3.238 \(\int \frac{\sqrt{a x^2+b x^3}}{x^4} \, dx\)

Optimal. Leaf size=84 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{4 a^{3/2}}-\frac{b \sqrt{a x^2+b x^3}}{4 a x^2}-\frac{\sqrt{a x^2+b x^3}}{2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.167857, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{4 a^{3/2}}-\frac{b \sqrt{a x^2+b x^3}}{4 a x^2}-\frac{\sqrt{a x^2+b x^3}}{2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^2 + b*x^3]/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 17.0402, size = 71, normalized size = 0.85 \[ - \frac{\sqrt{a x^{2} + b x^{3}}}{2 x^{3}} - \frac{b \sqrt{a x^{2} + b x^{3}}}{4 a x^{2}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a*x**2)**(1/2)/x**4,x)

[Out]

-sqrt(a*x**2 + b*x**3)/(2*x**3) - b*sqrt(a*x**2 + b*x**3)/(4*a*x**2) + b**2*atan
h(sqrt(a)*x/sqrt(a*x**2 + b*x**3))/(4*a**(3/2))

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Mathematica [A]  time = 0.0607862, size = 81, normalized size = 0.96 \[ \frac{\sqrt{x^2 (a+b x)} \left (b^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{a} \sqrt{a+b x} (2 a+b x)\right )}{4 a^{3/2} x^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 73, normalized size = 0.9 \[ -{\frac{1}{4\,{x}^{3}}\sqrt{b{x}^{3}+a{x}^{2}} \left ( \left ( bx+a \right ) ^{{\frac{3}{2}}}{a}^{{\frac{3}{2}}}-{\it Artanh} \left ({1\sqrt{bx+a}{\frac{1}{\sqrt{a}}}} \right ) a{b}^{2}{x}^{2}+\sqrt{bx+a}{a}^{{\frac{5}{2}}} \right ){\frac{1}{\sqrt{bx+a}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^(1/2)/x^4,x)

[Out]

-1/4*(b*x^3+a*x^2)^(1/2)*((b*x+a)^(3/2)*a^(3/2)-arctanh((b*x+a)^(1/2)/a^(1/2))*a
*b^2*x^2+(b*x+a)^(1/2)*a^(5/2))/x^3/(b*x+a)^(1/2)/a^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238143, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{a} b^{2} x^{3} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )}}{8 \, a^{2} x^{3}}, \frac{\sqrt{-a} b^{2} x^{3} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) - \sqrt{b x^{3} + a x^{2}}{\left (a b x + 2 \, a^{2}\right )}}{4 \, a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(sqrt(a)*b^2*x^3*log(((b*x^2 + 2*a*x)*sqrt(a) + 2*sqrt(b*x^3 + a*x^2)*a)/x^
2) - 2*sqrt(b*x^3 + a*x^2)*(a*b*x + 2*a^2))/(a^2*x^3), 1/4*(sqrt(-a)*b^2*x^3*arc
tan(a*x/(sqrt(b*x^3 + a*x^2)*sqrt(-a))) - sqrt(b*x^3 + a*x^2)*(a*b*x + 2*a^2))/(
a^2*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (a + b x\right )}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x**2)**(1/2)/x**4,x)

[Out]

Integral(sqrt(x**2*(a + b*x))/x**4, x)

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GIAC/XCAS [A]  time = 0.242657, size = 92, normalized size = 1.1 \[ -\frac{{\left (\frac{b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b x + a\right )}^{\frac{3}{2}} b^{3} + \sqrt{b x + a} a b^{3}}{a b^{2} x^{2}}\right )}{\rm sign}\left (x\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(b^3*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a) + ((b*x + a)^(3/2)*b^3 + s
qrt(b*x + a)*a*b^3)/(a*b^2*x^2))*sign(x)/b