Optimal. Leaf size=360 \[ -\frac{\left (-\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\left (\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac{d x}{\sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 1.26706, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1297, 191, 1692, 377, 205} \[ -\frac{\left (-\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\left (\frac{-a b e-2 a c d+b^2 d}{\sqrt{b^2-4 a c}}-a e+b d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac{d x}{\sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1297
Rule 191
Rule 1692
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx &=-\frac{\int \frac{a d+(b d-a e) 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{d^2 \int \frac{1}{\left (d+e x^2\right )^{3/2}} \, dx}{c d^2-b d e+a e^2}\\ &=\frac{d x}{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}-\frac{\int \left (\frac{b d-a e+\frac{-b^2 d+2 a c d+a b e}{\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{b d-a e-\frac{-b^2 d+2 a c d+a b e}{\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}{c d^2-b d e+a e^2}\\ &=\frac{d x}{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}-\frac{\left (b d-a e-\frac{b^2 d-2 a c d-a b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}} \, dx}{c d^2-b d e+a e^2}-\frac{\left (b d-a e+\frac{b^2 d-2 a c d-a b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}} \, dx}{c d^2-b d e+a e^2}\\ &=\frac{d x}{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}-\frac{\left (b d-a e-\frac{b^2 d-2 a c d-a b e}{\sqrt{b^2-4 a c}}\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 )}{c d^2-b d e+a e^2}-\frac{\left (b d-a e+\frac{b^2 d-2 a c d-a b e}{\sqrt{b^2-4 a c}}\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 )}{c d^2-b d e+a e^2}\\ &=\frac{d x}{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x^2}}-\frac{\left (b d-a e-\frac{b^2 d-2 a c d-a b 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 )}{\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{\left (b d-a e+\frac{b^2 d-2 a c d-a b 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 )}{\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 [B] time = 11.195, size = 7792, normalized size = 21.64 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.03, size = 338, normalized size = 0.9 \begin{align*}{\frac{x}{cd}{\frac{1}{\sqrt{e{x}^{2}+d}}}}-8\,{\frac{{e}^{3/2}a}{c \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}bd}{c \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}}{4\,a{e}^{2}-4\,deb+4\,c{d}^{2}}\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 ( \left ( ae-bd \right ){{\it \_R}}^{2}+2\,d \left ( -3\,ae+bd \right ){\it \_R}+a{d}^{2}e-b{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}}}} \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{x^{4}}{{\left (c x^{4} + b x^{2} + a\right )}{\left (e x^{2} + d\right )}^{\frac{3}{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{x^{4}}{\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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