Optimal. Leaf size=402 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right )}{4 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right )}{4 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right )}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right )}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{(b d-a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{(b f-a j) \log \left (a+b x^4\right )}{4 b^2}+\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{i x^3}{3 b}+\frac{j x^4}{4 b} \]
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Rubi [A] time = 0.566919, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.289, Rules used = {1885, 1887, 1168, 1162, 617, 204, 1165, 628, 1819, 1810, 635, 205, 260} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right )}{4 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (\sqrt{b} (b c-a g)-\sqrt{a} (b e-a i)\right )}{4 \sqrt{2} a^{3/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right )}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt{b} (b c-a g)+\sqrt{a} (b e-a i)\right )}{2 \sqrt{2} a^{3/4} b^{7/4}}+\frac{(b d-a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} b^{3/2}}+\frac{(b f-a j) \log \left (a+b x^4\right )}{4 b^2}+\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{i x^3}{3 b}+\frac{j x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 1885
Rule 1887
Rule 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+191 x^6+j x^7}{a+b x^4} \, dx &=\int \left (\frac{c+e x^2+g x^4+191 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+191 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{191 x^2}{b}+\frac{b c-a g-(191 a-b e) x^2}{b \left (a+b x^4\right )}\right ) \, dx\\ &=\frac{g x}{b}+\frac{191 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+(-191 a+b e) x^2}{a+b x^4} \, dx}{b}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{191 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 (191 a-b e-\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{2 b^2}+\frac{\left (191 a-b e+\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{2 b^2}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{191 x^3}{3 b}+\frac{j x^4}{4 b}-\frac{\left (191 a-b e-\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^2}-\frac{\left (191 a-b e-\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b^2}-\frac{\left (191 a-b e+\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\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} \sqrt [4]{a} b^{7/4}}-\frac{\left (191 a-b e+\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\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} \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{191 x^3}{3 b}+\frac{j x^4}{4 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 (191 a-b e+\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{\left (191 a-b e+\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{(b f-a j) \log \left (a+b x^4\right )}{4 b^2}-\frac{\left (191 a-b e-\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\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} \sqrt [4]{a} b^{7/4}}+\frac{\left (191 a-b e-\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\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} \sqrt [4]{a} b^{7/4}}\\ &=\frac{g x}{b}+\frac{h x^2}{2 b}+\frac{191 x^3}{3 b}+\frac{j x^4}{4 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 (191 a-b e-\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{\left (191 a-b e-\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{\left (191 a-b e+\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{\left (191 a-b e+\frac{\sqrt{b} (b c-a g)}{\sqrt{a}}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{(b f-a j) \log \left (a+b x^4\right )}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.308731, size = 445, normalized size = 1.11 \[ \frac{\frac{6 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (2 a^{5/4} \sqrt [4]{b} h+\sqrt{2} a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d-\sqrt{2} \sqrt{a} b e+\sqrt{2} a \sqrt{b} g-\sqrt{2} b^{3/2} c\right )}{a^{3/4}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (2 a^{5/4} \sqrt [4]{b} h-\sqrt{2} a^{3/2} i-2 \sqrt [4]{a} b^{5/4} d+\sqrt{2} \sqrt{a} b e-\sqrt{2} a \sqrt{b} g+\sqrt{2} b^{3/2} c\right )}{a^{3/4}}-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (a^{3/2} i-\sqrt{a} b e-a \sqrt{b} g+b^{3/2} c\right )}{a^{3/4}}+\frac{6 (b f-a j) \log \left (a+b x^4\right )}{\sqrt [4]{b}}+24 b^{3/4} g x+12 b^{3/4} h x^2+8 b^{3/4} i x^3+6 b^{3/4} j x^4}{24 b^{7/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 627, normalized size = 1.6 \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 = 1.0966, size = 780, normalized size = 1.94 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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