Optimal. Leaf size=220 \[ -\frac{2 x \left (32 a^2 c^2+\frac{b c \left (5 b^2-28 a c\right )}{x}-32 a b^2 c+5 b^4\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}+\frac{x \left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{3 a^3 \left (b^2-4 a c\right )^2}-\frac{5 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{7/2}}-\frac{2 x \left (-2 a c+b^2+\frac{b c}{x}\right )}{3 a \left (b^2-4 a c\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.193687, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1342, 740, 822, 806, 724, 206} \[ -\frac{2 x \left (32 a^2 c^2+\frac{b c \left (5 b^2-28 a c\right )}{x}-32 a b^2 c+5 b^4\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}+\frac{x \left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{3 a^3 \left (b^2-4 a c\right )^2}-\frac{5 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{7/2}}-\frac{2 x \left (-2 a c+b^2+\frac{b c}{x}\right )}{3 a \left (b^2-4 a c\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1342
Rule 740
Rule 822
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x+c x^2\right )^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-5 b^2+16 a c\right )-3 b c x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 a \left (b^2-4 a c\right )}\\ &=-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2}}-\frac{2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac{b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}-\frac{4 \operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (15 b^4-100 a b^2 c+128 a^2 c^2\right )+\frac{1}{2} b c \left (5 b^2-28 a c\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )}{3 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2}}-\frac{2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac{b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )}{2 a^3}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2}}-\frac{2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac{b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}-\frac{(5 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^3}\\ &=\frac{\left (15 b^4-100 a b^2 c+128 a^2 c^2\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x}{3 a^3 \left (b^2-4 a c\right )^2}-\frac{2 \left (b^2-2 a c+\frac{b c}{x}\right ) x}{3 a \left (b^2-4 a c\right ) \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2}}-\frac{2 \left (5 b^4-32 a b^2 c+32 a^2 c^2+\frac{b c \left (5 b^2-28 a c\right )}{x}\right ) x}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}-\frac{5 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^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.400968, size = 256, normalized size = 1.16 \[ \frac{2 \sqrt{a} \left (3 b^4 \left (a^2 x^4-30 a c x^2+5 c^2\right )-4 a b^2 c \left (6 a^2 x^4-12 a c x^2+25 c^2\right )+8 a^2 b c^2 x \left (32 a x^2+39 c\right )+16 a^2 c^2 \left (3 a^2 x^4+12 a c x^2+8 c^2\right )+10 b^5 \left (2 a x^3+3 c x\right )-2 a b^3 c x \left (74 a x^2+105 c\right )+15 b^6 x^2\right )-15 b \left (b^2-4 a c\right )^2 (x (a x+b)+c)^{3/2} \tanh ^{-1}\left (\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right )}{6 a^{7/2} x \left (b^2-4 a c\right )^2 (x (a x+b)+c) \sqrt{a+\frac{b x+c}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 376, normalized size = 1.7 \begin{align*}{\frac{a{x}^{2}+bx+c}{6\,{x}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( 96\,{a}^{13/2}{x}^{4}{c}^{2}-48\,{a}^{11/2}{x}^{4}{b}^{2}c+512\,{a}^{11/2}{x}^{3}b{c}^{2}+6\,{a}^{9/2}{x}^{4}{b}^{4}+384\,{a}^{11/2}{x}^{2}{c}^{3}-296\,{a}^{9/2}{x}^{3}{b}^{3}c+96\,{a}^{9/2}{x}^{2}{b}^{2}{c}^{2}+40\,{a}^{7/2}{x}^{3}{b}^{5}+624\,{a}^{9/2}xb{c}^{3}-180\,{a}^{7/2}{x}^{2}{b}^{4}c+256\,{a}^{9/2}{c}^{4}-420\,{a}^{7/2}x{b}^{3}{c}^{2}+30\,{a}^{5/2}{x}^{2}{b}^{6}-200\,{a}^{7/2}{b}^{2}{c}^{3}+60\,{a}^{5/2}x{b}^{5}c+30\,{a}^{5/2}{b}^{4}{c}^{2}-240\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) \left ( a{x}^{2}+bx+c \right ) ^{3/2}{a}^{4}b{c}^{2}+120\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) \left ( a{x}^{2}+bx+c \right ) ^{3/2}{a}^{3}{b}^{3}c-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) \left ( a{x}^{2}+bx+c \right ) ^{3/2}{a}^{2}{b}^{5} \right ){a}^{-{\frac{11}{2}}} \left ({\frac{a{x}^{2}+bx+c}{{x}^{2}}} \right ) ^{-{\frac{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.32664, size = 2286, normalized size = 10.39 \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{1}{\left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{5}{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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