Optimal. Leaf size=331 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (2 x (5 b d-a h)+3 x^2 (3 b e-a i)-a g+11 b c\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]
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Rubi [A] time = 0.567183, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {1858, 1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 (3 b e-a i)\right )}{256 a^{13/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}-5 a i+15 b e\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (2 x (5 b d-a h)+3 x^2 (3 b e-a i)-a g+11 b c\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+15 x^2 (3 b e-a i)\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]
Antiderivative was successfully verified.
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Rule 1858
Rule 1854
Rule 1855
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+205 x^6}{\left (a-b x^4\right )^4} \, dx &=\frac{x \left (b c+a g+(b d+a h) x+(205 a+b e) x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}-\frac{\int \frac{-b (11 b c-a g)-2 b (5 b d-a h) x+3 b (205 a-3 b e) x^2-8 b^2 f x^3}{\left (a-b x^4\right )^3} \, dx}{12 a b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(205 a+b e) x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac{8 a f+x \left (11 b c-a g+2 (5 b d-a h) x-3 (205 a-3 b e) x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac{\int \frac{7 b (11 b c-a g)+12 b (5 b d-a h) x-15 b (205 a-3 b e) x^2}{\left (a-b x^4\right )^2} \, dx}{96 a^2 b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(205 a+b e) x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac{x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (205 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{8 a f+x \left (11 b c-a g+2 (5 b d-a h) x-3 (205 a-3 b e) x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac{\int \frac{-21 b (11 b c-a g)-24 b (5 b d-a h) x+15 b (205 a-3 b e) x^2}{a-b x^4} \, dx}{384 a^3 b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(205 a+b e) x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac{x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (205 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{8 a f+x \left (11 b c-a g+2 (5 b d-a h) x-3 (205 a-3 b e) x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac{\int \left (-\frac{24 b (5 b d-a h) x}{a-b x^4}+\frac{-21 b (11 b c-a g)+15 b (205 a-3 b e) x^2}{a-b x^4}\right ) \, dx}{384 a^3 b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(205 a+b e) x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac{x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (205 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{8 a f+x \left (11 b c-a g+2 (5 b d-a h) x-3 (205 a-3 b e) x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac{\int \frac{-21 b (11 b c-a g)+15 b (205 a-3 b e) x^2}{a-b x^4} \, dx}{384 a^3 b^2}+\frac{(5 b d-a h) \int \frac{x}{a-b x^4} \, dx}{16 a^3 b}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(205 a+b e) x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac{x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (205 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{8 a f+x \left (11 b c-a g+2 (5 b d-a h) x-3 (205 a-3 b e) x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac{\left (5 (205 a-3 b e)-\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{256 a^3 b}-\frac{\left (5 (205 a-3 b e)+\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{256 a^3 b}+\frac{(5 b d-a h) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{32 a^3 b}\\ &=\frac{x \left (b c+a g+(b d+a h) x+(205 a+b e) x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac{x \left (7 (11 b c-a g)+12 (5 b d-a h) x-15 (205 a-3 b e) x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{8 a f+x \left (11 b c-a g+2 (5 b d-a h) x-3 (205 a-3 b e) x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac{\left (5 (205 a-3 b e)+\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}-\frac{\left (5 (205 a-3 b e)-\frac{7 \sqrt{b} (11 b c-a g)}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{13/4} b^{7/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.368419, size = 422, normalized size = 1.27 \[ \frac{\frac{128 a^3 b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^3}-\frac{16 a^2 b^{3/4} x (a (g+x (2 h+3 i x))-b (11 c+x (10 d+9 e x)))}{\left (a-b x^4\right )^2}+3 \sqrt [4]{a} \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i-40 \sqrt [4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right )-3 \sqrt [4]{a} \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-8 a^{5/4} \sqrt [4]{b} h+5 a^{3/2} i+40 \sqrt [4]{a} b^{5/4} d-15 \sqrt{a} b e+7 a \sqrt{b} g-77 b^{3/2} c\right )+6 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 a^{3/2} i-15 \sqrt{a} b e-7 a \sqrt{b} g+77 b^{3/2} c\right )-\frac{4 a b^{3/4} x (7 a g+3 a x (4 h+5 i x)-77 b c-15 b x (4 d+3 e x))}{a-b x^4}-24 \sqrt{a} \sqrt [4]{b} (a h-5 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{1536 a^4 b^{7/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 522, 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 [B] time = 1.10286, size = 1044, normalized size = 3.15 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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