3.454 \(\int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2}} \, dx\)

Optimal. Leaf size=220 \[ -\frac{5 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{7/2}}-\frac{2 x \left (32 a^2 c^2+\frac{b c \left (5 b^2-28 a c\right )}{x}-32 a b^2 c+5 b^4\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}+\frac{x \left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{3 a^3 \left (b^2-4 a c\right )^2}-\frac{2 x \left (-2 a c+b^2+\frac{b c}{x}\right )}{3 a \left (b^2-4 a c\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2}} \]

[Out]

((15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*Sqrt[a + c/x^2 + b/x]*x)/(3*a^3*(b^2 - 4*a
*c)^2) - (2*(b^2 - 2*a*c + (b*c)/x)*x)/(3*a*(b^2 - 4*a*c)*(a + c/x^2 + b/x)^(3/2
)) - (2*(5*b^4 - 32*a*b^2*c + 32*a^2*c^2 + (b*c*(5*b^2 - 28*a*c))/x)*x)/(3*a^2*(
b^2 - 4*a*c)^2*Sqrt[a + c/x^2 + b/x]) - (5*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt
[a + c/x^2 + b/x])])/(2*a^(7/2))

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Rubi [A]  time = 0.473482, 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 \[ -\frac{5 b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{2 a^{7/2}}-\frac{2 x \left (32 a^2 c^2+\frac{b c \left (5 b^2-28 a c\right )}{x}-32 a b^2 c+5 b^4\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}+\frac{x \left (128 a^2 c^2-100 a b^2 c+15 b^4\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{3 a^3 \left (b^2-4 a c\right )^2}-\frac{2 x \left (-2 a c+b^2+\frac{b c}{x}\right )}{3 a \left (b^2-4 a c\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c/x^2 + b/x)^(-5/2),x]

[Out]

((15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*Sqrt[a + c/x^2 + b/x]*x)/(3*a^3*(b^2 - 4*a
*c)^2) - (2*(b^2 - 2*a*c + (b*c)/x)*x)/(3*a*(b^2 - 4*a*c)*(a + c/x^2 + b/x)^(3/2
)) - (2*(5*b^4 - 32*a*b^2*c + 32*a^2*c^2 + (b*c*(5*b^2 - 28*a*c))/x)*x)/(3*a^2*(
b^2 - 4*a*c)^2*Sqrt[a + c/x^2 + b/x]) - (5*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt
[a + c/x^2 + b/x])])/(2*a^(7/2))

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Rubi in Sympy [A]  time = 67.4174, size = 206, normalized size = 0.94 \[ - \frac{2 x \left (- 2 a c + b^{2} + \frac{b c}{x}\right )}{3 a \left (- 4 a c + b^{2}\right ) \left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{3}{2}}} - \frac{4 x \left (16 a^{2} c^{2} - 16 a b^{2} c + \frac{5 b^{4}}{2} + \frac{b c \left (- 28 a c + 5 b^{2}\right )}{2 x}\right )}{3 a^{2} \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} + \frac{x \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}} \left (128 a^{2} c^{2} - 100 a b^{2} c + 15 b^{4}\right )}{3 a^{3} \left (- 4 a c + b^{2}\right )^{2}} - \frac{5 b \operatorname{atanh}{\left (\frac{2 a + \frac{b}{x}}{2 \sqrt{a} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} \right )}}{2 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+c/x**2+b/x)**(5/2),x)

[Out]

-2*x*(-2*a*c + b**2 + b*c/x)/(3*a*(-4*a*c + b**2)*(a + b/x + c/x**2)**(3/2)) - 4
*x*(16*a**2*c**2 - 16*a*b**2*c + 5*b**4/2 + b*c*(-28*a*c + 5*b**2)/(2*x))/(3*a**
2*(-4*a*c + b**2)**2*sqrt(a + b/x + c/x**2)) + x*sqrt(a + b/x + c/x**2)*(128*a**
2*c**2 - 100*a*b**2*c + 15*b**4)/(3*a**3*(-4*a*c + b**2)**2) - 5*b*atanh((2*a +
b/x)/(2*sqrt(a)*sqrt(a + b/x + c/x**2)))/(2*a**(7/2))

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Mathematica [A]  time = 0.527389, size = 235, normalized size = 1.07 \[ \frac{\frac{4 \left (5 a^2 b c^2 x+2 a^2 c^3-5 a b^3 c x-4 a b^2 c^2+b^5 x+b^4 c\right ) (x (a x+b)+c)}{b^2-4 a c}-\frac{4 \left (-80 a^3 b c^2 x-48 a^3 c^3+50 a^2 b^3 c x+56 a^2 b^2 c^2-7 a b^5 x-14 a b^4 c+b^6\right ) (x (a x+b)+c)^2}{\left (b^2-4 a c\right )^2}+6 a (x (a x+b)+c)^3-15 \sqrt{a} b (x (a x+b)+c)^{5/2} \log \left (2 \sqrt{a} \sqrt{x (a x+b)+c}+2 a x+b\right )}{6 a^4 x^5 \left (a+\frac{b x+c}{x^2}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c/x^2 + b/x)^(-5/2),x]

[Out]

