3.395 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=113 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}-\frac{5 b^3 \sqrt{a+b x^2}}{128 a x^2}-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8}-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6} \]

[Out]

(-5*b^2*Sqrt[a + b*x^2])/(64*x^4) - (5*b^3*Sqrt[a + b*x^2])/(128*a*x^2) - (5*b*(
a + b*x^2)^(3/2))/(48*x^6) - (a + b*x^2)^(5/2)/(8*x^8) + (5*b^4*ArcTanh[Sqrt[a +
b*x^2]/Sqrt[a]])/(128*a^(3/2))

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Rubi [A]  time = 0.181825, antiderivative size = 113, 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 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}-\frac{5 b^3 \sqrt{a+b x^2}}{128 a x^2}-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8}-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/x^9,x]

[Out]

(-5*b^2*Sqrt[a + b*x^2])/(64*x^4) - (5*b^3*Sqrt[a + b*x^2])/(128*a*x^2) - (5*b*(
a + b*x^2)^(3/2))/(48*x^6) - (a + b*x^2)^(5/2)/(8*x^8) + (5*b^4*ArcTanh[Sqrt[a +
b*x^2]/Sqrt[a]])/(128*a^(3/2))

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Rubi in Sympy [A]  time = 17.2222, size = 104, normalized size = 0.92 \[ - \frac{5 b^{2} \sqrt{a + b x^{2}}}{64 x^{4}} - \frac{5 b \left (a + b x^{2}\right )^{\frac{3}{2}}}{48 x^{6}} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{8 x^{8}} - \frac{5 b^{3} \sqrt{a + b x^{2}}}{128 a x^{2}} + \frac{5 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{128 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x**9,x)

[Out]

-5*b**2*sqrt(a + b*x**2)/(64*x**4) - 5*b*(a + b*x**2)**(3/2)/(48*x**6) - (a + b*
x**2)**(5/2)/(8*x**8) - 5*b**3*sqrt(a + b*x**2)/(128*a*x**2) + 5*b**4*atanh(sqrt
(a + b*x**2)/sqrt(a))/(128*a**(3/2))

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Mathematica [A]  time = 0.129644, size = 102, normalized size = 0.9 \[ \frac{5 b^4 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{128 a^{3/2}}-\frac{5 b^4 \log (x)}{128 a^{3/2}}+\left (-\frac{a^2}{8 x^8}-\frac{5 b^3}{128 a x^2}-\frac{17 a b}{48 x^6}-\frac{59 b^2}{192 x^4}\right ) \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/x^9,x]

[Out]

(-a^2/(8*x^8) - (17*a*b)/(48*x^6) - (59*b^2)/(192*x^4) - (5*b^3)/(128*a*x^2))*Sq
rt[a + b*x^2] - (5*b^4*Log[x])/(128*a^(3/2)) + (5*b^4*Log[a + Sqrt[a]*Sqrt[a + b
*x^2]])/(128*a^(3/2))

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Maple [A]  time = 0.023, size = 159, normalized size = 1.4 \[ -{\frac{1}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{b}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}}{192\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{4}}{384\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,{b}^{4}}{128\,{a}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x^9,x)

[Out]

-1/8/a/x^8*(b*x^2+a)^(7/2)+1/48*b/a^2/x^6*(b*x^2+a)^(7/2)+1/192*b^2/a^3/x^4*(b*x
^2+a)^(7/2)+1/128*b^3/a^4/x^2*(b*x^2+a)^(7/2)-1/128*b^4/a^4*(b*x^2+a)^(5/2)-5/38
4*b^4/a^3*(b*x^2+a)^(3/2)+5/128*b^4/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x
)-5/128*b^4/a^2*(b*x^2+a)^(1/2)

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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((b*x^2 + a)^(5/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275334, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{4} x^{8} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (15 \, b^{3} x^{6} + 118 \, a b^{2} x^{4} + 136 \, a^{2} b x^{2} + 48 \, a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{768 \, a^{\frac{3}{2}} x^{8}}, \frac{15 \, b^{4} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (15 \, b^{3} x^{6} + 118 \, a b^{2} x^{4} + 136 \, a^{2} b x^{2} + 48 \, a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{384 \, \sqrt{-a} a x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^9,x, algorithm="fricas")

[Out]

[1/768*(15*b^4*x^8*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2 + a)*a)/x^2) - 2*(
15*b^3*x^6 + 118*a*b^2*x^4 + 136*a^2*b*x^2 + 48*a^3)*sqrt(b*x^2 + a)*sqrt(a))/(a
^(3/2)*x^8), 1/384*(15*b^4*x^8*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (15*b^3*x^6 +
118*a*b^2*x^4 + 136*a^2*b*x^2 + 48*a^3)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^
8)]

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Sympy [A]  time = 27.9867, size = 150, normalized size = 1.33 \[ - \frac{a^{3}}{8 \sqrt{b} x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{23 a^{2} \sqrt{b}}{48 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{127 a b^{\frac{3}{2}}}{192 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{133 b^{\frac{5}{2}}}{384 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 b^{\frac{7}{2}}}{128 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{128 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x**9,x)

[Out]

-a**3/(8*sqrt(b)*x**9*sqrt(a/(b*x**2) + 1)) - 23*a**2*sqrt(b)/(48*x**7*sqrt(a/(b
*x**2) + 1)) - 127*a*b**(3/2)/(192*x**5*sqrt(a/(b*x**2) + 1)) - 133*b**(5/2)/(38
4*x**3*sqrt(a/(b*x**2) + 1)) - 5*b**(7/2)/(128*a*x*sqrt(a/(b*x**2) + 1)) + 5*b**
4*asinh(sqrt(a)/(sqrt(b)*x))/(128*a**(3/2))

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GIAC/XCAS [A]  time = 0.213009, size = 127, normalized size = 1.12 \[ -\frac{1}{384} \, b^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} + 73 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a - 55 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} + 15 \, \sqrt{b x^{2} + a} a^{3}}{a b^{4} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^9,x, algorithm="giac")

[Out]

-1/384*b^4*(15*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a) + (15*(b*x^2 + a)^(
7/2) + 73*(b*x^2 + a)^(5/2)*a - 55*(b*x^2 + a)^(3/2)*a^2 + 15*sqrt(b*x^2 + a)*a^
3)/(a*b^4*x^8))