3.377 \(\int \frac{\left (a+b x^2\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{3 b^3 \sqrt{a+b x^2}}{128 a^2 x^2}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8}-\frac{b \sqrt{a+b x^2}}{16 x^6} \]

[Out]

-(b*Sqrt[a + b*x^2])/(16*x^6) - (b^2*Sqrt[a + b*x^2])/(64*a*x^4) + (3*b^3*Sqrt[a
+ b*x^2])/(128*a^2*x^2) - (a + b*x^2)^(3/2)/(8*x^8) - (3*b^4*ArcTanh[Sqrt[a + b
*x^2]/Sqrt[a]])/(128*a^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.18348, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}+\frac{3 b^3 \sqrt{a+b x^2}}{128 a^2 x^2}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8}-\frac{b \sqrt{a+b x^2}}{16 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(3/2)/x^9,x]

[Out]

-(b*Sqrt[a + b*x^2])/(16*x^6) - (b^2*Sqrt[a + b*x^2])/(64*a*x^4) + (3*b^3*Sqrt[a
+ b*x^2])/(128*a^2*x^2) - (a + b*x^2)^(3/2)/(8*x^8) - (3*b^4*ArcTanh[Sqrt[a + b
*x^2]/Sqrt[a]])/(128*a^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.9348, size = 104, normalized size = 0.9 \[ - \frac{b \sqrt{a + b x^{2}}}{16 x^{6}} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{8 x^{8}} - \frac{b^{2} \sqrt{a + b x^{2}}}{64 a x^{4}} + \frac{3 b^{3} \sqrt{a + b x^{2}}}{128 a^{2} x^{2}} - \frac{3 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{128 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)/x**9,x)

[Out]

-b*sqrt(a + b*x**2)/(16*x**6) - (a + b*x**2)**(3/2)/(8*x**8) - b**2*sqrt(a + b*x
**2)/(64*a*x**4) + 3*b**3*sqrt(a + b*x**2)/(128*a**2*x**2) - 3*b**4*atanh(sqrt(a
+ b*x**2)/sqrt(a))/(128*a**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.128904, size = 102, normalized size = 0.88 \[ -\frac{3 b^4 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{128 a^{5/2}}+\frac{3 b^4 \log (x)}{128 a^{5/2}}+\left (\frac{3 b^3}{128 a^2 x^2}-\frac{b^2}{64 a x^4}-\frac{a}{8 x^8}-\frac{3 b}{16 x^6}\right ) \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(3/2)/x^9,x]

[Out]

(-a/(8*x^8) - (3*b)/(16*x^6) - b^2/(64*a*x^4) + (3*b^3)/(128*a^2*x^2))*Sqrt[a +
b*x^2] + (3*b^4*Log[x])/(128*a^(5/2)) - (3*b^4*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])
/(128*a^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 142, normalized size = 1.2 \[ -{\frac{1}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{b}{16\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{3\,{b}^{4}}{128\,{a}^{3}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)/x^9,x)

[Out]

-1/8/a/x^8*(b*x^2+a)^(5/2)+1/16*b/a^2/x^6*(b*x^2+a)^(5/2)-1/64*b^2/a^3/x^4*(b*x^
2+a)^(5/2)-1/128*b^3/a^4/x^2*(b*x^2+a)^(5/2)+1/128*b^4/a^4*(b*x^2+a)^(3/2)-3/128
*b^4/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+3/128*b^4/a^3*(b*x^2+a)^(1/2)

_______________________________________________________________________________________

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((b*x^2 + a)^(3/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.271716, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{4} x^{8} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \, b^{3} x^{6} - 2 \, a b^{2} x^{4} - 24 \, a^{2} b x^{2} - 16 \, a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{256 \, a^{\frac{5}{2}} x^{8}}, -\frac{3 \, b^{4} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \, b^{3} x^{6} - 2 \, a b^{2} x^{4} - 24 \, a^{2} b x^{2} - 16 \, a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{128 \, \sqrt{-a} a^{2} x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)/x^9,x, algorithm="fricas")

[Out]

[1/256*(3*b^4*x^8*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2) + 2*(3
*b^3*x^6 - 2*a*b^2*x^4 - 24*a^2*b*x^2 - 16*a^3)*sqrt(b*x^2 + a)*sqrt(a))/(a^(5/2
)*x^8), -1/128*(3*b^4*x^8*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (3*b^3*x^6 - 2*a*b^
2*x^4 - 24*a^2*b*x^2 - 16*a^3)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^8)]

_______________________________________________________________________________________

Sympy [A]  time = 28.5302, size = 148, normalized size = 1.28 \[ - \frac{a^{2}}{8 \sqrt{b} x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 a \sqrt{b}}{16 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{13 b^{\frac{3}{2}}}{64 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{\frac{5}{2}}}{128 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{7}{2}}}{128 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{128 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)/x**9,x)

[Out]

-a**2/(8*sqrt(b)*x**9*sqrt(a/(b*x**2) + 1)) - 5*a*sqrt(b)/(16*x**7*sqrt(a/(b*x**
2) + 1)) - 13*b**(3/2)/(64*x**5*sqrt(a/(b*x**2) + 1)) + b**(5/2)/(128*a*x**3*sqr
t(a/(b*x**2) + 1)) + 3*b**(7/2)/(128*a**2*x*sqrt(a/(b*x**2) + 1)) - 3*b**4*asinh
(sqrt(a)/(sqrt(b)*x))/(128*a**(5/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211433, size = 127, normalized size = 1.09 \[ \frac{1}{128} \, b^{4}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 11 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a - 11 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} + 3 \, \sqrt{b x^{2} + a} a^{3}}{a^{2} b^{4} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)/x^9,x, algorithm="giac")

[Out]

1/128*b^4*(3*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*x^2 + a)^(7
/2) - 11*(b*x^2 + a)^(5/2)*a - 11*(b*x^2 + a)^(3/2)*a^2 + 3*sqrt(b*x^2 + a)*a^3)
/(a^2*b^4*x^8))