Optimal. Leaf size=602 \[ \frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}+\frac{19}{96 a^2 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac{1}{8 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.483648, antiderivative size = 602, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1112, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}+\frac{19}{96 a^2 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac{1}{8 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 290
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (19 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^4} \, dx}{16 a \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (95 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^3} \, dx}{64 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1045 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^2} \, dx}{512 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7315 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{2048 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{2048 a^5 d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 b \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{1024 a^5 d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 b \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d-\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{2048 a^{11/2} d^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 b \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d+\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{2048 a^{11/2} d^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7315 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{4096 \sqrt{2} a^{23/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7315 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{4096 \sqrt{2} a^{23/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{4096 a^{11/2} \sqrt{b} d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{4096 a^{11/2} \sqrt{b} d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (7315 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{23/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (7315 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{23/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1045}{1024 a^4 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a d (d x)^{3/2} \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{19}{96 a^2 d (d x)^{3/2} \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{95}{256 a^3 d (d x)^{3/2} \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 \left (a+b x^2\right )}{3072 a^5 d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{4096 \sqrt{2} a^{23/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0248275, size = 54, normalized size = 0.09 \[ -\frac{2 x \left (a+b x^2\right )^5 \, _2F_1\left (-\frac{3}{4},5;\frac{1}{4};-\frac{b x^2}{a}\right )}{3 a^5 (d x)^{5/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.243, size = 1183, normalized size = 2. \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 [A] time = 1.88879, size = 1133, normalized size = 1.88 \begin{align*} -\frac{87780 \,{\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{17} b d^{7} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{3}{4}} - \sqrt{a^{12} d^{6} \sqrt{-\frac{b^{3}}{a^{23} d^{10}}} + b^{2} d x} a^{17} d^{7} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{3}{4}}}{b^{3}}\right ) + 21945 \,{\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} \log \left (7315 \, a^{6} d^{3} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} + 7315 \, \sqrt{d x} b\right ) - 21945 \,{\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} \log \left (-7315 \, a^{6} d^{3} \left (-\frac{b^{3}}{a^{23} d^{10}}\right )^{\frac{1}{4}} + 7315 \, \sqrt{d x} b\right ) + 4 \,{\left (7315 \, b^{4} x^{8} + 26125 \, a b^{3} x^{6} + 33345 \, a^{2} b^{2} x^{4} + 16967 \, a^{3} b x^{2} + 2048 \, a^{4}\right )} \sqrt{d x}}{12288 \,{\left (a^{5} b^{4} d^{3} x^{10} + 4 \, a^{6} b^{3} d^{3} x^{8} + 6 \, a^{7} b^{2} d^{3} x^{6} + 4 \, a^{8} b d^{3} x^{4} + a^{9} d^{3} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d x\right )^{\frac{5}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.48317, size = 593, normalized size = 0.99 \begin{align*} -\frac{7315 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{4096 \, a^{6} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{7315 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{4096 \, a^{6} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{7315 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{8192 \, a^{6} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{7315 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \log \left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{8192 \, a^{6} d^{3} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{2}{3 \, \sqrt{d x} a^{5} d^{2} x \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{5267 \, \sqrt{d x} b^{4} d^{6} x^{6} + 17933 \, \sqrt{d x} a b^{3} d^{6} x^{4} + 21057 \, \sqrt{d x} a^{2} b^{2} d^{6} x^{2} + 8775 \, \sqrt{d x} a^{3} b d^{6}}{3072 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{5} d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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