3.248 \(\int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=81 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{4 \sqrt{a}}-\frac{3 b \sqrt{a x^2+b x^3}}{4 x^2}-\frac{\left (a x^2+b x^3\right )^{3/2}}{2 x^5} \]

[Out]

(-3*b*Sqrt[a*x^2 + b*x^3])/(4*x^2) - (a*x^2 + b*x^3)^(3/2)/(2*x^5) - (3*b^2*ArcT
anh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/(4*Sqrt[a])

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Rubi [A]  time = 0.160324, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{4 \sqrt{a}}-\frac{3 b \sqrt{a x^2+b x^3}}{4 x^2}-\frac{\left (a x^2+b x^3\right )^{3/2}}{2 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^3)^(3/2)/x^6,x]

[Out]

(-3*b*Sqrt[a*x^2 + b*x^3])/(4*x^2) - (a*x^2 + b*x^3)^(3/2)/(2*x^5) - (3*b^2*ArcT
anh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/(4*Sqrt[a])

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Rubi in Sympy [A]  time = 16.3087, size = 75, normalized size = 0.93 \[ - \frac{3 b \sqrt{a x^{2} + b x^{3}}}{4 x^{2}} - \frac{\left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{2 x^{5}} - \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a*x**2)**(3/2)/x**6,x)

[Out]

-3*b*sqrt(a*x**2 + b*x**3)/(4*x**2) - (a*x**2 + b*x**3)**(3/2)/(2*x**5) - 3*b**2
*atanh(sqrt(a)*x/sqrt(a*x**2 + b*x**3))/(4*sqrt(a))

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Mathematica [A]  time = 0.069556, size = 82, normalized size = 1.01 \[ -\frac{\sqrt{x^2 (a+b x)} \left (3 b^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\sqrt{a} \sqrt{a+b x} (2 a+5 b x)\right )}{4 \sqrt{a} x^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^2 + b*x^3)^(3/2)/x^6,x]

[Out]

-(Sqrt[x^2*(a + b*x)]*(Sqrt[a]*Sqrt[a + b*x]*(2*a + 5*b*x) + 3*b^2*x^2*ArcTanh[S
qrt[a + b*x]/Sqrt[a]]))/(4*Sqrt[a]*x^3*Sqrt[a + b*x])

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Maple [A]  time = 0.017, size = 74, normalized size = 0.9 \[ -{\frac{1}{4\,{x}^{5}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){x}^{2}{b}^{2}+5\, \left ( bx+a \right ) ^{3/2}\sqrt{a}-3\,\sqrt{bx+a}{a}^{3/2} \right ) \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^(3/2)/x^6,x)

[Out]

-1/4*(b*x^3+a*x^2)^(3/2)*(3*arctanh((b*x+a)^(1/2)/a^(1/2))*x^2*b^2+5*(b*x+a)^(3/
2)*a^(1/2)-3*(b*x+a)^(1/2)*a^(3/2))/x^5/(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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25484, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{2} x^{3} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{3} + a x^{2}}{\left (5 \, a b x + 2 \, a^{2}\right )}}{8 \, a x^{3}}, -\frac{3 \, \sqrt{-a} b^{2} x^{3} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) + \sqrt{b x^{3} + a x^{2}}{\left (5 \, a b x + 2 \, a^{2}\right )}}{4 \, a x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x^2)^(3/2)/x^6,x, algorithm="fricas")

[Out]

[1/8*(3*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)*(5*a*b*x + 2*a^2))/(a*x^3), -1/4*(3*sqrt(-a)*b^2*x^
3*arctan(a*x/(sqrt(b*x^3 + a*x^2)*sqrt(-a))) + sqrt(b*x^3 + a*x^2)*(5*a*b*x + 2*
a^2))/(a*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x**2)**(3/2)/x**6,x)

[Out]

Integral((x**2*(a + b*x))**(3/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.248436, size = 95, normalized size = 1.17 \[ \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{\sqrt{-a}} - \frac{5 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}{\rm sign}\left (x\right ) - 3 \, \sqrt{b x + a} a b^{3}{\rm sign}\left (x\right )}{b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x^2)^(3/2)/x^6,x, algorithm="giac")

[Out]

1/4*(3*b^3*arctan(sqrt(b*x + a)/sqrt(-a))*sign(x)/sqrt(-a) - (5*(b*x + a)^(3/2)*
b^3*sign(x) - 3*sqrt(b*x + a)*a*b^3*sign(x))/(b^2*x^2))/b