3.382 \(\int \frac{\left (a+b x^2\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=61 \[ b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{3/2}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0627503, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{3/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(3/2)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.93087, size = 51, normalized size = 0.84 \[ b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - \frac{b \sqrt{a + b x^{2}}}{x} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0477523, size = 55, normalized size = 0.9 \[ b^{3/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )-\frac{\sqrt{a+b x^2} \left (a+4 b x^2\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(3/2)/x^4,x]

[Out]

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

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Maple [A]  time = 0.007, size = 92, normalized size = 1.5 \[ -{\frac{1}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,b}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}x}{a}\sqrt{b{x}^{2}+a}}+{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246801, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b^{\frac{3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (4 \, b x^{2} + a\right )} \sqrt{b x^{2} + a}}{6 \, x^{3}}, \frac{3 \, \sqrt{-b} b x^{3} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left (4 \, b x^{2} + a\right )} \sqrt{b x^{2} + a}}{3 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.38371, size = 78, normalized size = 1.28 \[ - \frac{a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3} - \frac{b^{\frac{3}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{2} + b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216687, size = 154, normalized size = 2.52 \[ -\frac{1}{2} \, b^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{4 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a b^{\frac{3}{2}} - 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{\frac{3}{2}} + 2 \, a^{3} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)/x^4,x, algorithm="giac")

[Out]

-1/2*b^(3/2)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 4/3*(3*(sqrt(b)*x - sqrt(b*x^
2 + a))^4*a*b^(3/2) - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(3/2) + 2*a^3*b^(3
/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3