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

Optimal. Leaf size=142 \[ \frac{3 a b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.115137, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{3 a b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.0299, size = 112, normalized size = 0.79 \[ \frac{3 a^{2} b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + 3 a b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{3 b \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0308332, size = 56, normalized size = 0.39 \[ \frac{\sqrt{(a+b x)^2} \left (-2 a^3+6 a^2 b x \log (x)+6 a b^2 x^2+b^3 x^3\right )}{2 x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(-2*a^3 + 6*a*b^2*x^2 + b^3*x^3 + 6*a^2*b*x*Log[x]))/(2*x*(a
+ b*x))

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Maple [A]  time = 0.015, size = 53, normalized size = 0.4 \[{\frac{{b}^{3}{x}^{3}+6\,{a}^{2}b\ln \left ( x \right ) x+6\,a{b}^{2}{x}^{2}-2\,{a}^{3}}{2\, \left ( bx+a \right ) ^{3}x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((b*x+a)^2)^(3/2)*(b^3*x^3+6*a^2*b*ln(x)*x+6*a*b^2*x^2-2*a^3)/(b*x+a)^3/x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242865, size = 49, normalized size = 0.35 \[ \frac{b^{3} x^{3} + 6 \, a b^{2} x^{2} + 6 \, a^{2} b x \log \left (x\right ) - 2 \, a^{3}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^3*x^3 + 6*a*b^2*x^2 + 6*a^2*b*x*log(x) - 2*a^3)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209113, size = 77, normalized size = 0.54 \[ \frac{1}{2} \, b^{3} x^{2}{\rm sign}\left (b x + a\right ) + 3 \, a b^{2} x{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{a^{3}{\rm sign}\left (b x + a\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^3*x^2*sign(b*x + a) + 3*a*b^2*x*sign(b*x + a) + 3*a^2*b*ln(abs(x))*sign(b*
x + a) - a^3*sign(b*x + a)/x