Optimal. Leaf size=133 \[ \frac{x \left (3 b^2-8 a c\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{a^2 \left (b^2-4 a c\right )}-\frac{3 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{5/2}}-\frac{2 x \left (-2 a c+b^2+\frac{b c}{x}\right )}{a \left (b^2-4 a c\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}} \]
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Rubi [A] time = 0.0992594, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1342, 740, 806, 724, 206} \[ \frac{x \left (3 b^2-8 a c\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{a^2 \left (b^2-4 a c\right )}-\frac{3 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{5/2}}-\frac{2 x \left (-2 a c+b^2+\frac{b c}{x}\right )}{a \left (b^2-4 a c\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 1342
Rule 740
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-3 b^2+8 a c\right )-b c x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )}{a \left (b^2-4 a c\right )}\\ &=\frac{\left (3 b^2-8 a c\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x}{a^2 \left (b^2-4 a c\right )}-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )}{2 a^2}\\ &=\frac{\left (3 b^2-8 a c\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x}{a^2 \left (b^2-4 a c\right )}-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+\frac{b}{x}}{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )}{a^2}\\ &=\frac{\left (3 b^2-8 a c\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x}{a^2 \left (b^2-4 a c\right )}-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{a \left (b^2-4 a c\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}-\frac{3 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )}{2 a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.174029, size = 138, normalized size = 1.04 \[ -\frac{2 \sqrt{a} \left (-b^2 \left (a x^2+3 c\right )+10 a b c x+4 a c \left (a x^2+2 c\right )-3 b^3 x\right )+3 b \left (b^2-4 a c\right ) \sqrt{x (a x+b)+c} \tanh ^{-1}\left (\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right )}{2 a^{5/2} x \left (b^2-4 a c\right ) \sqrt{a+\frac{b x+c}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 197, normalized size = 1.5 \begin{align*}{\frac{a{x}^{2}+bx+c}{2\,{x}^{3} \left ( 4\,ac-{b}^{2} \right ) } \left ( 8\,{a}^{7/2}{x}^{2}c-2\,{a}^{5/2}{x}^{2}{b}^{2}+20\,{a}^{5/2}xbc-6\,{a}^{3/2}x{b}^{3}+16\,{a}^{5/2}{c}^{2}-6\,{a}^{3/2}{b}^{2}c-12\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) \sqrt{a{x}^{2}+bx+c}{a}^{2}bc+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) \sqrt{a{x}^{2}+bx+c}a{b}^{3} \right ){a}^{-{\frac{7}{2}}} \left ({\frac{a{x}^{2}+bx+c}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \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 (a + \frac{b}{x} + \frac{c}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30451, size = 1002, normalized size = 7.53 \begin{align*} \left [\frac{3 \,{\left (b^{3} c - 4 \, a b c^{2} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2} +{\left (b^{4} - 4 \, a b^{2} c\right )} x\right )} \sqrt{a} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c + 4 \,{\left (2 \, a x^{2} + b x\right )} \sqrt{a} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}\right ) + 4 \,{\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{3} +{\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2} +{\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x\right )} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}}{4 \,{\left (a^{3} b^{2} c - 4 \, a^{4} c^{2} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x\right )}}, \frac{3 \,{\left (b^{3} c - 4 \, a b c^{2} +{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2} +{\left (b^{4} - 4 \, a b^{2} c\right )} x\right )} \sqrt{-a} \arctan \left (\frac{{\left (2 \, a x^{2} + b x\right )} \sqrt{-a} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}}{2 \,{\left (a^{2} x^{2} + a b x + a c\right )}}\right ) + 2 \,{\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{3} +{\left (3 \, a b^{3} - 10 \, a^{2} b c\right )} x^{2} +{\left (3 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x\right )} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}}{2 \,{\left (a^{3} b^{2} c - 4 \, a^{4} c^{2} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{2} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x\right )}}\right ] \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}{\left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{3}{2}}}\, 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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