3.595 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=250 \[ \frac{b^5 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{10 a^2 b^3 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2 \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(6*x^6*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(4*x^4*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])/(x^2*(a + b*x^2)) + (5*a*b^4*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*(a
+ b*x^2)) + (b^5*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*(a + b*x^2)) + (10*a^2
*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[x])/(a + b*x^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.204852, 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 \[ \frac{b^5 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{10 a^2 b^3 \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{6 x^6 \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 x^4 \left (a+b x^2\right )}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^7,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(6*x^6*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4])/(4*x^4*(a + b*x^2)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4])/(x^2*(a + b*x^2)) + (5*a*b^4*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(2*(a
+ b*x^2)) + (b^5*x^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*(a + b*x^2)) + (10*a^2
*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*Log[x])/(a + b*x^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.6674, size = 207, normalized size = 0.83 \[ \frac{10 a^{2} b^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (x \right )}}{a + b x^{2}} + 5 a b^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} - \frac{5 a b^{2} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2 x^{2}} + \frac{5 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{12 x^{6}} + \frac{5 b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{6 x^{2}} - \frac{7 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**7,x)

[Out]

10*a**2*b**3*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)*log(x)/(a + b*x**2) + 5*a*b**3*
sqrt(a**2 + 2*a*b*x**2 + b**2*x**4) - 5*a*b**2*(a + b*x**2)*sqrt(a**2 + 2*a*b*x*
*2 + b**2*x**4)/(2*x**2) + 5*a*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/
2)/(12*x**6) + 5*b**2*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(6*x**2) - 7*(a**2
+ 2*a*b*x**2 + b**2*x**4)**(5/2)/(12*x**6)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0421725, size = 85, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (-2 a^5-15 a^4 b x^2-60 a^3 b^2 x^4+120 a^2 b^3 x^6 \log (x)+30 a b^4 x^8+3 b^5 x^{10}\right )}{12 x^6 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^7,x]

[Out]

(Sqrt[(a + b*x^2)^2]*(-2*a^5 - 15*a^4*b*x^2 - 60*a^3*b^2*x^4 + 30*a*b^4*x^8 + 3*
b^5*x^10 + 120*a^2*b^3*x^6*Log[x]))/(12*x^6*(a + b*x^2))

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 82, normalized size = 0.3 \[{\frac{3\,{b}^{5}{x}^{10}+30\,a{b}^{4}{x}^{8}+120\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{6}-60\,{a}^{3}{b}^{2}{x}^{4}-15\,{a}^{4}b{x}^{2}-2\,{a}^{5}}{12\, \left ( b{x}^{2}+a \right ) ^{5}{x}^{6}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^7,x)

[Out]

1/12*((b*x^2+a)^2)^(5/2)*(3*b^5*x^10+30*a*b^4*x^8+120*a^2*b^3*ln(x)*x^6-60*a^3*b
^2*x^4-15*a^4*b*x^2-2*a^5)/(b*x^2+a)^5/x^6

_______________________________________________________________________________________

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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.269117, size = 82, normalized size = 0.33 \[ \frac{3 \, b^{5} x^{10} + 30 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} \log \left (x\right ) - 60 \, a^{3} b^{2} x^{4} - 15 \, a^{4} b x^{2} - 2 \, a^{5}}{12 \, x^{6}} \]

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^7,x, algorithm="fricas")

[Out]

1/12*(3*b^5*x^10 + 30*a*b^4*x^8 + 120*a^2*b^3*x^6*log(x) - 60*a^3*b^2*x^4 - 15*a
^4*b*x^2 - 2*a^5)/x^6

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}{x^{7}}\, dx \]

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**7,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**7, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271616, size = 173, normalized size = 0.69 \[ \frac{1}{4} \, b^{5} x^{4}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{2} \, a b^{4} x^{2}{\rm sign}\left (b x^{2} + a\right ) + 5 \, a^{2} b^{3}{\rm ln}\left (x^{2}\right ){\rm sign}\left (b x^{2} + a\right ) - \frac{110 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 60 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 15 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 2 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{12 \, x^{6}} \]

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^7,x, algorithm="giac")

[Out]

1/4*b^5*x^4*sign(b*x^2 + a) + 5/2*a*b^4*x^2*sign(b*x^2 + a) + 5*a^2*b^3*ln(x^2)*
sign(b*x^2 + a) - 1/12*(110*a^2*b^3*x^6*sign(b*x^2 + a) + 60*a^3*b^2*x^4*sign(b*
x^2 + a) + 15*a^4*b*x^2*sign(b*x^2 + a) + 2*a^5*sign(b*x^2 + a))/x^6