((4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3 + b^5*x - 5*a*b^3*c*x + 5*a^2*b*c^2*x)*(c +
x*(b + a*x)))/(b^2 - 4*a*c) - (4*(b^6 - 14*a*b^4*c + 56*a^2*b^2*c^2 - 48*a^3*c^
3 - 7*a*b^5*x + 50*a^2*b^3*c*x - 80*a^3*b*c^2*x)*(c + x*(b + a*x))^2)/(b^2 - 4*a
*c)^2 + 6*a*(c + x*(b + a*x))^3 - 15*Sqrt[a]*b*(c + x*(b + a*x))^(5/2)*Log[b + 2
*a*x + 2*Sqrt[a]*Sqrt[c + x*(b + a*x)]])/(6*a^4*x^5*(a + (c + b*x)/x^2)^(5/2))

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Maple [A]  time = 0.016, size = 376, normalized size = 1.7 \[ -{\frac{a{x}^{2}+bx+c}{6\,{x}^{5} \left ( 4\,ac-{b}^{2} \right ) ^{2}} \left ( -96\,{a}^{13/2}{x}^{4}{c}^{2}+48\,{a}^{11/2}{x}^{4}{b}^{2}c-512\,{a}^{11/2}{x}^{3}b{c}^{2}-6\,{a}^{9/2}{x}^{4}{b}^{4}-384\,{a}^{11/2}{x}^{2}{c}^{3}+296\,{a}^{9/2}{x}^{3}{b}^{3}c-96\,{a}^{9/2}{x}^{2}{b}^{2}{c}^{2}-40\,{a}^{7/2}{x}^{3}{b}^{5}-624\,{a}^{9/2}xb{c}^{3}+180\,{a}^{7/2}{x}^{2}{b}^{4}c-256\,{a}^{9/2}{c}^{4}+420\,{a}^{7/2}x{b}^{3}{c}^{2}-30\,{a}^{5/2}{x}^{2}{b}^{6}+200\,{a}^{7/2}{b}^{2}{c}^{3}-60\,{a}^{5/2}x{b}^{5}c+240\, \left ( a{x}^{2}+bx+c \right ) ^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{4}b{c}^{2}-120\, \left ( a{x}^{2}+bx+c \right ) ^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}{b}^{3}c+15\, \left ( a{x}^{2}+bx+c \right ) ^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2}{b}^{5}-30\,{a}^{5/2}{b}^{4}{c}^{2} \right ){a}^{-{\frac{11}{2}}} \left ({\frac{a{x}^{2}+bx+c}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)^(5/2),x)

[Out]

-1/6*(a*x^2+b*x+c)*(-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)*x*b*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+240*(a*x^2+b*x+c)^(3/2)*ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*a^(1/2)+2*a*x+b)/a^
(1/2))*a^4*b*c^2-120*(a*x^2+b*x+c)^(3/2)*ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*a^(1/2)+2
*a*x+b)/a^(1/2))*a^3*b^3*c+15*(a*x^2+b*x+c)^(3/2)*ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*
a^(1/2)+2*a*x+b)/a^(1/2))*a^2*b^5-30*a^(5/2)*b^4*c^2)/a^(11/2)/((a*x^2+b*x+c)/x^
2)^(5/2)/x^5/(4*a*c-b^2)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x + c/x^2)^(-5/2),x, algorithm="maxima")

[Out]

integrate((a + b/x + c/x^2)^(-5/2), x)

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Fricas [A]  time = 0.422722, 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]

[1/12*(15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^
4*b*c^2)*x^4 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*x^3 + (b^7 - 6*a*b^5*c +
32*a^3*b*c^3)*x^2 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*x)*sqrt(a)*log(-(8
*a^2*x^2 + 8*a*b*x + b^2 + 4*a*c)*sqrt(a) + 4*(2*a^2*x^2 + a*b*x)*sqrt((a*x^2 +
b*x + c)/x^2)) + 4*(3*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*x^5 + 4*(5*a^2*b^5 -
37*a^3*b^3*c + 64*a^4*b*c^2)*x^4 + 3*(5*a*b^6 - 30*a^2*b^4*c + 16*a^3*b^2*c^2 +
64*a^4*c^3)*x^3 + 6*(5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^3)*x^2 + (15*a*b^4*
c^2 - 100*a^2*b^2*c^3 + 128*a^3*c^4)*x)*sqrt((a*x^2 + b*x + c)/x^2))/(a^4*b^4*c^
2 - 8*a^5*b^2*c^3 + 16*a^6*c^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*x^4 + 2*(a
^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^3 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*
x^2 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x), 1/6*(15*(b^5*c^2 - 8*a*b^
3*c^3 + 16*a^2*b*c^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^4 + 2*(a*b^6 - 8
*a^2*b^4*c + 16*a^3*b^2*c^2)*x^3 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^2 + 2*(b^6
*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*x)*sqrt(-a)*arctan(1/2*(2*a*x + b)*sqrt(-a)/(
a*x*sqrt((a*x^2 + b*x + c)/x^2))) + 2*(3*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*x^
5 + 4*(5*a^2*b^5 - 37*a^3*b^3*c + 64*a^4*b*c^2)*x^4 + 3*(5*a*b^6 - 30*a^2*b^4*c
+ 16*a^3*b^2*c^2 + 64*a^4*c^3)*x^3 + 6*(5*a*b^5*c - 35*a^2*b^3*c^2 + 52*a^3*b*c^
3)*x^2 + (15*a*b^4*c^2 - 100*a^2*b^2*c^3 + 128*a^3*c^4)*x)*sqrt((a*x^2 + b*x + c
)/x^2))/(a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*
a^8*c^2)*x^4 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^3 + (a^4*b^6 - 6*a^5*b
^4*c + 32*a^7*c^3)*x^2 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x + c/x^2)^(-5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError