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

Optimal. Leaf size=143 \[ \frac{3 a^2 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.10831, antiderivative size = 143, 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 \[ \frac{3 a^2 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^2*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (3*a*b^2*x^2*Sqrt[a^2 + 2*
a*b*x + b^2*x^2])/(2*(a + b*x)) + (b^3*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a
+ b*x)) + (a^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 = 14.2079, size = 109, normalized size = 0.76 \[ \frac{a^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + a^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{a \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3} \]

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,x)

[Out]

a**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a + b*x) + a**2*sqrt(a**2 + 2*a*b*
x + b**2*x**2) + a*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/6 + (a**2 + 2*
a*b*x + b**2*x**2)**(3/2)/3

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Mathematica [A]  time = 0.0320345, size = 52, normalized size = 0.36 \[ \frac{\sqrt{(a+b x)^2} \left (6 a^3 \log (x)+b x \left (18 a^2+9 a b x+2 b^2 x^2\right )\right )}{6 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 51, normalized size = 0.4 \[{\frac{2\,{b}^{3}{x}^{3}+9\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( x \right ) +18\,{a}^{2}bx}{6\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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,x)

[Out]

1/6*((b*x+a)^2)^(3/2)*(2*b^3*x^3+9*a*b^2*x^2+6*a^3*ln(x)+18*a^2*b*x)/(b*x+a)^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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232962, size = 42, normalized size = 0.29 \[ \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \]

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,x, algorithm="fricas")

[Out]

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

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

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,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/x, x)

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GIAC/XCAS [A]  time = 0.210229, size = 76, normalized size = 0.53 \[ \frac{1}{3} \, b^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b x{\rm sign}\left (b x + a\right ) + a^{3}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) \]

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,x, algorithm="giac")

[Out]

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