Optimal. Leaf size=161 \[ -\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{7/2}}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8} \]
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Rubi [A] time = 0.149705, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1114, 744, 806, 720, 724, 206} \[ -\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{7/2}}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 744
Rule 806
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2+c x^4}}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}-\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{5 b}{2}+c x\right ) \sqrt{a+b x+c x^2}}{x^4} \, dx,x,x^2\right )}{8 a}\\ &=-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}+\frac{\left (5 b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{32 a^2}\\ &=-\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}-\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{256 a^3}\\ &=-\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}+\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{128 a^3}\\ &=-\frac{\left (5 b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{128 a^3 x^4}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{8 a x^8}+\frac{5 b \left (a+b x^2+c x^4\right )^{3/2}}{48 a^2 x^6}+\frac{\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{256 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0950067, size = 141, normalized size = 0.88 \[ \frac{3 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )-\frac{2 \sqrt{a} \sqrt{a+b x^2+c x^4} \left (8 a^2 x^2 \left (b+3 c x^2\right )+48 a^3-2 a b x^4 \left (5 b+26 c x^2\right )+15 b^3 x^6\right )}{x^8}}{768 a^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.161, size = 387, normalized size = 2.4 \begin{align*} -{\frac{1}{8\,a{x}^{8}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,b}{48\,{a}^{2}{x}^{6}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{4}}{128\,{a}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,{b}^{4}}{256}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,{b}^{3}c{x}^{2}}{128\,{a}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,{b}^{2}c}{64\,{a}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{2}c}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{c}{16\,{a}^{2}{x}^{4}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{bc}{32\,{x}^{2}{a}^{3}} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{b{c}^{2}{x}^{2}}{32\,{a}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{c}^{2}}{16\,{a}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{c}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \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 = 2.03444, size = 743, normalized size = 4.61 \begin{align*} \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{a} x^{8} \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \,{\left ({\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{6} + 8 \, a^{3} b x^{2} - 2 \,{\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{4} + 48 \, a^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}}{1536 \, a^{4} x^{8}}, -\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-a} x^{8} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \,{\left ({\left (15 \, a b^{3} - 52 \, a^{2} b c\right )} x^{6} + 8 \, a^{3} b x^{2} - 2 \,{\left (5 \, a^{2} b^{2} - 12 \, a^{3} c\right )} x^{4} + 48 \, a^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}}{768 \, a^{4} x^{8}}\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{\sqrt{a + b x^{2} + c x^{4}}}{x^{9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{4} + b x^{2} + a}}{x^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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