Optimal. Leaf size=339 \[ -\frac{2 c^2 \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2}}-\frac{2 c^2 \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a \sqrt{\sqrt{b^2-4 a c}+b} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2}}+\frac{e x (c d-b e)}{a d \sqrt{d+e x^2} \left (e (a e-b d)+c d^2\right )}+\frac{-d-2 e x^2}{a d^2 x \sqrt{d+e x^2}} \]
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Rubi [A] time = 2.83666, antiderivative size = 462, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {1301, 271, 191, 6728, 264, 1692, 377, 205} \[ -\frac{c \left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{c \left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{2 e^3 x}{d^2 \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac{e^2}{d x \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x^2} (c d-b e)}{a d x \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1301
Rule 271
Rule 191
Rule 6728
Rule 264
Rule 1692
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx &=\frac{\int \frac{c d-b e-c e x^2}{x^2 \sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{c d^2-b d e+a e^2}+\frac{e^2 \int \frac{1}{x^2 \left (d+e x^2\right )^{3/2}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac{e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt{d+e x^2}}+\frac{\int \left (\frac{c d-b e}{a x^2 \sqrt{d+e x^2}}+\frac{-b c d+b^2 e-a c e-c (c d-b e) x^2}{a \sqrt{d+e x^2} \left (a+b x^2+c x^4\right )}\right ) \, dx}{c d^2-b d e+a e^2}-\frac{\left (2 e^3\right ) \int \frac{1}{\left (d+e x^2\right )^{3/2}} \, dx}{d \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt{d+e x^2}}-\frac{2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}+\frac{\int \frac{-b c d+b^2 e-a c e-c (c d-b e) x^2}{\sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a \left (c d^2-b d e+a e^2\right )}+\frac{(c d-b e) \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{a \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt{d+e x^2}}-\frac{2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}-\frac{(c d-b e) \sqrt{d+e x^2}}{a d \left (c d^2-b d e+a e^2\right ) x}+\frac{\int \left (\frac{-c (c d-b e)+\frac{c \left (-b c d+b^2 e-2 a c e\right )}{\sqrt{b^2-4 a c}}}{\left (b-\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}}+\frac{-c (c d-b e)-\frac{c \left (-b c d+b^2 e-2 a c e\right )}{\sqrt{b^2-4 a c}}}{\left (b+\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}}\right ) \, dx}{a \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt{d+e x^2}}-\frac{2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}-\frac{(c d-b e) \sqrt{d+e x^2}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac{\left (c \left (c d-b e-\frac{b c d-b^2 e+2 a c e}{\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 \left (c d^2-b d e+a e^2\right )}-\frac{\left (c \left (c d-b e+\frac{b c d-b^2 e+2 a c e}{\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 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt{d+e x^2}}-\frac{2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}-\frac{(c d-b e) \sqrt{d+e x^2}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac{\left (c \left (c d-b e-\frac{b c d-b^2 e+2 a c e}{\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 \left (c d^2-b d e+a e^2\right )}-\frac{\left (c \left (c d-b e+\frac{b c d-b^2 e+2 a c e}{\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 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt{d+e x^2}}-\frac{2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}-\frac{(c d-b e) \sqrt{d+e x^2}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac{c \left (c d-b e+\frac{b c d-b^2 e+2 a c e}{\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 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac{c \left (c d-b e-\frac{b c d-b^2 e+2 a c e}{\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 \sqrt{b+\sqrt{b^2-4 a c}} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end{align*}
Mathematica [C] time = 8.77517, size = 2158, normalized size = 6.37 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.032, size = 387, normalized size = 1.1 \begin{align*} -8\,{\frac{{e}^{3/2}b}{a \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}+8\,{\frac{\sqrt{e}cd}{a \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}-2\,{\frac{\sqrt{e}}{a \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) }\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{ \left ( c \left ( be-cd \right ){{\it \_R}}^{2}+2\, \left ( -2\,{e}^{2}ac+2\,{b}^{2}{e}^{2}-3\,bcde+{c}^{2}{d}^{2} \right ){\it \_R}+bc{d}^{2}e-{c}^{2}{d}^{3} \right ) \ln \left ( \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{2}-{\it \_R} \right ) }{{{\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}}}}-{\frac{1}{adx}{\frac{1}{\sqrt{e{x}^{2}+d}}}}-2\,{\frac{ex}{a{d}^{2}\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 )}{\left (e x^{2} + d\right )}^{\frac{3}{2}} x^{2}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (d + e x^{2}\right )^{\frac{3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \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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