3.449 \(\int \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2} \, dx\)

Optimal. Leaf size=204 \[ \frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )+\frac{5 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{128 c^{3/2}}-\frac{5 \left (\frac{2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{64 c}+x \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{5/2}-\frac{5}{24} \left (7 b+\frac{6 c}{x}\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2} \]

[Out]

(-5*(a + c/x^2 + b/x)^(3/2)*(7*b + (6*c)/x))/24 - (5*Sqrt[a + c/x^2 + b/x]*(b*(b
^2 + 44*a*c) + (2*c*(b^2 + 12*a*c))/x))/(64*c) + (a + c/x^2 + b/x)^(5/2)*x + (5*
a^(3/2)*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/x])])/2 + (5*(b^4 -
24*a*b^2*c - 48*a^2*c^2)*ArcTanh[(b + (2*c)/x)/(2*Sqrt[c]*Sqrt[a + c/x^2 + b/x])
])/(128*c^(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]

(-5*(a + c/x^2 + b/x)^(3/2)*(7*b + (6*c)/x))/24 - (5*Sqrt[a + c/x^2 + b/x]*(b*(b
^2 + 44*a*c) + (2*c*(b^2 + 12*a*c))/x))/(64*c) + (a + c/x^2 + b/x)^(5/2)*x + (5*
a^(3/2)*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/x])])/2 + (5*(b^4 -
24*a*b^2*c - 48*a^2*c^2)*ArcTanh[(b + (2*c)/x)/(2*Sqrt[c]*Sqrt[a + c/x^2 + b/x])
])/(128*c^(3/2))

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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]

5*a**(3/2)*b*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + b/x + c/x**2)))/2 + x*(a + b/
x + c/x**2)**(5/2) - 5*(7*b + 6*c/x)*(a + b/x + c/x**2)**(3/2)/24 - 5*(b*(44*a*c
+ b**2)/2 + c*(12*a*c + b**2)/x)*sqrt(a + b/x + c/x**2)/(32*c) + 5*(-48*a**2*c*
*2 - 24*a*b**2*c + b**4)*atanh((b + 2*c/x)/(2*sqrt(c)*sqrt(a + b/x + c/x**2)))/(
128*c**(3/2))

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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]

[Out]

(x*(a + (c + b*x)/x^2)^(5/2)*(-2*Sqrt[c]*Sqrt[c + x*(b + a*x)]*(48*c^3 + 15*b^3*
x^3 + 8*c^2*x*(17*b + 27*a*x) + 2*c*x^2*(59*b^2 + 278*a*b*x - 96*a^2*x^2)) - 15*
(b^4 - 24*a*b^2*c - 48*a^2*c^2)*x^4*Log[x] + 960*a^(3/2)*b*c^(3/2)*x^4*Log[b + 2
*a*x + 2*Sqrt[a]*Sqrt[c + x*(b + a*x)]] + 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*x^4
*Log[2*c + b*x + 2*Sqrt[c]*Sqrt[c + x*(b + a*x)]]))/(384*c^(3/2)*(c + x*(b + a*x
))^(5/2))

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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)

[Out]

-1/384*((a*x^2+b*x+c)/x^2)^(5/2)*x*(720*a^(7/2)*ln((2*c+b*x+2*c^(1/2)*(a*x^2+b*x
+c)^(1/2))/x)*c^(9/2)*x^4+96*(a*x^2+b*x+c)^(7/2)*c^3*a^(3/2)+6*(a*x^2+b*x+c)^(5/
2)*a^(5/2)*x^5*b^3-144*(a*x^2+b*x+c)^(5/2)*a^(7/2)*x^4*c^2+144*(a*x^2+b*x+c)^(7/
2)*a^(5/2)*x^2*c^2-240*(a*x^2+b*x+c)^(3/2)*a^(7/2)*x^4*c^3-720*(a*x^2+b*x+c)^(1/
2)*a^(7/2)*x^4*c^4-6*(a*x^2+b*x+c)^(7/2)*a^(3/2)*x^3*b^3+6*(a*x^2+b*x+c)^(5/2)*a
^(3/2)*x^4*b^4-660*(a*x^2+b*x+c)^(1/2)*a^(5/2)*x^4*b^2*c^3-960*a^3*ln(1/2*(2*(a*
x^2+b*x+c)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^4*b*c^4-4*(a*x^2+b*x+c)^(7/2)*a^(3/
2)*x^2*b^2*c+10*(a*x^2+b*x+c)^(3/2)*a^(3/2)*x^4*b^4*c-16*(a*x^2+b*x+c)^(7/2)*a^(
3/2)*x*b*c^2+30*(a*x^2+b*x+c)^(1/2)*a^(3/2)*x^4*b^4*c^2+360*a^(5/2)*ln((2*c+b*x+
2*c^(1/2)*(a*x^2+b*x+c)^(1/2))/x)*c^(7/2)*x^4*b^2-15*a^(3/2)*ln((2*c+b*x+2*c^(1/
2)*(a*x^2+b*x+c)^(1/2))/x)*c^(5/2)*x^4*b^4-600*(a*x^2+b*x+c)^(1/2)*a^(7/2)*x^5*b
*c^3-152*(a*x^2+b*x+c)^(5/2)*a^(7/2)*x^5*b*c+152*(a*x^2+b*x+c)^(7/2)*a^(5/2)*x^3
*b*c-148*(a*x^2+b*x+c)^(5/2)*a^(5/2)*x^4*b^2*c-280*(a*x^2+b*x+c)^(3/2)*a^(7/2)*x
^5*b*c^2+10*(a*x^2+b*x+c)^(3/2)*a^(5/2)*x^5*b^3*c-260*(a*x^2+b*x+c)^(3/2)*a^(5/2
)*x^4*b^2*c^2+30*(a*x^2+b*x+c)^(1/2)*a^(5/2)*x^5*b^3*c^2)/(a*x^2+b*x+c)^(5/2)/c^
4/a^(3/2)

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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")

[Out]

Exception raised: ValueError

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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")

[Out]

[1/768*(960*a^(3/2)*b*c^2*x^3*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^
2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) - 15*(b^4 - 24*a*b^2*c - 48*a^2*c^
2)*sqrt(c)*x^3*log(-((8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2)*sqrt(c) - 4*(b*c*x^2
+ 2*c^2*x)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) + 4*(192*a^2*c^2*x^4 - 136*b*c^3*x
- 48*c^4 - (15*b^3*c + 556*a*b*c^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*sqrt((
a*x^2 + b*x + c)/x^2))/(c^2*x^3), 1/768*(1920*sqrt(-a)*a*b*c^2*x^3*arctan(1/2*(2
*a*x + b)/(sqrt(-a)*x*sqrt((a*x^2 + b*x + c)/x^2))) - 15*(b^4 - 24*a*b^2*c - 48*
a^2*c^2)*sqrt(c)*x^3*log(-((8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2)*sqrt(c) - 4*(b*
c*x^2 + 2*c^2*x)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) + 4*(192*a^2*c^2*x^4 - 136*b*
c^3*x - 48*c^4 - (15*b^3*c + 556*a*b*c^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*
sqrt((a*x^2 + b*x + c)/x^2))/(c^2*x^3), 1/384*(480*a^(3/2)*b*c^2*x^3*log(-8*a^2*
x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x
^2)) - 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*sqrt(-c)*x^3*arctan(1/2*(b*x + 2*c)*sq
rt(-c)/(c*x*sqrt((a*x^2 + b*x + c)/x^2))) + 2*(192*a^2*c^2*x^4 - 136*b*c^3*x - 4
8*c^4 - (15*b^3*c + 556*a*b*c^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*sqrt((a*x
^2 + b*x + c)/x^2))/(c^2*x^3), 1/384*(960*sqrt(-a)*a*b*c^2*x^3*arctan(1/2*(2*a*x
+ b)/(sqrt(-a)*x*sqrt((a*x^2 + b*x + c)/x^2))) - 15*(b^4 - 24*a*b^2*c - 48*a^2*
c^2)*sqrt(-c)*x^3*arctan(1/2*(b*x + 2*c)*sqrt(-c)/(c*x*sqrt((a*x^2 + b*x + c)/x^
2))) + 2*(192*a^2*c^2*x^4 - 136*b*c^3*x - 48*c^4 - (15*b^3*c + 556*a*b*c^2)*x^3
- 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c^2*x^3)]

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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)

[Out]

Integral((a + b/x + c/x**2)**(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")

[Out]

Timed out