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

Optimal. Leaf size=163 \[ \frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a^2 b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{b^3 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]

[Out]

(3*a^2*b*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*(a + b*x^2)) + (3*a*b^2*x^4*Sqr
t[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*(a + b*x^2)) + (b^3*x^6*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(6*(a + b*x^2)) + (a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[x])/(a + b*
x^2)

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Rubi [A]  time = 0.134498, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{3 a b^2 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a^2 b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{b^3 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^2*b*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*(a + b*x^2)) + (3*a*b^2*x^4*Sqr
t[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*(a + b*x^2)) + (b^3*x^6*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(6*(a + b*x^2)) + (a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[x])/(a + b*
x^2)

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Rubi in Sympy [A]  time = 16.7836, size = 117, normalized size = 0.72 \[ \frac{a^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (x \right )}}{a + b x^{2}} + \frac{a^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2} + \frac{a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{4} + \frac{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)*log(x)/(a + b*x**2) + a**2*sqrt(a**2 +
2*a*b*x**2 + b**2*x**4)/2 + a*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/4
+ (a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/6

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Mathematica [A]  time = 0.034978, size = 60, normalized size = 0.37 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (12 a^3 \log (x)+b x^2 \left (18 a^2+9 a b x^2+2 b^2 x^4\right )\right )}{12 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^2)^2]*(b*x^2*(18*a^2 + 9*a*b*x^2 + 2*b^2*x^4) + 12*a^3*Log[x]))/(
12*(a + b*x^2))

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Maple [A]  time = 0.012, size = 57, normalized size = 0.4 \[{\frac{2\,{b}^{3}{x}^{6}+9\,a{x}^{4}{b}^{2}+18\,{a}^{2}b{x}^{2}+12\,{a}^{3}\ln \left ( x \right ) }{12\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*((b*x^2+a)^2)^(3/2)*(2*b^3*x^6+9*a*x^4*b^2+18*a^2*b*x^2+12*a^3*ln(x))/(b*x^
2+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^4 + 2*a*b*x^2 + a^2)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272004, size = 45, normalized size = 0.28 \[ \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x,x, algorithm="fricas")

[Out]

1/6*b^3*x^6 + 3/4*a*b^2*x^4 + 3/2*a^2*b*x^2 + a^3*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270201, size = 92, normalized size = 0.56 \[ \frac{1}{6} \, b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{4} \, a b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{2} \, a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{2} \, a^{3}{\rm ln}\left (x^{2}\right ){\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/x,x, algorithm="giac")

[Out]

1/6*b^3*x^6*sign(b*x^2 + a) + 3/4*a*b^2*x^4*sign(b*x^2 + a) + 3/2*a^2*b*x^2*sign
(b*x^2 + a) + 1/2*a^3*ln(x^2)*sign(b*x^2 + a)