Optimal. Leaf size=241 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (2 x (3 b d-a h)-a g+7 b c+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]
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Rubi [A] time = 0.338917, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {1858, 1854, 1876, 275, 208, 1167, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (2 x (3 b d-a h)-a g+7 b c+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1858
Rule 1854
Rule 1876
Rule 275
Rule 208
Rule 1167
Rule 205
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{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^{5/2} \sqrt{b}}+\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e-3 a g\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^{5/2} \sqrt{b}}+\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{\left (21 b c-5 \sqrt{a} \sqrt{b} e-3 a g\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac{\left (21 b c+5 \sqrt{a} \sqrt{b} e-3 a g\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.28654, size = 309, normalized size = 1.28 \[ \frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} h-5 \sqrt{a} b^{3/4} e-12 \sqrt [4]{a} b d+3 a \sqrt [4]{b} g-21 b^{5/4} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} h+5 \sqrt{a} b^{3/4} e-12 \sqrt [4]{a} b d-3 a \sqrt [4]{b} g+21 b^{5/4} c\right )+\frac{16 a^{7/4} \sqrt{b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^2}+\frac{4 a^{3/4} \sqrt{b} x (-a (g+2 h x)+7 b c+b x (6 d+5 e x))}{a-b x^4}+2 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt{a} \sqrt{b} e-3 a g+21 b c\right )-4 \sqrt [4]{a} (a h-3 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{128 a^{11/4} b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 389, normalized size = 1.6 \begin{align*} -{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{2}} \left ({\frac{5\,be{x}^{7}}{32\,{a}^{2}}}-{\frac{ \left ( ah-3\,bd \right ){x}^{6}}{16\,{a}^{2}}}-{\frac{ \left ( ag-7\,bc \right ){x}^{5}}{32\,{a}^{2}}}-{\frac{9\,e{x}^{3}}{32\,a}}-{\frac{ \left ( ah+5\,bd \right ){x}^{2}}{16\,ab}}-{\frac{ \left ( 3\,ag+11\,bc \right ) x}{32\,ab}}-{\frac{f}{8\,b}} \right ) }-{\frac{3\,g}{64\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{3\,g}{128\,b{a}^{2}}\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{21\,c}{128\,{a}^{3}}\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{h}{32\,ab}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,d}{32\,{a}^{2}}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{5\,e}{64\,b{a}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{5\,e}{128\,b{a}^{2}}\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 [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 [B] time = 1.12877, size = 656, normalized size = 2.72 \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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