Optimal. Leaf size=266 \[ \frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}-\frac{21 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{21 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2} \]
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Rubi [A] time = 0.230498, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {1855, 1876, 211, 1165, 628, 1162, 617, 204, 275, 205} \[ \frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}-\frac{21 c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{21 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 1855
Rule 1876
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 275
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x}{\left (a+b x^4\right )^3} \, dx &=\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2}-\frac{\int \frac{-7 c-6 d x}{\left (a+b x^4\right )^2} \, dx}{8 a}\\ &=\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{\int \frac{21 c+12 d x}{a+b x^4} \, dx}{32 a^2}\\ &=\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{\int \left (\frac{21 c}{a+b x^4}+\frac{12 d x}{a+b x^4}\right ) \, dx}{32 a^2}\\ &=\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{(21 c) \int \frac{1}{a+b x^4} \, dx}{32 a^2}+\frac{(3 d) \int \frac{x}{a+b x^4} \, dx}{8 a^2}\\ &=\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{(21 c) \int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx}{64 a^{5/2}}+\frac{(21 c) \int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx}{64 a^{5/2}}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{16 a^2}\\ &=\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{(21 c) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^{5/2} \sqrt{b}}+\frac{(21 c) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^{5/2} \sqrt{b}}-\frac{(21 c) \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^{11/4} \sqrt [4]{b}}-\frac{(21 c) \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^{11/4} \sqrt [4]{b}}\\ &=\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{21 c \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{(21 c) \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^{11/4} \sqrt [4]{b}}-\frac{(21 c) \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^{11/4} \sqrt [4]{b}}\\ &=\frac{x (c+d x)}{8 a \left (a+b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a+b x^4\right )}+\frac{3 d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}-\frac{21 c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{21 c \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{21 c \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{11/4} \sqrt [4]{b}}\\ \end{align*}
Mathematica [A] time = 0.207356, size = 249, normalized size = 0.94 \[ \frac{\frac{32 a^{7/4} x (c+d x)}{\left (a+b x^4\right )^2}+\frac{8 a^{3/4} x (7 c+6 d x)}{a+b x^4}-\frac{6 \left (8 \sqrt [4]{a} d+7 \sqrt{2} \sqrt [4]{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{b}}+\frac{6 \left (7 \sqrt{2} \sqrt [4]{b} c-8 \sqrt [4]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt{b}}-\frac{21 \sqrt{2} c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}+\frac{21 \sqrt{2} c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{b}}}{256 a^{11/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 222, normalized size = 0.8 \begin{align*}{\frac{cx}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{21\,c\sqrt{2}}{256\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,c\sqrt{2}}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}+a \right ) ^{2}}}+{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}+a \right ) }}+{\frac{3\,d}{16\,{a}^{2}}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}} \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 [A] time = 2.15468, size = 192, normalized size = 0.72 \begin{align*} \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{2} + 4718592 t^{2} a^{6} b d^{2} - 2709504 t a^{3} b c^{2} d + 20736 a d^{4} + 194481 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 67108864 t^{3} a^{9} b d^{2} - 9633792 t^{2} a^{6} b c^{2} d - 589824 t a^{4} d^{4} - 2765952 t a^{3} b c^{4} + 423360 a c^{2} d^{3}}{193536 a c d^{4} - 453789 b c^{5}} \right )} \right )\right )} + \frac{11 a c x + 10 a d x^{2} + 7 b c x^{5} + 6 b d x^{6}}{32 a^{4} + 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08334, size = 346, normalized size = 1.3 \begin{align*} \frac{21 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b} - \frac{21 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{3} b} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b d + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b d + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac{6 \, b d x^{6} + 7 \, b c x^{5} + 10 \, a d x^{2} + 11 \, a c x}{32 \,{\left (b x^{4} + a\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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