Optimal. Leaf size=556 \[ \frac{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.429213, antiderivative size = 556, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1112, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 290
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d x} \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}{\sqrt{d x} \left (a b+b^2 x^2\right )^5} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (15 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \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{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (55 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \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{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (385 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \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{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 \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^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 \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^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 \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^{9/2} d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 \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^{9/2} d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 \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^{9/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 \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^{9/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (1155 \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^{19/4} b^{5/4} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (1155 \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^{19/4} b^{5/4} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \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^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \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^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (1155 \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^{19/4} b^{5/4} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (1155 \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^{19/4} b^{5/4} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{385 \sqrt{d x}}{1024 a^4 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{8 a d \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \sqrt{d x}}{32 a^2 d \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{55 \sqrt{d x}}{256 a^3 d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \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^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \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^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1155 \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^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 \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^{19/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.137275, size = 319, normalized size = 0.57 \[ \frac{\sqrt{x} \left (a+b x^2\right ) \left (3080 a^{3/4} \sqrt{x} \left (a+b x^2\right )^3+1760 a^{7/4} \sqrt{x} \left (a+b x^2\right )^2+1280 a^{11/4} \sqrt{x} \left (a+b x^2\right )+1024 a^{15/4} \sqrt{x}-\frac{1155 \sqrt{2} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{1155 \sqrt{2} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}-\frac{2310 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{2310 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}\right )}{8192 a^{19/4} \sqrt{d x} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.227, size = 1133, 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 = 1.70475, size = 946, normalized size = 1.7 \begin{align*} \frac{4620 \,{\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac{1}{a^{19} b d^{2}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{a^{10} d^{2} \sqrt{-\frac{1}{a^{19} b d^{2}}} + d x} a^{14} b d \left (-\frac{1}{a^{19} b d^{2}}\right )^{\frac{3}{4}} - \sqrt{d x} a^{14} b d \left (-\frac{1}{a^{19} b d^{2}}\right )^{\frac{3}{4}}\right ) + 1155 \,{\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac{1}{a^{19} b d^{2}}\right )^{\frac{1}{4}} \log \left (a^{5} d \left (-\frac{1}{a^{19} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) - 1155 \,{\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\right )} \left (-\frac{1}{a^{19} b d^{2}}\right )^{\frac{1}{4}} \log \left (-a^{5} d \left (-\frac{1}{a^{19} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) + 4 \,{\left (385 \, b^{3} x^{6} + 1375 \, a b^{2} x^{4} + 1755 \, a^{2} b x^{2} + 893 \, a^{3}\right )} \sqrt{d x}}{4096 \,{\left (a^{4} b^{4} d x^{8} + 4 \, a^{5} b^{3} d x^{6} + 6 \, a^{6} b^{2} d x^{4} + 4 \, a^{7} b d x^{2} + a^{8} d\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}{\sqrt{d x} \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 = 1.41937, size = 556, normalized size = 1. \begin{align*} \frac{1155 \, \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^{5} b d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{1155 \, \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^{5} b d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{1155 \, \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^{5} b d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{1155 \, \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^{5} b d \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{385 \, \sqrt{d x} b^{3} d^{7} x^{6} + 1375 \, \sqrt{d x} a b^{2} d^{7} x^{4} + 1755 \, \sqrt{d x} a^{2} b d^{7} x^{2} + 893 \, \sqrt{d x} a^{3} d^{7}}{1024 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{4} \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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