Optimal. Leaf size=463 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (3 \sqrt{b} (a g+7 b c)-\sqrt{a} (3 a i+5 b e)\right )}{128 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (3 \sqrt{b} (a g+7 b c)-\sqrt{a} (3 a i+5 b e)\right )}{128 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt{b} (a g+7 b c)+\sqrt{a} (3 a i+5 b e)\right )}{64 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (3 \sqrt{b} (a g+7 b c)+\sqrt{a} (3 a i+5 b e)\right )}{64 \sqrt{2} a^{11/4} b^{7/4}}+\frac{(a h+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{4 a f-x \left (2 x (a h+3 b d)+x^2 (3 a i+5 b e)+a g+7 b c\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]
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Rubi [A] time = 0.685778, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {1858, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (3 \sqrt{b} (a g+7 b c)-\sqrt{a} (3 a i+5 b e)\right )}{128 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (3 \sqrt{b} (a g+7 b c)-\sqrt{a} (3 a i+5 b e)\right )}{128 \sqrt{2} a^{11/4} b^{7/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 \sqrt{b} (a g+7 b c)+\sqrt{a} (3 a i+5 b e)\right )}{64 \sqrt{2} a^{11/4} b^{7/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (3 \sqrt{b} (a g+7 b c)+\sqrt{a} (3 a i+5 b e)\right )}{64 \sqrt{2} a^{11/4} b^{7/4}}+\frac{(a h+3 b d) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{4 a f-x \left (2 x (a h+3 b d)+x^2 (3 a i+5 b e)+a g+7 b c\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{8 a b \left (a+b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1858
Rule 1854
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x+e x^2+f x^3+g x^4+h x^5+202 x^6}{\left (a+b x^4\right )^3} \, dx &=\frac{x \left (b c-a g+(b d-a h) x-(202 a-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-b (606 a+5 b 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-(202 a-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+(606 a+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+b (606 a+5 b 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-(202 a-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+(606 a+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)+b (606 a+5 b 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-(202 a-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+(606 a+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)+b (606 a+5 b 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-(202 a-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+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}-\frac{\left (606 a+5 b e-\frac{3 \sqrt{b} (7 b c+a g)}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{64 a^2 b^2}+\frac{\left (606 a+5 b e+\frac{3 \sqrt{b} (7 b c+a g)}{\sqrt{a}}\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{64 a^2 b^2}+\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-(202 a-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+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{(3 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{\left (606 a+5 b e-\frac{3 \sqrt{b} (7 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}{128 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (606 a+5 b e-\frac{3 \sqrt{b} (7 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}{128 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (606 a+5 b e+\frac{3 \sqrt{b} (7 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}{128 a^2 b^2}+\frac{\left (606 a+5 b e+\frac{3 \sqrt{b} (7 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}{128 a^2 b^2}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(202 a-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+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{(3 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{\left (606 a+5 b e-\frac{3 \sqrt{b} (7 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 )}{128 \sqrt{2} a^{9/4} b^{7/4}}-\frac{\left (606 a+5 b e-\frac{3 \sqrt{b} (7 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 )}{128 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (606 a+5 b e+\frac{3 \sqrt{b} (7 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 )}{64 \sqrt{2} a^{9/4} b^{7/4}}-\frac{\left (606 a+5 b e+\frac{3 \sqrt{b} (7 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 )}{64 \sqrt{2} a^{9/4} b^{7/4}}\\ &=\frac{x \left (b c-a g+(b d-a h) x-(202 a-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+(606 a+5 b e) x^2\right )}{32 a^2 b \left (a+b x^4\right )}+\frac{(3 b d+a h) \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}-\frac{\left (606 a+5 b e+\frac{3 \sqrt{b} (7 b c+a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (606 a+5 b e+\frac{3 \sqrt{b} (7 b c+a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{9/4} b^{7/4}}+\frac{\left (606 a+5 b e-\frac{3 \sqrt{b} (7 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 )}{128 \sqrt{2} a^{9/4} b^{7/4}}-\frac{\left (606 a+5 b e-\frac{3 \sqrt{b} (7 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 )}{128 \sqrt{2} a^{9/4} b^{7/4}}\\ \end{align*}
Mathematica [A] time = 0.530798, size = 473, normalized size = 1.02 \[ \frac{-\frac{32 a^{7/4} b^{3/4} (a (f+x (g+x (h+i x)))-b x (c+x (d+e x)))}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} b^{3/4} x (a g+a x (2 h+3 i x)+7 b c+b x (6 d+5 e x))}{a+b x^4}-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (8 a^{5/4} \sqrt [4]{b} h+3 \sqrt{2} a^{3/2} i+24 \sqrt [4]{a} b^{5/4} d+5 \sqrt{2} \sqrt{a} b e+3 \sqrt{2} a \sqrt{b} g+21 \sqrt{2} b^{3/2} c\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-8 a^{5/4} \sqrt [4]{b} h+3 \sqrt{2} a^{3/2} i-24 \sqrt [4]{a} b^{5/4} d+5 \sqrt{2} \sqrt{a} b e+3 \sqrt{2} a \sqrt{b} g+21 \sqrt{2} b^{3/2} c\right )+\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (3 a^{3/2} i+5 \sqrt{a} b e-3 a \sqrt{b} g-21 b^{3/2} c\right )+\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right ) \left (-3 a^{3/2} i-5 \sqrt{a} b e+3 a \sqrt{b} g+21 b^{3/2} c\right )}{256 a^{11/4} b^{7/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 716, 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 = 1.12375, size = 892, normalized size = 1.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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