3.593 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=250 \[ \frac{b^5 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac{5 a b^4 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac{5 a^2 b^3 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*x^2*(a + b*x^2)) + (5*a^3*b^2*x^2*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])/(a + b*x^2) + (5*a^2*b^3*x^4*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(2*(a + b*x^2)) + (5*a*b^4*x^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(6*(a
+ b*x^2)) + (b^5*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*(a + b*x^2)) + (5*a^4*
b*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 = 0.207094, antiderivative size = 250, 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 \[ \frac{b^5 x^8 \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac{5 a b^4 x^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac{5 a^2 b^3 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^3,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*x^2*(a + b*x^2)) + (5*a^3*b^2*x^2*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])/(a + b*x^2) + (5*a^2*b^3*x^4*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4])/(2*(a + b*x^2)) + (5*a*b^4*x^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(6*(a
+ b*x^2)) + (b^5*x^8*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(8*(a + b*x^2)) + (5*a^4*
b*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 = 26.0027, size = 199, normalized size = 0.8 \[ \frac{5 a^{4} b \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (x \right )}}{a + b x^{2}} + \frac{5 a^{3} b \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2} + \frac{5 a^{2} b \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{4} + \frac{5 a b \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{6} - \frac{5 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{8 x^{2}} + \frac{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{8 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**3,x)

[Out]

5*a**4*b*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)*log(x)/(a + b*x**2) + 5*a**3*b*sqrt
(a**2 + 2*a*b*x**2 + b**2*x**4)/2 + 5*a**2*b*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2
+ b**2*x**4)/4 + 5*a*b*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/6 - 5*a*(a + b*x*
*2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(8*x**2) + (a**2 + 2*a*b*x**2 + b**2*
x**4)**(5/2)/(8*x**2)

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Mathematica [A]  time = 0.0491727, size = 85, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (-12 a^5+120 a^4 b x^2 \log (x)+120 a^3 b^2 x^4+60 a^2 b^3 x^6+20 a b^4 x^8+3 b^5 x^{10}\right )}{24 x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^3,x]

[Out]

(Sqrt[(a + b*x^2)^2]*(-12*a^5 + 120*a^3*b^2*x^4 + 60*a^2*b^3*x^6 + 20*a*b^4*x^8
+ 3*b^5*x^10 + 120*a^4*b*x^2*Log[x]))/(24*x^2*(a + b*x^2))

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Maple [A]  time = 0.017, size = 82, normalized size = 0.3 \[{\frac{3\,{b}^{5}{x}^{10}+20\,a{b}^{4}{x}^{8}+60\,{a}^{2}{b}^{3}{x}^{6}+120\,{a}^{3}{b}^{2}{x}^{4}+120\,{a}^{4}b\ln \left ( x \right ){x}^{2}-12\,{a}^{5}}{24\, \left ( b{x}^{2}+a \right ) ^{5}{x}^{2}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^3,x)

[Out]

1/24*((b*x^2+a)^2)^(5/2)*(3*b^5*x^10+20*a*b^4*x^8+60*a^2*b^3*x^6+120*a^3*b^2*x^4
+120*a^4*b*ln(x)*x^2-12*a^5)/(b*x^2+a)^5/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^4 + 2*a*b*x^2 + a^2)^(5/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269034, size = 82, normalized size = 0.33 \[ \frac{3 \, b^{5} x^{10} + 20 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} \log \left (x\right ) - 12 \, a^{5}}{24 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^3,x, algorithm="fricas")

[Out]

1/24*(3*b^5*x^10 + 20*a*b^4*x^8 + 60*a^2*b^3*x^6 + 120*a^3*b^2*x^4 + 120*a^4*b*x
^2*log(x) - 12*a^5)/x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**3,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**3, x)

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GIAC/XCAS [A]  time = 0.272321, size = 169, normalized size = 0.68 \[ \frac{1}{8} \, b^{5} x^{8}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{6} \, a b^{4} x^{6}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{2} \, a^{2} b^{3} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 5 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{2} \, a^{4} b{\rm ln}\left (x^{2}\right ){\rm sign}\left (b x^{2} + a\right ) - \frac{5 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + a^{5}{\rm sign}\left (b x^{2} + a\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^3,x, algorithm="giac")

[Out]

1/8*b^5*x^8*sign(b*x^2 + a) + 5/6*a*b^4*x^6*sign(b*x^2 + a) + 5/2*a^2*b^3*x^4*si
gn(b*x^2 + a) + 5*a^3*b^2*x^2*sign(b*x^2 + a) + 5/2*a^4*b*ln(x^2)*sign(b*x^2 + a
) - 1/2*(5*a^4*b*x^2*sign(b*x^2 + a) + a^5*sign(b*x^2 + a))/x^2