3.402 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=82 \[ b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b^2 \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0885073, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b^2 \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.9482, size = 70, normalized size = 0.85 \[ b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - \frac{b^{2} \sqrt{a + b x^{2}}}{x} - \frac{b \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.0561669, size = 68, normalized size = 0.83 \[ b^{5/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )-\frac{\sqrt{a+b x^2} \left (3 a^2+11 a b x^2+23 b^2 x^4\right )}{15 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/x^6,x]

[Out]

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

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Maple [A]  time = 0.012, size = 130, normalized size = 1.6 \[ -{\frac{1}{5\,a{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,b}{15\,{a}^{2}{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{b}^{2}}{15\,{a}^{3}x} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,{b}^{3}x}{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{3}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{3}x}{a}\sqrt{b{x}^{2}+a}}+{b}^{{\frac{5}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x^6,x)

[Out]

-1/5/a/x^5*(b*x^2+a)^(7/2)-2/15*b/a^2/x^3*(b*x^2+a)^(7/2)-8/15*b^2/a^3/x*(b*x^2+
a)^(7/2)+8/15*b^3/a^3*x*(b*x^2+a)^(5/2)+2/3*b^3/a^2*x*(b*x^2+a)^(3/2)+b^3/a*x*(b
*x^2+a)^(1/2)+b^(5/2)*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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247394, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{\frac{5}{2}} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (23 \, b^{2} x^{4} + 11 \, a b x^{2} + 3 \, a^{2}\right )} \sqrt{b x^{2} + a}}{30 \, x^{5}}, \frac{15 \, \sqrt{-b} b^{2} x^{5} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left (23 \, b^{2} x^{4} + 11 \, a b x^{2} + 3 \, a^{2}\right )} \sqrt{b x^{2} + a}}{15 \, x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[1/30*(15*b^(5/2)*x^5*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(23*b^
2*x^4 + 11*a*b*x^2 + 3*a^2)*sqrt(b*x^2 + a))/x^5, 1/15*(15*sqrt(-b)*b^2*x^5*arct
an(b*x/(sqrt(b*x^2 + a)*sqrt(-b))) - (23*b^2*x^4 + 11*a*b*x^2 + 3*a^2)*sqrt(b*x^
2 + a))/x^5]

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Sympy [A]  time = 12.9194, size = 105, normalized size = 1.28 \[ - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{11 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 x^{2}} - \frac{23 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15} - \frac{b^{\frac{5}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{2} + b^{\frac{5}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x**6,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226094, size = 227, normalized size = 2.77 \[ -\frac{1}{2} \, b^{\frac{5}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (45 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a b^{\frac{5}{2}} - 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} b^{\frac{5}{2}} + 140 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} b^{\frac{5}{2}} - 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} b^{\frac{5}{2}} + 23 \, a^{5} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^6,x, algorithm="giac")

[Out]

-1/2*b^(5/2)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/15*(45*(sqrt(b)*x - sqrt(b*
x^2 + a))^8*a*b^(5/2) - 90*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(5/2) + 140*(sq
rt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(5/2) - 70*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^
4*b^(5/2) + 23*a^5*b^(5/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5