Optimal. Leaf size=203 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]
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Rubi [A] time = 0.27428, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {1858, 1876, 275, 208, 1167, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}-3 a i+b e\right )}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1858
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+193 x^6}{\left (a-b x^4\right )^2} \, dx &=\frac{x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac{\int \frac{-b (3 b c-a g)-2 b (b d-a h) x+b (579 a-b e) x^2}{a-b x^4} \, dx}{4 a b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac{\int \left (-\frac{2 b (b d-a h) x}{a-b x^4}+\frac{-b (3 b c-a g)+b (579 a-b e) x^2}{a-b x^4}\right ) \, dx}{4 a b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac{\int \frac{-b (3 b c-a g)+b (579 a-b e) x^2}{a-b x^4} \, dx}{4 a b^2}+\frac{(b d-a h) \int \frac{x}{a-b x^4} \, dx}{2 a b}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}-\frac{\left (579 a-b e-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{8 a b}-\frac{\left (579 a-b e+\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{8 a b}+\frac{(b d-a h) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{4 a b}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(193 a+b e) x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac{\left (579 a-b e+\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{5/4} b^{7/4}}-\frac{\left (579 a-b e-\frac{\sqrt{b} (3 b c-a g)}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{5/4} b^{7/4}}+\frac{(b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.221807, size = 302, normalized size = 1.49 \[ \frac{\frac{4 a^{3/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{a-b x^4}+\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d-\sqrt{a} b e+a \sqrt{b} g-3 b^{3/2} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (2 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d+\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+3 b^{3/2} c\right )-2 \sqrt [4]{a} \sqrt [4]{b} (a h-b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{16 a^{7/4} b^{7/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 409, normalized size = 2. \begin{align*}{\frac{1}{b{x}^{4}-a} \left ( -{\frac{ \left ( ai+be \right ){x}^{3}}{4\,ab}}-{\frac{ \left ( ah+bd \right ){x}^{2}}{4\,ab}}-{\frac{ \left ( ag+bc \right ) x}{4\,ab}}-{\frac{f}{4\,b}} \right ) }-{\frac{g}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{g}{16\,ab}\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{3\,c}{16\,{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{h}{8\,b}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{d}{8\,a}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,i}{8\,{b}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{e}{8\,ab}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{3\,i}{16\,{b}^{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}}}}}}+{\frac{e}{16\,ab}\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 = 1.09331, size = 848, normalized size = 4.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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