Optimal. Leaf size=600 \[ \frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.463117, antiderivative size = 600, 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, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 288
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(d x)^{19/2}}{\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)^{19/2}}{\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)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (17 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{15/2}}{\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)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (221 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{192 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (663 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{7/2}}{\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{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3315 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac{(d x)^{3/2}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3315 a d^{10} \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^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3315 a d^9 \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^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3315 \sqrt{a} d^8 \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 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3315 \sqrt{a} d^8 \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 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3315 \sqrt [4]{a} d^{19/2} \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} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3315 \sqrt [4]{a} d^{19/2} \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} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3315 \sqrt{a} d^{10} \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 b^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3315 \sqrt{a} d^{10} \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 b^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \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} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \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} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (3315 \sqrt [4]{a} d^{19/2} \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} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (3315 \sqrt [4]{a} d^{19/2} \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} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \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} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \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} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \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} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \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} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.29358, size = 384, normalized size = 0.64 \[ \frac{(d x)^{19/2} \left (a+b x^2\right ) \left (39829504 a^2 b^{9/4} x^{9/2}+32587776 a^3 b^{5/4} x^{5/2}-1166880 a^2 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2-848640 a^3 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )+10183680 a^4 \sqrt [4]{b} \sqrt{x}+21446656 a b^{13/4} x^{13/2}-2042040 a \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^3+765765 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-765765 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+1531530 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-1531530 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+3784704 b^{17/4} x^{17/2}\right )}{1892352 b^{21/4} x^{19/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.24, size = 1202, 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 = 1.69105, size = 964, normalized size = 1.61 \begin{align*} -\frac{39780 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \arctan \left (-\frac{\left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{3}{4}} \sqrt{d x} b^{16} d^{9} - \sqrt{d^{19} x + \sqrt{-\frac{a d^{38}}{b^{21}}} b^{10}} \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{3}{4}} b^{16}}{a d^{38}}\right ) + 9945 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt{d x} d^{9} + 3315 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}\right ) - 9945 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt{d x} d^{9} - 3315 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}\right ) - 4 \,{\left (6144 \, b^{4} d^{9} x^{8} + 31501 \, a b^{3} d^{9} x^{6} + 52819 \, a^{2} b^{2} d^{9} x^{4} + 37791 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt{d x}}{12288 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\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 = 1.38807, size = 578, normalized size = 0.96 \begin{align*} -\frac{1}{24576} \, d^{8}{\left (\frac{19890 \, \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 )}{b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{19890 \, \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 )}{b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{9945 \, \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 )}{b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{9945 \, \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 )}{b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{49152 \, \sqrt{d x} d}{b^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (6925 \, \sqrt{d x} a b^{3} d^{9} x^{6} + 15955 \, \sqrt{d x} a^{2} b^{2} d^{9} x^{4} + 13215 \, \sqrt{d x} a^{3} b d^{9} x^{2} + 3801 \, \sqrt{d x} a^{4} d^{9}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5} \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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