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

Optimal. Leaf size=141 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.114714, antiderivative size = 141, 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 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (3*a^2*b*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(x*(a + b*x)) + (b^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x)
+ (3*a*b^2*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 = 14.3769, size = 117, normalized size = 0.83 \[ \frac{3 a b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + 3 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} - \frac{3 b \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 x} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{2 x^{2}} \]

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**3,x)

[Out]

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

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Mathematica [A]  time = 0.0263868, size = 55, normalized size = 0.39 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3+6 a^2 b x-6 a b^2 x^2 \log (x)-2 b^3 x^3\right )}{2 x^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 54, normalized size = 0.4 \[{\frac{6\,a{b}^{2}\ln \left ( x \right ){x}^{2}+2\,{b}^{3}{x}^{3}-6\,{a}^{2}bx-{a}^{3}}{2\, \left ( bx+a \right ) ^{3}{x}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230782, size = 50, normalized size = 0.35 \[ \frac{2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} \log \left (x\right ) - 6 \, a^{2} b x - a^{3}}{2 \, x^{2}} \]

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^3,x, algorithm="fricas")

[Out]

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

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

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**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.210406, size = 76, normalized size = 0.54 \[ b^{3} x{\rm sign}\left (b x + a\right ) + 3 \, a b^{2}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{6 \, a^{2} b x{\rm sign}\left (b x + a\right ) + a^{3}{\rm sign}\left (b x + a\right )}{2 \, x^{2}} \]

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^3,x, algorithm="giac")

[Out]

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