3.392 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=80 \[ -\frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{\left (a+b x^2\right )^{5/2}}{2 x^2}+\frac{5}{6} b \left (a+b x^2\right )^{3/2}+\frac{5}{2} a b \sqrt{a+b x^2} \]

[Out]

(5*a*b*Sqrt[a + b*x^2])/2 + (5*b*(a + b*x^2)^(3/2))/6 - (a + b*x^2)^(5/2)/(2*x^2
) - (5*a^(3/2)*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2

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Rubi [A]  time = 0.130733, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{\left (a+b x^2\right )^{5/2}}{2 x^2}+\frac{5}{6} b \left (a+b x^2\right )^{3/2}+\frac{5}{2} a b \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/x^3,x]

[Out]

(5*a*b*Sqrt[a + b*x^2])/2 + (5*b*(a + b*x^2)^(3/2))/6 - (a + b*x^2)^(5/2)/(2*x^2
) - (5*a^(3/2)*b*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/2

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Rubi in Sympy [A]  time = 11.9321, size = 73, normalized size = 0.91 \[ - \frac{5 a^{\frac{3}{2}} b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2} + \frac{5 a b \sqrt{a + b x^{2}}}{2} + \frac{5 b \left (a + b x^{2}\right )^{\frac{3}{2}}}{6} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x**3,x)

[Out]

-5*a**(3/2)*b*atanh(sqrt(a + b*x**2)/sqrt(a))/2 + 5*a*b*sqrt(a + b*x**2)/2 + 5*b
*(a + b*x**2)**(3/2)/6 - (a + b*x**2)**(5/2)/(2*x**2)

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Mathematica [A]  time = 0.108891, size = 76, normalized size = 0.95 \[ \frac{1}{6} \left (-15 a^{3/2} b \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+15 a^{3/2} b \log (x)+\sqrt{a+b x^2} \left (-\frac{3 a^2}{x^2}+14 a b+2 b^2 x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/x^3,x]

[Out]

(Sqrt[a + b*x^2]*(14*a*b - (3*a^2)/x^2 + 2*b^2*x^2) + 15*a^(3/2)*b*Log[x] - 15*a
^(3/2)*b*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/6

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Maple [A]  time = 0.008, size = 88, normalized size = 1.1 \[ -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{b}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,b}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,b}{2}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{5\,ab}{2}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x^3,x)

[Out]

-1/2/a/x^2*(b*x^2+a)^(7/2)+1/2*b/a*(b*x^2+a)^(5/2)+5/6*b*(b*x^2+a)^(3/2)-5/2*b*a
^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+5/2*a*b*(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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250088, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{3}{2}} b x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, b^{2} x^{4} + 14 \, a b x^{2} - 3 \, a^{2}\right )} \sqrt{b x^{2} + a}}{12 \, x^{2}}, -\frac{15 \, \sqrt{-a} a b x^{2} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (2 \, b^{2} x^{4} + 14 \, a b x^{2} - 3 \, a^{2}\right )} \sqrt{b x^{2} + a}}{6 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^3,x, algorithm="fricas")

[Out]

[1/12*(15*a^(3/2)*b*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*
(2*b^2*x^4 + 14*a*b*x^2 - 3*a^2)*sqrt(b*x^2 + a))/x^2, -1/6*(15*sqrt(-a)*a*b*x^2
*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))) - (2*b^2*x^4 + 14*a*b*x^2 - 3*a^2)*sqrt(b*
x^2 + a))/x^2]

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Sympy [A]  time = 11.4189, size = 112, normalized size = 1.4 \[ - \frac{a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}}}{2 x^{2}} + \frac{7 a^{\frac{3}{2}} b \sqrt{1 + \frac{b x^{2}}{a}}}{3} + \frac{5 a^{\frac{3}{2}} b \log{\left (\frac{b x^{2}}{a} \right )}}{4} - \frac{5 a^{\frac{3}{2}} b \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )}}{2} + \frac{\sqrt{a} b^{2} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x**3,x)

[Out]

-a**(5/2)*sqrt(1 + b*x**2/a)/(2*x**2) + 7*a**(3/2)*b*sqrt(1 + b*x**2/a)/3 + 5*a*
*(3/2)*b*log(b*x**2/a)/4 - 5*a**(3/2)*b*log(sqrt(1 + b*x**2/a) + 1)/2 + sqrt(a)*
b**2*x**2*sqrt(1 + b*x**2/a)/3

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GIAC/XCAS [A]  time = 0.211039, size = 99, normalized size = 1.24 \[ \frac{1}{6} \,{\left (\frac{15 \, a^{2} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} + 12 \, \sqrt{b x^{2} + a} a - \frac{3 \, \sqrt{b x^{2} + a} a^{2}}{b x^{2}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^3,x, algorithm="giac")

[Out]

1/6*(15*a^2*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 2*(b*x^2 + a)^(3/2) + 12
*sqrt(b*x^2 + a)*a - 3*sqrt(b*x^2 + a)*a^2/(b*x^2))*b