Optimal. Leaf size=443 \[ -\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{a^3 d x}-\frac{c \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{c \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 b e \sqrt{d+e x^2}}{3 a^2 d^2 x}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}-\frac{8 e^2 \sqrt{d+e x^2}}{15 a d^3 x}+\frac{4 e \sqrt{d+e x^2}}{15 a d^2 x^3}-\frac{\sqrt{d+e x^2}}{5 a d x^5} \]
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Rubi [A] time = 1.43202, antiderivative size = 443, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1303, 271, 264, 1692, 377, 205} \[ -\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{a^3 d x}-\frac{c \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{c \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 b e \sqrt{d+e x^2}}{3 a^2 d^2 x}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}-\frac{8 e^2 \sqrt{d+e x^2}}{15 a d^3 x}+\frac{4 e \sqrt{d+e x^2}}{15 a d^2 x^3}-\frac{\sqrt{d+e x^2}}{5 a d x^5} \]
Antiderivative was successfully verified.
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Rule 1303
Rule 271
Rule 264
Rule 1692
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^6 \sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac{1}{a x^6 \sqrt{d+e x^2}}-\frac{b}{a^2 x^4 \sqrt{d+e x^2}}+\frac{b^2-a c}{a^3 x^2 \sqrt{d+e x^2}}+\frac{-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x^2}{a^3 \sqrt{d+e x^2} \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac{\int \frac{-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x^2}{\sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a^3}+\frac{\int \frac{1}{x^6 \sqrt{d+e x^2}} \, dx}{a}-\frac{b \int \frac{1}{x^4 \sqrt{d+e x^2}} \, dx}{a^2}+\frac{\left (b^2-a c\right ) \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{a^3}\\ &=-\frac{\sqrt{d+e x^2}}{5 a d x^5}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}-\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{a^3 d x}+\frac{\int \left (\frac{-\frac{b c \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-c \left (b^2-a c\right )}{\left (b-\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}}+\frac{\frac{b c \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-c \left (b^2-a c\right )}{\left (b+\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}}\right ) \, dx}{a^3}-\frac{(4 e) \int \frac{1}{x^4 \sqrt{d+e x^2}} \, dx}{5 a d}+\frac{(2 b e) \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{3 a^2 d}\\ &=-\frac{\sqrt{d+e x^2}}{5 a d x^5}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}+\frac{4 e \sqrt{d+e x^2}}{15 a d^2 x^3}-\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{a^3 d x}-\frac{2 b e \sqrt{d+e x^2}}{3 a^2 d^2 x}-\frac{\left (c \left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}} \, dx}{a^3}-\frac{\left (c \left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}} \, dx}{a^3}+\frac{\left (8 e^2\right ) \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{15 a d^2}\\ &=-\frac{\sqrt{d+e x^2}}{5 a d x^5}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}+\frac{4 e \sqrt{d+e x^2}}{15 a d^2 x^3}-\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{a^3 d x}-\frac{2 b e \sqrt{d+e x^2}}{3 a^2 d^2 x}-\frac{8 e^2 \sqrt{d+e x^2}}{15 a d^3 x}-\frac{\left (c \left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+\sqrt{b^2-4 a c}-\left (-2 c d+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{a^3}-\frac{\left (c \left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b-\sqrt{b^2-4 a c}-\left (-2 c d+\left (b-\sqrt{b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{a^3}\\ &=-\frac{\sqrt{d+e x^2}}{5 a d x^5}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}+\frac{4 e \sqrt{d+e x^2}}{15 a d^2 x^3}-\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{a^3 d x}-\frac{2 b e \sqrt{d+e x^2}}{3 a^2 d^2 x}-\frac{8 e^2 \sqrt{d+e x^2}}{15 a d^3 x}-\frac{c \left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} x}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}-\frac{c \left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} x}{\sqrt{b+\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{b+\sqrt{b^2-4 a c}} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\\ \end{align*}
Mathematica [A] time = 1.79496, size = 383, normalized size = 0.86 \[ -\frac{\frac{a^2 \sqrt{d+e x^2} \left (3 d^2-4 d e x^2+8 e^2 x^4\right )}{d^3 x^5}+\frac{15 \left (b^2-a c\right ) \sqrt{d+e x^2}}{d x}+\frac{15 c \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{15 c \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{5 a b \left (d-2 e x^2\right ) \sqrt{d+e x^2}}{d^2 x^3}}{15 a^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 350, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{a}^{3}}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,deb+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{c \left ( ac-{b}^{2} \right ){{\it \_R}}^{2}+2\, \left ( 4\,abce-a{c}^{2}d-2\,{b}^{3}e+{b}^{2}cd \right ){\it \_R}+a{c}^{2}{d}^{2}-{b}^{2}c{d}^{2}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{2}-{\it \_R} \right ) }}-{\frac{-ac+{b}^{2}}{{a}^{3}dx}\sqrt{e{x}^{2}+d}}-{\frac{1}{5\,ad{x}^{5}}\sqrt{e{x}^{2}+d}}+{\frac{4\,e}{15\,a{d}^{2}{x}^{3}}\sqrt{e{x}^{2}+d}}-{\frac{8\,{e}^{2}}{15\,a{d}^{3}x}\sqrt{e{x}^{2}+d}}+{\frac{b}{3\,{a}^{2}d{x}^{3}}\sqrt{e{x}^{2}+d}}-{\frac{2\,be}{3\,{a}^{2}{d}^{2}x}\sqrt{e{x}^{2}+d}} \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{1}{{\left (c x^{4} + b x^{2} + a\right )} \sqrt{e x^{2} + d} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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