3.157 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=145 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.116313, antiderivative size = 145, 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 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^4,x]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (3*a^2*b*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*
(a + b*x)) + (b^3*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 = 18.3613, size = 121, normalized size = 0.83 \[ - \frac{a b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x \left (a + b x\right )} + \frac{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} - \frac{b \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 x^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} \]

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**4,x)

[Out]

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

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Mathematica [A]  time = 0.0329192, size = 57, normalized size = 0.39 \[ -\frac{\sqrt{(a+b x)^2} \left (a \left (2 a^2+9 a b x+18 b^2 x^2\right )-6 b^3 x^3 \log (x)\right )}{6 x^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(a*(2*a^2 + 9*a*b*x + 18*b^2*x^2) - 6*b^3*x^3*Log[x]))/(6*x^
3*(a + b*x))

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Maple [A]  time = 0.01, size = 54, normalized size = 0.4 \[{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-18\,a{b}^{2}{x}^{2}-9\,{a}^{2}bx-2\,{a}^{3}}{6\, \left ( bx+a \right ) ^{3}{x}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^4,x)

[Out]

1/6*((b*x+a)^2)^(3/2)*(6*b^3*ln(x)*x^3-18*a*b^2*x^2-9*a^2*b*x-2*a^3)/(b*x+a)^3/x
^3

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235117, size = 50, normalized size = 0.34 \[ \frac{6 \, b^{3} x^{3} \log \left (x\right ) - 18 \, a b^{2} x^{2} - 9 \, a^{2} b x - 2 \, a^{3}}{6 \, x^{3}} \]

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^4,x, algorithm="fricas")

[Out]

1/6*(6*b^3*x^3*log(x) - 18*a*b^2*x^2 - 9*a^2*b*x - 2*a^3)/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{4}}\, dx \]

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**4,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.209596, size = 80, normalized size = 0.55 \[ b^{3}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{18 \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 9 \, a^{2} b x{\rm sign}\left (b x + a\right ) + 2 \, a^{3}{\rm sign}\left (b x + a\right )}{6 \, x^{3}} \]

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^4,x, algorithm="giac")

[Out]

b^3*ln(abs(x))*sign(b*x + a) - 1/6*(18*a*b^2*x^2*sign(b*x + a) + 9*a^2*b*x*sign(
b*x + a) + 2*a^3*sign(b*x + a))/x^3