Optimal. Leaf size=554 \[ -\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.422205, antiderivative size = 554, 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, 288, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 288
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(d x)^{15/2}}{\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{(d x)^{15/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (13 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{11/2}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (39 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{7/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{3/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d^8 \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 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d^7 \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 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d^6 \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 \sqrt{a} b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d^6 \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 \sqrt{a} b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (195 d^{15/2} \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^{3/4} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (195 d^{15/2} \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^{3/4} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d^8 \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 \sqrt{a} b^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d^8 \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 \sqrt{a} b^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \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^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \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^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (195 d^{15/2} \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^{3/4} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (195 d^{15/2} \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^{3/4} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{195 d^7 \sqrt{d x}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{13/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13 d^3 (d x)^{9/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{39 d^5 (d x)^{5/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \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^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \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^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{195 d^{15/2} \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^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{195 d^{15/2} \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^{3/4} b^{17/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.282188, size = 366, normalized size = 0.66 \[ \frac{(d x)^{15/2} \left (a+b x^2\right ) \left (-1916928 a^2 b^{5/4} x^{5/2}+49920 a^2 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )-\frac{45045 \sqrt{2} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{45045 \sqrt{2} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}-\frac{90090 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{90090 \sqrt{2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-599040 a^3 \sqrt [4]{b} \sqrt{x}-2342912 a b^{9/4} x^{9/2}+120120 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^3+68640 a \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2-1261568 b^{13/4} x^{13/2}\right )}{1892352 b^{17/4} x^{15/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.245, size = 1134, normalized size = 2.1 \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 = 1.698, size = 975, normalized size = 1.76 \begin{align*} \frac{2340 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{3}{4}} \sqrt{d x} a^{2} b^{13} d^{7} - \sqrt{d^{15} x + \sqrt{-\frac{d^{30}}{a^{3} b^{17}}} a^{2} b^{8}} \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{3}{4}} a^{2} b^{13}}{d^{30}}\right ) + 585 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} \log \left (195 \, \sqrt{d x} d^{7} + 195 \, \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} a b^{4}\right ) - 585 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )} \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} \log \left (195 \, \sqrt{d x} d^{7} - 195 \, \left (-\frac{d^{30}}{a^{3} b^{17}}\right )^{\frac{1}{4}} a b^{4}\right ) - 4 \,{\left (1853 \, b^{3} d^{7} x^{6} + 3107 \, a b^{2} d^{7} x^{4} + 2223 \, a^{2} b d^{7} x^{2} + 585 \, a^{3} d^{7}\right )} \sqrt{d x}}{12288 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} \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 = 1.4427, size = 552, normalized size = 1. \begin{align*} \frac{1}{24576} \, d^{6}{\left (\frac{1170 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 )}{a b^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{1170 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 )}{a b^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{585 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 )}{a b^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{585 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \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 )}{a b^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (1853 \, \sqrt{d x} b^{3} d^{9} x^{6} + 3107 \, \sqrt{d x} a b^{2} d^{9} x^{4} + 2223 \, \sqrt{d x} a^{2} b d^{9} x^{2} + 585 \, \sqrt{d x} a^{3} d^{9}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{4} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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