3.247 \(\int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=73 \[ \frac{3 b \sqrt{a x^2+b x^3}}{x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )-\frac{\left (a x^2+b x^3\right )^{3/2}}{x^4} \]

[Out]

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

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Rubi [A]  time = 0.159179, antiderivative size = 73, 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 \[ \frac{3 b \sqrt{a x^2+b x^3}}{x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )-\frac{\left (a x^2+b x^3\right )^{3/2}}{x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^3)^(3/2)/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.951, size = 65, normalized size = 0.89 \[ - 3 \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )} + \frac{3 b \sqrt{a x^{2} + b x^{3}}}{x} - \frac{\left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a*x**2)**(3/2)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0751285, size = 66, normalized size = 0.9 \[ -\frac{\sqrt{a+b x} \left (\sqrt{a+b x} (a-2 b x)+3 \sqrt{a} b x \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right )}{\sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^2 + b*x^3)^(3/2)/x^5,x]

[Out]

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

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Maple [A]  time = 0.016, size = 72, normalized size = 1. \[ -{\frac{1}{{x}^{4}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( \sqrt{bx+a}{a}^{{\frac{3}{2}}}-2\,\sqrt{bx+a}xb\sqrt{a}+3\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) xab \right ) \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^(3/2)/x^5,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240975, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b x^{2} \log \left (\frac{b x^{2} + 2 \, a x - 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{a}}{x^{2}}\right ) + 2 \, \sqrt{b x^{3} + a x^{2}}{\left (2 \, b x - a\right )}}{2 \, x^{2}}, -\frac{3 \, \sqrt{-a} b x^{2} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}}}{\sqrt{-a} x}\right ) - \sqrt{b x^{3} + a x^{2}}{\left (2 \, b x - a\right )}}{x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x^2)^(3/2)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x**2)**(3/2)/x**5,x)

[Out]

Integral((x**2*(a + b*x))**(3/2)/x**5, x)

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GIAC/XCAS [A]  time = 0.243206, size = 84, normalized size = 1.15 \[ \frac{\frac{3 \, a b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{\sqrt{-a}} + 2 \, \sqrt{b x + a} b^{2}{\rm sign}\left (x\right ) - \frac{\sqrt{b x + a} a b{\rm sign}\left (x\right )}{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x^2)^(3/2)/x^5,x, algorithm="giac")

[Out]

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