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

Optimal. Leaf size=161 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac{3 a b^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^3*(a + b*x^2)) - (3*a^2*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2)) + (3*a*b^2*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x
^4])/(a + b*x^2) + (b^3*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2))

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Rubi [A]  time = 0.125021, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}+\frac{3 a b^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*x^3*(a + b*x^2)) - (3*a^2*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(x*(a + b*x^2)) + (3*a*b^2*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x
^4])/(a + b*x^2) + (b^3*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2))

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Rubi in Sympy [A]  time = 24.2465, size = 134, normalized size = 0.83 \[ \frac{16 a b^{2} x \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{3 \left (a + b x^{2}\right )} + \frac{2 a \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{x^{3}} + \frac{8 b^{2} x \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{3} - \frac{7 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{3 x^{3}} \]

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**4,x)

[Out]

16*a*b**2*x*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(3*(a + b*x**2)) + 2*a*(a + b*x*
*2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/x**3 + 8*b**2*x*sqrt(a**2 + 2*a*b*x**2 +
b**2*x**4)/3 - 7*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(3*x**3)

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Mathematica [A]  time = 0.0250204, size = 59, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (a^3+9 a^2 b x^2-9 a b^2 x^4-b^3 x^6\right )}{3 x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x^2)^2]*(a^3 + 9*a^2*b*x^2 - 9*a*b^2*x^4 - b^3*x^6))/(3*x^3*(a + b
*x^2))

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Maple [A]  time = 0.009, size = 56, normalized size = 0.4 \[ -{\frac{-{b}^{3}{x}^{6}-9\,a{x}^{4}{b}^{2}+9\,{a}^{2}b{x}^{2}+{a}^{3}}{3\,{x}^{3} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^4,x)

[Out]

-1/3*(-b^3*x^6-9*a*b^2*x^4+9*a^2*b*x^2+a^3)*((b*x^2+a)^2)^(3/2)/x^3/(b*x^2+a)^3

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Maxima [A]  time = 0.687994, size = 49, normalized size = 0.3 \[ \frac{b^{3} x^{6} + 9 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} - a^{3}}{3 \, x^{3}} \]

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^4,x, algorithm="maxima")

[Out]

1/3*(b^3*x^6 + 9*a*b^2*x^4 - 9*a^2*b*x^2 - a^3)/x^3

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Fricas [A]  time = 0.259887, size = 49, normalized size = 0.3 \[ \frac{b^{3} x^{6} + 9 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} - a^{3}}{3 \, x^{3}} \]

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^4,x, algorithm="fricas")

[Out]

1/3*(b^3*x^6 + 9*a*b^2*x^4 - 9*a^2*b*x^2 - a^3)/x^3

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

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**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.269993, size = 90, normalized size = 0.56 \[ \frac{1}{3} \, b^{3} x^{3}{\rm sign}\left (b x^{2} + a\right ) + 3 \, a b^{2} x{\rm sign}\left (b x^{2} + a\right ) - \frac{9 \, a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + a^{3}{\rm sign}\left (b x^{2} + a\right )}{3 \, x^{3}} \]

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^4,x, algorithm="giac")

[Out]

1/3*b^3*x^3*sign(b*x^2 + a) + 3*a*b^2*x*sign(b*x^2 + a) - 1/3*(9*a^2*b*x^2*sign(
b*x^2 + a) + a^3*sign(b*x^2 + a))/x^3