Optimal. Leaf size=346 \[ -\frac{\left (-c d \left (d \sqrt{b^2-4 a c}-4 a e\right )+a e^2 \sqrt{b^2-4 a c}-b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )+a e^2 \sqrt{b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{a} \]
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Rubi [A] time = 1.74378, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1251, 897, 1287, 206, 1166, 208} \[ -\frac{\left (-c d \left (d \sqrt{b^2-4 a c}-4 a e\right )+a e^2 \sqrt{b^2-4 a c}-b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )+a e^2 \sqrt{b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 897
Rule 1287
Rule 206
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2}}{x \left (a+b x^2+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2}}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (-\frac{d}{e}+\frac{x^2}{e}\right ) \left (\frac{c d^2-b d e+a e^2}{e^2}-\frac{(2 c d-b e) x^2}{e^2}+\frac{c x^4}{e^2}\right )} \, dx,x,\sqrt{d+e x^2}\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{d^2 e}{a \left (d-x^2\right )}+\frac{e \left (d \left (c d^2-b d e+a e^2\right )-\left (c d^2-a e^2\right ) x^2\right )}{a \left (c d^2-b d e+a e^2-(2 c d-b e) x^2+c x^4\right )}\right ) \, dx,x,\sqrt{d+e x^2}\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{d \left (c d^2-b d e+a e^2\right )+\left (-c d^2+a e^2\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x^2}\right )}{a}-\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{d-x^2} \, dx,x,\sqrt{d+e x^2}\right )}{a}\\ &=-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{a}+\frac{\left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-4 a e\right )-b \left (c d^2+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x^2}\right )}{2 a \sqrt{b^2-4 a c}}+\frac{\left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+4 a e\right )+b \left (c d^2+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x^2}\right )}{2 a \sqrt{b^2-4 a c}}\\ &=-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{a}-\frac{\left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d-4 a e\right )-b \left (c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}-\frac{\left (a \sqrt{b^2-4 a c} e^2-c d \left (\sqrt{b^2-4 a c} d+4 a e\right )+b \left (c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{\sqrt{2} a \sqrt{c} \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 = 1.37316, size = 333, normalized size = 0.96 \[ -\frac{\frac{\left (-c d \left (d \sqrt{b^2-4 a c}+4 a e\right )+a e^2 \sqrt{b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\left (c d \left (d \sqrt{b^2-4 a c}-4 a e\right )-a e^2 \sqrt{b^2-4 a c}+b \left (a e^2+c d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}}{\sqrt{2} a \sqrt{c} \sqrt{b^2-4 a c}}-\frac{d^{3/2} \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )}{a}+\frac{d^{3/2} \log (x)}{a} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 388, normalized size = 1.1 \begin{align*}{\frac{{x}^{3}}{6\,a}{e}^{{\frac{3}{2}}}}-{\frac{e{x}^{2}}{8\,a}\sqrt{e{x}^{2}+d}}+{\frac{3\,dx}{4\,a}\sqrt{e}}+{\frac{7}{24\,a} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,d}{8\,a}\sqrt{e{x}^{2}+d}}-{\frac{5\,{d}^{2}}{8\,a} \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{-1}}-{\frac{{d}^{3}}{24\,a} \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{-3}}-{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{6}+ \left ( 16\,a{e}^{2}-8\,deb+6\,c{d}^{2} \right ){{\it \_Z}}^{4}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){{\it \_Z}}^{2}+c{d}^{4} \right ) }{\frac{ \left ( -a{e}^{2}+c{d}^{2} \right ){{\it \_R}}^{6}+d \left ( -5\,a{e}^{2}+4\,deb-3\,c{d}^{2} \right ){{\it \_R}}^{4}+{d}^{2} \left ( 5\,a{e}^{2}-4\,deb+3\,c{d}^{2} \right ){{\it \_R}}^{2}+a{d}^{3}{e}^{2}-c{d}^{5}}{{{\it \_R}}^{7}c+3\,{{\it \_R}}^{5}be-3\,{{\it \_R}}^{5}cd+8\,{{\it \_R}}^{3}a{e}^{2}-4\,{{\it \_R}}^{3}bde+3\,{{\it \_R}}^{3}c{d}^{2}+{\it \_R}\,b{d}^{2}e-{\it \_R}\,c{d}^{3}}\ln \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x-{\it \_R} \right ) }}-{\frac{1}{a}{d}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{e{x}^{2}+d} \right ) } \right ) } \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 (e x^{2} + d\right )}^{\frac{3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )} x}\,{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{\left (d + e x^{2}\right )^{\frac{3}{2}}}{x \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(-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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