3.401 \(\int \frac{\left (a+b x^2\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=86 \[ \frac{5}{2} a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )+\frac{5}{2} b^2 x \sqrt{a+b x^2}-\frac{5 b \left (a+b x^2\right )^{3/2}}{3 x}-\frac{\left (a+b x^2\right )^{5/2}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.079175, antiderivative size = 86, 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 \[ \frac{5}{2} a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )+\frac{5}{2} b^2 x \sqrt{a+b x^2}-\frac{5 b \left (a+b x^2\right )^{3/2}}{3 x}-\frac{\left (a+b x^2\right )^{5/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.21725, size = 78, normalized size = 0.91 \[ \frac{5 a b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2} + \frac{5 b^{2} x \sqrt{a + b x^{2}}}{2} - \frac{5 b \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0683628, size = 73, normalized size = 0.85 \[ \left (-\frac{a^2}{3 x^3}-\frac{7 a b}{3 x}+\frac{b^2 x}{2}\right ) \sqrt{a+b x^2}+\frac{5}{2} a b^{3/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/x^4,x]

[Out]

(-a^2/(3*x^3) - (7*a*b)/(3*x) + (b^2*x)/2)*Sqrt[a + b*x^2] + (5*a*b^(3/2)*Log[b*
x + Sqrt[b]*Sqrt[a + b*x^2]])/2

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Maple [A]  time = 0.007, size = 110, normalized size = 1.3 \[ -{\frac{1}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{4\,b}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{4\,{b}^{2}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{b}^{2}x}{3\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}x}{2}\sqrt{b{x}^{2}+a}}+{\frac{5\,a}{2}{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249989, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a b^{\frac{3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (3 \, b^{2} x^{4} - 14 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt{b x^{2} + a}}{12 \, x^{3}}, \frac{15 \, a \sqrt{-b} b x^{3} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) +{\left (3 \, b^{2} x^{4} - 14 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt{b x^{2} + a}}{6 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.4392, size = 112, normalized size = 1.3 \[ - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{7 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3} - \frac{5 a b^{\frac{3}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{4} + \frac{5 a b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )}}{2} + \frac{b^{\frac{5}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219529, size = 178, normalized size = 2.07 \[ \frac{1}{2} \, \sqrt{b x^{2} + a} b^{2} x - \frac{5}{4} \, a b^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} b^{\frac{3}{2}} - 12 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} b^{\frac{3}{2}} + 7 \, a^{4} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/x^4,x, algorithm="giac")

[Out]

1/2*sqrt(b*x^2 + a)*b^2*x - 5/4*a*b^(3/2)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) +
2/3*(9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(3/2) - 12*(sqrt(b)*x - sqrt(b*x^2
+ a))^2*a^3*b^(3/2) + 7*a^4*b^(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3