{\int \frac{-b (7 b c+a g)-2 b (3 b d+a h) x-5 b^2 e x^2-4 b^2 f x^3}{\left (a+b x^4\right )^2} \, dx}{8 a b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac{4 a f-x \left (7 b c+a g+2 (3 b d+a h) x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{\int \frac{3 b (7 b c+a g)+4 b (3 b d+a h) x+5 b^2 e x^2}{a+b x^4} \, dx}{32 a^2 b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac{4 a f-x \left (7 b c+a g+2 (3 b d+a h) x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{\int \left (\frac{4 b (3 b d+a h) x}{a+b x^4}+\frac{3 b (7 b c+a g)+5 b^2 e x^2}{a+b x^4}\right ) \, dx}{32 a^2 b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac{4 a f-x \left (7 b c+a g+2 (3 b d+a h) x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{\int \frac{3 b (7 b c+a g)+5 b^2 e x^2}{a+b x^4} \, dx}{32 a^2 b^2}+\frac{(3 b d+a h) \int \frac{x}{a+b x^4} \, dx}{8 a^2 b}\\ &=\frac{x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac{4 a f-x \left (7 b c+a g+2 (3 b d+a h) x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{\left (21 b c-5 \sqrt{a} \sqrt{b} e+3 a g\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{64 a^{5/2} b^{3/2}}+\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e+3 a g\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{64 a^{5/2} b^{3/2}}+\frac{(3 b d+a h) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{16 a^2 b}\\ &=\frac{x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac{4 a f-x \left (7 b c+a g+2 (3 b d+a h) x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{(3 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{\left (21 b c-5 \sqrt{a} \sqrt{b} e+3 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}{128 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\left (21 b c-5 \sqrt{a} \sqrt{b} e+3 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}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e+3 a g\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^{5/2} b^{3/2}}+\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e+3 a g\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^{5/2} b^{3/2}}\\ &=\frac{x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac{4 a f-x \left (7 b c+a g+2 (3 b d+a h) x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{(3 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{\left (21 b c-5 \sqrt{a} \sqrt{b} e+3 a g\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\left (21 b c-5 \sqrt{a} \sqrt{b} e+3 a g\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e+3 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 )}{64 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e+3 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 )}{64 \sqrt{2} a^{11/4} b^{5/4}}\\ &=\frac{x \left (b c-a g+(b d-a h) x+b e x^2+b f x^3\right )}{8 a b \left (a+b x^4\right )^2}-\frac{4 a f-x \left (7 b c+a g+2 (3 b d+a h) x+5 b e x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{(3 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e+3 a g\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e+3 a g\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}-\frac{\left (21 b c-5 \sqrt{a} \sqrt{b} e+3 a g\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\left (21 b c-5 \sqrt{a} \sqrt{b} e+3 a g\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} b^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.364447, size = 411, normalized size = 1. \[ \frac{-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (8 a^{5/4} h+5 \sqrt{2} \sqrt{a} b^{3/4} e+24 \sqrt [4]{a} b d+3 \sqrt{2} a \sqrt [4]{b} g+21 \sqrt{2} b^{5/4} c\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-8 a^{5/4} h+5 \sqrt{2} \sqrt{a} b^{3/4} e-24 \sqrt [4]{a} b d+3 \sqrt{2} a \sqrt [4]{b} g+21 \sqrt{2} b^{5/4} c\right )-\frac{32 a^{7/4} \sqrt{b} (a (f+x (g+h x))-b x (c+x (d+e x)))}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} \sqrt{b} x (a (g+2 h x)+7 b c+b x (6 d+5 e x))}{a+b x^4}+\sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g-21 b c\right )+\sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-5 \sqrt{a} \sqrt{b} e+3 a g+21 b c\right )}{256 a^{11/4} b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 561, normalized size = 1.4 \begin{align*} \text{result too large to display} \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 = 1.10831, size = 620, normalized size = 1.5 \begin{align*} \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 4 \, \sqrt{2} \sqrt{a b} a b h + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 5 \, \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 )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (12 \, \sqrt{2} \sqrt{a b} b^{2} d + 4 \, \sqrt{2} \sqrt{a b} a b h + 21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g + 5 \, \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 )}{128 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \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 )}{256 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (21 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b g - 5 \, \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 )}{256 \, a^{3} b^{3}} + \frac{5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} + 2 \, a b h x^{6} + 7 \, b^{2} c x^{5} + a b g x^{5} + 9 \, a b x^{3} e + 10 \, a b d x^{2} - 2 \, a^{2} h x^{2} + 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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