Optimal. Leaf size=145 \[ -\frac{3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{8 \sqrt{c}}+x \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2}-\frac{3}{4} \left (3 b+\frac{2 c}{x}\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}+\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right ) \]
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Rubi [A] time = 0.134451, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {1342, 732, 814, 843, 621, 206, 724} \[ -\frac{3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{8 \sqrt{c}}+x \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2}-\frac{3}{4} \left (3 b+\frac{2 c}{x}\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}+\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right ) \]
Antiderivative was successfully verified.
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Rule 1342
Rule 732
Rule 814
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} x-\frac{3}{2} \operatorname{Subst}\left (\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3}{4} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \left (3 b+\frac{2 c}{x}\right )+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} x+\frac{3 \operatorname{Subst}\left (\int \frac{-4 a b c-c \left (b^2+4 a c\right ) x}{x \sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )}{8 c}\\ &=-\frac{3}{4} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \left (3 b+\frac{2 c}{x}\right )+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} x-\frac{1}{2} (3 a b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (3 \left (b^2+4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3}{4} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \left (3 b+\frac{2 c}{x}\right )+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} x+(3 a 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 )-\frac{1}{4} \left (3 \left (b^2+4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+\frac{2 c}{x}}{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )\\ &=-\frac{3}{4} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \left (3 b+\frac{2 c}{x}\right )+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} x+\frac{3}{2} \sqrt{a} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )-\frac{3 \left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )}{8 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.258137, size = 163, normalized size = 1.12 \[ \frac{\sqrt{a+\frac{b x+c}{x^2}} \left (-3 x^2 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{b x+2 c}{2 \sqrt{c} \sqrt{x (a x+b)+c}}\right )+12 \sqrt{a} b \sqrt{c} x^2 \tanh ^{-1}\left (\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right )-2 \sqrt{c} (x (5 b-4 a x)+2 c) \sqrt{x (a x+b)+c}\right )}{8 \sqrt{c} x \sqrt{x (a x+b)+c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 334, normalized size = 2.3 \begin{align*} -{\frac{x}{8\,{c}^{2}} \left ({\frac{a{x}^{2}+bx+c}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 12\,{a}^{5/2}{c}^{5/2}\ln \left ({\frac{2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c}}{x}} \right ){x}^{2}-2\,{a}^{5/2} \left ( a{x}^{2}+bx+c \right ) ^{3/2}{x}^{3}b-4\,{a}^{5/2} \left ( a{x}^{2}+bx+c \right ) ^{3/2}{x}^{2}c-6\,{a}^{5/2}\sqrt{a{x}^{2}+bx+c}{x}^{3}bc+3\,{a}^{3/2}{c}^{3/2}\ln \left ({\frac{2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c}}{x}} \right ){x}^{2}{b}^{2}-12\,{a}^{5/2}\sqrt{a{x}^{2}+bx+c}{x}^{2}{c}^{2}+2\,{a}^{3/2} \left ( a{x}^{2}+bx+c \right ) ^{5/2}xb-2\,{a}^{3/2} \left ( a{x}^{2}+bx+c \right ) ^{3/2}{x}^{2}{b}^{2}+4\, \left ( a{x}^{2}+bx+c \right ) ^{5/2}c{a}^{3/2}-6\,{a}^{3/2}\sqrt{a{x}^{2}+bx+c}{x}^{2}{b}^{2}c-12\,{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}b{c}^{2} \right ) \left ( a{x}^{2}+bx+c \right ) ^{-{\frac{3}{2}}}{a}^{-{\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{\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 = 2.57067, size = 1689, normalized size = 11.65 \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 \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(-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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