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

Optimal. Leaf size=83 \[ \frac{15}{8} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{\left (a+b x^2\right )^{5/2}}{x}+\frac{5}{4} b x \left (a+b x^2\right )^{3/2}+\frac{15}{8} a b x \sqrt{a+b x^2} \]

[Out]

(15*a*b*x*Sqrt[a + b*x^2])/8 + (5*b*x*(a + b*x^2)^(3/2))/4 - (a + b*x^2)^(5/2)/x
+ (15*a^2*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/8

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Rubi [A]  time = 0.073047, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{15}{8} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{\left (a+b x^2\right )^{5/2}}{x}+\frac{5}{4} b x \left (a+b x^2\right )^{3/2}+\frac{15}{8} a b x \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/x^2,x]

[Out]

(15*a*b*x*Sqrt[a + b*x^2])/8 + (5*b*x*(a + b*x^2)^(3/2))/4 - (a + b*x^2)^(5/2)/x
+ (15*a^2*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/8

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Rubi in Sympy [A]  time = 7.74218, size = 76, normalized size = 0.92 \[ \frac{15 a^{2} \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8} + \frac{15 a b x \sqrt{a + b x^{2}}}{8} + \frac{5 b x \left (a + b x^{2}\right )^{\frac{3}{2}}}{4} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*a**2*sqrt(b)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/8 + 15*a*b*x*sqrt(a + b*x**2)/
8 + 5*b*x*(a + b*x**2)**(3/2)/4 - (a + b*x**2)**(5/2)/x

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Mathematica [A]  time = 0.0600864, size = 73, normalized size = 0.88 \[ \sqrt{a+b x^2} \left (-\frac{a^2}{x}+\frac{9 a b x}{8}+\frac{b^2 x^3}{4}\right )+\frac{15}{8} a^2 \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*(-(a^2/x) + (9*a*b*x)/8 + (b^2*x^3)/4) + (15*a^2*Sqrt[b]*Log[b*x
+ Sqrt[b]*Sqrt[a + b*x^2]])/8

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Maple [A]  time = 0.007, size = 85, normalized size = 1. \[ -{\frac{1}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{bx}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,bx}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{15\,abx}{8}\sqrt{b{x}^{2}+a}}+{\frac{15\,{a}^{2}}{8}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x^2,x)

[Out]

-1/a/x*(b*x^2+a)^(7/2)+b/a*x*(b*x^2+a)^(5/2)+5/4*b*x*(b*x^2+a)^(3/2)+15/8*a*b*x*
(b*x^2+a)^(1/2)+15/8*b^(1/2)*a^2*ln(x*b^(1/2)+(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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255858, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (2 \, b^{2} x^{4} + 9 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{16 \, x}, \frac{15 \, a^{2} \sqrt{-b} x \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) +{\left (2 \, b^{2} x^{4} + 9 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{8 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(15*a^2*sqrt(b)*x*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(2*b
^2*x^4 + 9*a*b*x^2 - 8*a^2)*sqrt(b*x^2 + a))/x, 1/8*(15*a^2*sqrt(-b)*x*arctan(b*
x/(sqrt(b*x^2 + a)*sqrt(-b))) + (2*b^2*x^4 + 9*a*b*x^2 - 8*a^2)*sqrt(b*x^2 + a))
/x]

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Sympy [A]  time = 13.0899, size = 117, normalized size = 1.41 \[ - \frac{a^{\frac{5}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{a^{\frac{3}{2}} b x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 \sqrt{a} b^{2} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{15 a^{2} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8} + \frac{b^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(5/2)/(x*sqrt(1 + b*x**2/a)) + a**(3/2)*b*x/(8*sqrt(1 + b*x**2/a)) + 11*sqrt
(a)*b**2*x**3/(8*sqrt(1 + b*x**2/a)) + 15*a**2*sqrt(b)*asinh(sqrt(b)*x/sqrt(a))/
8 + b**3*x**5/(4*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.215292, size = 117, normalized size = 1.41 \[ -\frac{15}{16} \, a^{2} \sqrt{b}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \, a^{3} \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} + \frac{1}{8} \,{\left (2 \, b^{2} x^{2} + 9 \, a b\right )} \sqrt{b x^{2} + a} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/16*a^2*sqrt(b)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2*a^3*sqrt(b)/((sqrt(b)
*x - sqrt(b*x^2 + a))^2 - a) + 1/8*(2*b^2*x^2 + 9*a*b)*sqrt(b*x^2 + a)*x