Optimal. Leaf size=155 \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2}}+\frac{\left (8 a c+b^2+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.184235, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1114, 734, 814, 843, 621, 206, 724} \[ -\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2}}+\frac{\left (8 a c+b^2+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 734
Rule 814
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{(-2 a-b x) \sqrt{a+b x+c x^2}}{x} \, dx,x,x^2\right )\\ &=\frac{\left (b^2+8 a c+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2 c-\frac{1}{2} b \left (b^2-12 a c\right ) x}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{16 c}\\ &=\frac{\left (b^2+8 a c+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )-\frac{\left (b \left (b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{32 c}\\ &=\frac{\left (b^2+8 a c+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2}-a^2 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )-\frac{\left (b \left (b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{16 c}\\ &=\frac{\left (b^2+8 a c+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c}+\frac{1}{6} \left (a+b x^2+c x^4\right )^{3/2}-\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.140945, size = 143, normalized size = 0.92 \[ \frac{1}{96} \left (-48 a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )-\frac{3 b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{c^{3/2}}+\frac{2 \sqrt{a+b x^2+c x^4} \left (8 c \left (4 a+c x^4\right )+3 b^2+14 b c x^2\right )}{c}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.168, size = 192, normalized size = 1.2 \begin{align*}{\frac{c{x}^{4}}{6}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{7\,b{x}^{2}}{24}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{2}}{16\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{b}^{3}}{32}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,ab}{8}\ln \left ({ \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{2\,a}{3}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{1}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49747, size = 1715, normalized size = 11.06 \begin{align*} \text{result too large to display} \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}\, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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