Optimal. Leaf size=204 \[ \frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )+\frac{5 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{128 c^{3/2}}-\frac{5 \left (\frac{2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{64 c}+x \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{5/2}-\frac{5}{24} \left (7 b+\frac{6 c}{x}\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2} \]
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Rubi [A] time = 0.607295, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )+\frac{5 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{128 c^{3/2}}-\frac{5 \left (\frac{2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{64 c}+x \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{5/2}-\frac{5}{24} \left (7 b+\frac{6 c}{x}\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(a + c/x^2 + b/x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 61.5345, size = 178, normalized size = 0.87 \[ \frac{5 a^{\frac{3}{2}} b \operatorname{atanh}{\left (\frac{2 a + \frac{b}{x}}{2 \sqrt{a} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} \right )}}{2} + x \left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{5}{2}} - \frac{5 \left (7 b + \frac{6 c}{x}\right ) \left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{3}{2}}}{24} - \frac{5 \left (\frac{b \left (44 a c + b^{2}\right )}{2} + \frac{c \left (12 a c + b^{2}\right )}{x}\right ) \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}}{32 c} + \frac{5 \left (- 48 a^{2} c^{2} - 24 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + \frac{2 c}{x}}{2 \sqrt{c} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} \right )}}{128 c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+c/x**2+b/x)**(5/2),x)
[Out]
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Mathematica [A] time = 1.02075, size = 233, normalized size = 1.14 \[ \frac{x \left (a+\frac{b x+c}{x^2}\right )^{5/2} \left (960 a^{3/2} b c^{3/2} x^4 \log \left (2 \sqrt{a} \sqrt{x (a x+b)+c}+2 a x+b\right )-15 x^4 \log (x) \left (-48 a^2 c^2-24 a b^2 c+b^4\right )+15 x^4 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \log \left (2 \sqrt{c} \sqrt{x (a x+b)+c}+b x+2 c\right )-2 \sqrt{c} \sqrt{x (a x+b)+c} \left (2 c x^2 \left (-96 a^2 x^2+278 a b x+59 b^2\right )+8 c^2 x (27 a x+17 b)+15 b^3 x^3+48 c^3\right )\right )}{384 c^{3/2} (x (a x+b)+c)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c/x^2 + b/x)^(5/2),x]
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Maple [B] time = 0.02, size = 701, normalized size = 3.4 \[ -{\frac{x}{384\,{c}^{4}} \left ({\frac{a{x}^{2}+bx+c}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( 720\,{a}^{7/2}\ln \left ({\frac{2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c}}{x}} \right ){c}^{9/2}{x}^{4}+96\, \left ( a{x}^{2}+bx+c \right ) ^{7/2}{c}^{3}{a}^{3/2}+6\, \left ( a{x}^{2}+bx+c \right ) ^{5/2}{a}^{5/2}{x}^{5}{b}^{3}-144\, \left ( a{x}^{2}+bx+c \right ) ^{5/2}{a}^{7/2}{x}^{4}{c}^{2}+144\, \left ( a{x}^{2}+bx+c \right ) ^{7/2}{a}^{5/2}{x}^{2}{c}^{2}-240\, \left ( a{x}^{2}+bx+c \right ) ^{3/2}{a}^{7/2}{x}^{4}{c}^{3}-720\,\sqrt{a{x}^{2}+bx+c}{a}^{7/2}{x}^{4}{c}^{4}-6\, \left ( a{x}^{2}+bx+c \right ) ^{7/2}{a}^{3/2}{x}^{3}{b}^{3}+6\, \left ( a{x}^{2}+bx+c \right ) ^{5/2}{a}^{3/2}{x}^{4}{b}^{4}-660\,\sqrt{a{x}^{2}+bx+c}{a}^{5/2}{x}^{4}{b}^{2}{c}^{3}-960\,{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{4}b{c}^{4}-4\, \left ( a{x}^{2}+bx+c \right ) ^{7/2}{a}^{3/2}{x}^{2}{b}^{2}c+10\, \left ( a{x}^{2}+bx+c \right ) ^{3/2}{a}^{3/2}{x}^{4}{b}^{4}c-16\, \left ( a{x}^{2}+bx+c \right ) ^{7/2}{a}^{3/2}xb{c}^{2}+30\,\sqrt{a{x}^{2}+bx+c}{a}^{3/2}{x}^{4}{b}^{4}{c}^{2}+360\,{a}^{5/2}\ln \left ({\frac{2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c}}{x}} \right ){c}^{7/2}{x}^{4}{b}^{2}-15\,{a}^{3/2}\ln \left ({\frac{2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c}}{x}} \right ){c}^{5/2}{x}^{4}{b}^{4}-600\,\sqrt{a{x}^{2}+bx+c}{a}^{7/2}{x}^{5}b{c}^{3}-152\, \left ( a{x}^{2}+bx+c \right ) ^{5/2}{a}^{7/2}{x}^{5}bc+152\, \left ( a{x}^{2}+bx+c \right ) ^{7/2}{a}^{5/2}{x}^{3}bc-148\, \left ( a{x}^{2}+bx+c \right ) ^{5/2}{a}^{5/2}{x}^{4}{b}^{2}c-280\, \left ( a{x}^{2}+bx+c \right ) ^{3/2}{a}^{7/2}{x}^{5}b{c}^{2}+10\, \left ( a{x}^{2}+bx+c \right ) ^{3/2}{a}^{5/2}{x}^{5}{b}^{3}c-260\, \left ( a{x}^{2}+bx+c \right ) ^{3/2}{a}^{5/2}{x}^{4}{b}^{2}{c}^{2}+30\,\sqrt{a{x}^{2}+bx+c}{a}^{5/2}{x}^{5}{b}^{3}{c}^{2} \right ) \left ( a{x}^{2}+bx+c \right ) ^{-{\frac{5}{2}}}{a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+c/x^2+b/x)^(5/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x + c/x^2)^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.423722, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x + c/x^2)^(5/2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+c/x**2+b/x)**(5/2),x)
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x + c/x^2)^(5/2),x, algorithm="giac")
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