Optimal. Leaf size=268 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (2 x (3 b d-a h)+x^2 (5 b e-3 a i)-a g+7 b c\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]
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Rubi [A] time = 0.434678, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {1858, 1854, 1876, 275, 208, 1167, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}-3 a i+5 b e\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}+\frac{x \left (2 x (3 b d-a h)+x^2 (5 b e-3 a i)-a g+7 b c\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+x^2 (a i+b e)+a g+b c+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1858
Rule 1854
Rule 1876
Rule 275
Rule 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+199 x^6}{\left (a-b x^4\right )^3} \, dx &=\frac{x \left (b c+a g+(b d+a h) x+(199 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 (597 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+(199 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-(597 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 (597 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+(199 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-(597 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 (597 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+(199 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-(597 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 (597 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+(199 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-(597 a-5 b e) x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac{\left (597 a-5 b e-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^2 b}-\frac{\left (597 a-5 b e+\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{64 a^2 b}+\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+(199 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-(597 a-5 b e) x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac{\left (597 a-5 b e+\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}-\frac{\left (597 a-5 b e-\frac{3 \sqrt{b} (7 b c-a g)}{\sqrt{a}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{7/4}}+\frac{(3 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.352227, size = 359, normalized size = 1.34 \[ \frac{\frac{16 a^{7/4} b^{3/4} (a (f+x (g+x (h+i x)))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^2}-\frac{4 a^{3/4} b^{3/4} x (a (g+x (2 h+3 i x))-b (7 c+x (6 d+5 e x)))}{a-b x^4}+\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h+3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d-5 \sqrt{a} b e+3 a \sqrt{b} g-21 b^{3/2} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} \sqrt [4]{b} h-3 a^{3/2} i-12 \sqrt [4]{a} b^{5/4} d+5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (3 a^{3/2} i-5 \sqrt{a} b e-3 a \sqrt{b} g+21 b^{3/2} c\right )-4 \sqrt [4]{a} \sqrt [4]{b} (a h-3 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{128 a^{11/4} b^{7/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 472, normalized size = 1.8 \begin{align*} -{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{2}} \left ( -{\frac{ \left ( 3\,ai-5\,be \right ){x}^{7}}{32\,{a}^{2}}}-{\frac{ \left ( ah-3\,bd \right ){x}^{6}}{16\,{a}^{2}}}-{\frac{ \left ( ag-7\,bc \right ){x}^{5}}{32\,{a}^{2}}}-{\frac{ \left ( ai+9\,be \right ){x}^{3}}{32\,ab}}-{\frac{ \left ( ah+5\,bd \right ){x}^{2}}{16\,ab}}-{\frac{ \left ( 3\,ag+11\,bc \right ) x}{32\,ab}}-{\frac{f}{8\,b}} \right ) }-{\frac{3\,g}{64\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{3\,g}{128\,b{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{21\,c}{128\,{a}^{3}}\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}{32\,ab}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,d}{32\,{a}^{2}}\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}{64\,{b}^{2}a}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,e}{64\,b{a}^{2}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{3\,i}{128\,{b}^{2}a}\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{5\,e}{128\,b{a}^{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}}}}}} \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 = 1.12048, size = 942, normalized size = 3.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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