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

Optimal. Leaf size=219 \[ \frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.163308, antiderivative size = 219, 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 \[ \frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a b^4 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{10 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{5 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.7138, size = 184, normalized size = 0.84 \[ \frac{5 a b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + 5 b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} - \frac{5 b^{3} \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 x} - \frac{5 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{6 x^{2}} - \frac{5 b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 x^{3}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0387352, size = 79, normalized size = 0.36 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^5+20 a^4 b x+60 a^3 b^2 x^2+120 a^2 b^3 x^3-60 a b^4 x^4 \log (x)-12 b^5 x^5\right )}{12 x^4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 76, normalized size = 0.4 \[{\frac{60\,a{b}^{4}\ln \left ( x \right ){x}^{4}+12\,{b}^{5}{x}^{5}-120\,{a}^{2}{b}^{3}{x}^{3}-60\,{a}^{3}{b}^{2}{x}^{2}-20\,{a}^{4}bx-3\,{a}^{5}}{12\, \left ( bx+a \right ) ^{5}{x}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*((b*x+a)^2)^(5/2)*(60*a*b^4*ln(x)*x^4+12*b^5*x^5-120*a^2*b^3*x^3-60*a^3*b^2
*x^2-20*a^4*b*x-3*a^5)/(b*x+a)^5/x^4

_______________________________________________________________________________________

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)^(5/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.225232, size = 80, normalized size = 0.37 \[ \frac{12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2094, size = 123, normalized size = 0.56 \[ b^{5} x{\rm sign}\left (b x + a\right ) + 5 \, a b^{4}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{120 \, a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 60 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 20 \, a^{4} b x{\rm sign}\left (b x + a\right ) + 3 \, a^{5}{\rm sign}\left (b x + a\right )}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^5*x*sign(b*x + a) + 5*a*b^4*ln(abs(x))*sign(b*x + a) - 1/12*(120*a^2*b^3*x^3*s
ign(b*x + a) + 60*a^3*b^2*x^2*sign(b*x + a) + 20*a^4*b*x*sign(b*x + a) + 3*a^5*s
ign(b*x + a))/x^4