Optimal. Leaf size=447 \[ -\frac{\sqrt [4]{c} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 a^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 b \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a^2 x}-\frac{2 b \sqrt{c} x \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{2 b \sqrt [4]{c} \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 a^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\left (b^2-20 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a x^3}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}-\frac{3 \left (b+10 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 x^5} \]
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Rubi [A] time = 0.394906, antiderivative size = 447, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1117, 1271, 1281, 1197, 1103, 1195} \[ \frac{2 b \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a^2 x}-\frac{2 b \sqrt{c} x \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{c} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 a^{7/4} \sqrt{a+b x^2+c x^4}}+\frac{2 b \sqrt [4]{c} \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 a^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\left (b^2-20 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a x^3}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}-\frac{3 \left (b+10 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 x^5} \]
Antiderivative was successfully verified.
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Rule 1117
Rule 1271
Rule 1281
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx &=-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac{3}{7} \int \frac{\left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{x^6} \, dx\\ &=-\frac{3 \left (b+10 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 x^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac{3}{35} \int \frac{b^2-20 a c-8 b c x^2}{x^4 \sqrt{a+b x^2+c x^4}} \, dx\\ &=-\frac{\left (b^2-20 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a x^3}-\frac{3 \left (b+10 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 x^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}-\frac{\int \frac{2 b \left (b^2-8 a c\right )+c \left (b^2-20 a c\right ) x^2}{x^2 \sqrt{a+b x^2+c x^4}} \, dx}{35 a}\\ &=-\frac{\left (b^2-20 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a x^3}+\frac{2 b \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a^2 x}-\frac{3 \left (b+10 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 x^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac{\int \frac{-a c \left (b^2-20 a c\right )-2 b c \left (b^2-8 a c\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{35 a^2}\\ &=-\frac{\left (b^2-20 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a x^3}+\frac{2 b \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a^2 x}-\frac{3 \left (b+10 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 x^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac{\left (2 b \sqrt{c} \left (b^2-8 a c\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{35 a^{3/2}}-\frac{\left (\sqrt{c} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{35 a^{3/2}}\\ &=-\frac{\left (b^2-20 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a x^3}+\frac{2 b \left (b^2-8 a c\right ) \sqrt{a+b x^2+c x^4}}{35 a^2 x}-\frac{2 b \sqrt{c} \left (b^2-8 a c\right ) x \sqrt{a+b x^2+c x^4}}{35 a^2 \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \left (b+10 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 x^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac{2 b \sqrt [4]{c} \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 a^{7/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{c} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 a^{7/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 1.59206, size = 572, normalized size = 1.28 \[ \frac{i x^7 \left (-20 a^2 c^2+b^3 \sqrt{b^2-4 a c}+9 a b^2 c-8 a b c \sqrt{b^2-4 a c}-b^4\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )-2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (3 a^2 \left (3 b^2 x^4+13 b c x^6+5 c^2 x^8\right )+a^3 \left (13 b x^2+20 c x^4\right )+5 a^4+a b x^6 \left (-b^2+17 b c x^2+16 c^2 x^4\right )-2 b^3 x^8 \left (b+c x^2\right )\right )-i b x^7 \left (b^2-8 a c\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{70 a^2 x^7 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.231, size = 495, normalized size = 1.1 \begin{align*} -{\frac{a}{7\,{x}^{7}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{8\,b}{35\,{x}^{5}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{15\,ac+{b}^{2}}{35\,a{x}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{2\,b \left ( 8\,ac-{b}^{2} \right ) }{35\,{a}^{2}x}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{\sqrt{2}}{4} \left ({c}^{2}-{\frac{c \left ( 15\,ac+{b}^{2} \right ) }{35\,a}} \right ) \sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{bc \left ( 8\,ac-{b}^{2} \right ) \sqrt{2}}{35\,a}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}} \left ({\it EllipticF} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{8}}, x\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{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x^{8}}\, 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{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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