Optimal. Leaf size=205 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{(a h+b d) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{(a j+b f) \log \left (a-b x^4\right )}{4 b^2}-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{i x^3}{3 b}-\frac{j x^4}{4 b} \]
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Rubi [A] time = 0.313444, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.196, Rules used = {1885, 1887, 1167, 205, 208, 1819, 1810, 635, 260} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{\sqrt{b} (a g+b c)}{\sqrt{a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{(a h+b d) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{(a j+b f) \log \left (a-b x^4\right )}{4 b^2}-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{i x^3}{3 b}-\frac{j x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 1885
Rule 1887
Rule 1167
Rule 205
Rule 208
Rule 1819
Rule 1810
Rule 635
Rule 260
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+188 x^6+j x^7}{a-b x^4} \, dx &=\int \left (\frac{c+e x^2+g x^4+188 x^6}{a-b x^4}+\frac{x \left (d+f x^2+h x^4+j x^6\right )}{a-b x^4}\right ) \, dx\\ &=\int \frac{c+e x^2+g x^4+188 x^6}{a-b x^4} \, dx+\int \frac{x \left (d+f x^2+h x^4+j x^6\right )}{a-b x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+f x+h x^2+j x^3}{a-b x^2} \, dx,x,x^2\right )+\int \left (-\frac{g}{b}-\frac{188 x^2}{b}+\frac{b c+a g+(188 a+b e) x^2}{b \left (a-b x^4\right )}\right ) \, dx\\ &=-\frac{g x}{b}-\frac{188 x^3}{3 b}+\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{h}{b}-\frac{j x}{b}+\frac{b d+a h+(b f+a j) x}{b \left (a-b x^2\right )}\right ) \, dx,x,x^2\right )+\frac{\int \frac{b c+a g+(188 a+b e) x^2}{a-b x^4} \, dx}{b}\\ &=-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{188 x^3}{3 b}-\frac{j x^4}{4 b}+\frac{\operatorname{Subst}\left (\int \frac{b d+a h+(b f+a j) x}{a-b x^2} \, dx,x,x^2\right )}{2 b}+\frac{\left (188 a+b e-\frac{\sqrt{b} (b c+a g)}{\sqrt{a}}\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{2 b}+\frac{\left (188 a+b e+\frac{\sqrt{b} (b c+a g)}{\sqrt{a}}\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{2 b}\\ &=-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{188 x^3}{3 b}-\frac{j x^4}{4 b}-\frac{\left (188 a+b e-\frac{\sqrt{b} (b c+a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{\left (188 a+b e+\frac{\sqrt{b} (b c+a g)}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{(b d+a h) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{2 b}+\frac{(b f+a j) \operatorname{Subst}\left (\int \frac{x}{a-b x^2} \, dx,x,x^2\right )}{2 b}\\ &=-\frac{g x}{b}-\frac{h x^2}{2 b}-\frac{188 x^3}{3 b}-\frac{j x^4}{4 b}-\frac{\left (188 a+b e-\frac{\sqrt{b} (b c+a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{\left (188 a+b e+\frac{\sqrt{b} (b c+a g)}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac{(b d+a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}-\frac{(b f+a j) \log \left (a-b x^4\right )}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.351059, size = 318, normalized size = 1.55 \[ \frac{-\frac{3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{5/4} \sqrt [4]{b} h+a^{3/2} i+\sqrt [4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-a^{5/4} \sqrt [4]{b} h+a^{3/2} i-\sqrt [4]{a} b^{5/4} d+\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (a^{3/2} (-i)-\sqrt{a} b e+a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{3 \sqrt [4]{b} (a h+b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{a}}-\frac{3 (a j+b f) \log \left (a-b x^4\right )}{\sqrt [4]{b}}-12 b^{3/4} g x-6 b^{3/4} h x^2-4 b^{3/4} i x^3-3 b^{3/4} j x^4}{12 b^{7/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 393, normalized size = 1.9 \begin{align*} -{\frac{j{x}^{4}}{4\,b}}-{\frac{i{x}^{3}}{3\,b}}-{\frac{h{x}^{2}}{2\,b}}-{\frac{gx}{b}}+{\frac{g}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{g}{4\,b}\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}{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{ah}{4\,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}{4}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{ai}{2\,{b}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{e}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{ai}{4\,{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}{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}}}}}}-{\frac{\ln \left ( b{x}^{4}-a \right ) aj}{4\,{b}^{2}}}-{\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 [B] time = 1.10172, size = 828, normalized size = 4.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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