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

Optimal. Leaf size=72 \[ a^{5/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a^2 \sqrt{a+b x^2}+\frac{1}{3} a \left (a+b x^2\right )^{3/2}+\frac{1}{5} \left (a+b x^2\right )^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.126076, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ a^{5/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a^2 \sqrt{a+b x^2}+\frac{1}{3} a \left (a+b x^2\right )^{3/2}+\frac{1}{5} \left (a+b x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.9501, size = 60, normalized size = 0.83 \[ - a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + a^{2} \sqrt{a + b x^{2}} + \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.074612, size = 72, normalized size = 1. \[ -a^{5/2} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+a^{5/2} \log (x)+\frac{1}{15} \sqrt{a+b x^2} \left (23 a^2+11 a b x^2+3 b^2 x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*(23*a^2 + 11*a*b*x^2 + 3*b^2*x^4))/15 + a^(5/2)*Log[x] - a^(5/2
)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]]

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Maple [A]  time = 0.007, size = 66, normalized size = 0.9 \[{\frac{1}{5} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{a}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +{a}^{2}\sqrt{b{x}^{2}+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24829, size = 1, normalized size = 0.01 \[ \left [\frac{1}{2} \, a^{\frac{5}{2}} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + \frac{1}{15} \,{\left (3 \, b^{2} x^{4} + 11 \, a b x^{2} + 23 \, a^{2}\right )} \sqrt{b x^{2} + a}, -\sqrt{-a} a^{2} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) + \frac{1}{15} \,{\left (3 \, b^{2} x^{4} + 11 \, a b x^{2} + 23 \, a^{2}\right )} \sqrt{b x^{2} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*a^(5/2)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 1/15*(3*b^2*x
^4 + 11*a*b*x^2 + 23*a^2)*sqrt(b*x^2 + a), -sqrt(-a)*a^2*arctan(a/(sqrt(b*x^2 +
a)*sqrt(-a))) + 1/15*(3*b^2*x^4 + 11*a*b*x^2 + 23*a^2)*sqrt(b*x^2 + a)]

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Sympy [A]  time = 11.8058, size = 105, normalized size = 1.46 \[ \frac{23 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}}}{15} + \frac{a^{\frac{5}{2}} \log{\left (\frac{b x^{2}}{a} \right )}}{2} - a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )} + \frac{11 a^{\frac{3}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}{15} + \frac{\sqrt{a} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

23*a**(5/2)*sqrt(1 + b*x**2/a)/15 + a**(5/2)*log(b*x**2/a)/2 - a**(5/2)*log(sqrt
(1 + b*x**2/a) + 1) + 11*a**(3/2)*b*x**2*sqrt(1 + b*x**2/a)/15 + sqrt(a)*b**2*x*
*4*sqrt(1 + b*x**2/a)/5

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GIAC/XCAS [A]  time = 0.210943, size = 84, normalized size = 1.17 \[ \frac{a^{3} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{1}{5} \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} + \frac{1}{3} \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a + \sqrt{b x^{2} + a} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^3*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 1/5*(b*x^2 + a)^(5/2) + 1/3*(b*x
^2 + a)^(3/2)*a + sqrt(b*x^2 + a)*a^2