3.174 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=223 \[ \frac{b^5 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.164758, antiderivative size = 223, 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 \[ \frac{b^5 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 26.9355, size = 184, normalized size = 0.83 \[ - \frac{a b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x \left (a + b x\right )} + \frac{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} - \frac{b^{3} \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 x^{2}} - \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 x^{4}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 x^{5}} \]

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**6,x)

[Out]

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

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Mathematica [A]  time = 0.0445052, size = 79, normalized size = 0.35 \[ -\frac{\sqrt{(a+b x)^2} \left (a \left (12 a^4+75 a^3 b x+200 a^2 b^2 x^2+300 a b^3 x^3+300 b^4 x^4\right )-60 b^5 x^5 \log (x)\right )}{60 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a*(12*a^4 + 75*a^3*b*x + 200*a^2*b^2*x^2 + 300*a*b^3*x^3 +
300*b^4*x^4) - 60*b^5*x^5*Log[x]))/(60*x^5*(a + b*x))

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Maple [A]  time = 0.017, size = 76, normalized size = 0.3 \[{\frac{60\,{b}^{5}\ln \left ( x \right ){x}^{5}-300\,a{b}^{4}{x}^{4}-300\,{a}^{2}{b}^{3}{x}^{3}-200\,{a}^{3}{b}^{2}{x}^{2}-75\,{a}^{4}bx-12\,{a}^{5}}{60\,{x}^{5} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^6,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(60*b^5*ln(x)*x^5-300*a*b^4*x^4-300*a^2*b^3*x^3-200*a^3*b
^2*x^2-75*a^4*b*x-12*a^5)/x^5/(b*x+a)^5

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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^2 + 2*a*b*x + a^2)^(5/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22191, size = 80, normalized size = 0.36 \[ \frac{60 \, b^{5} x^{5} \log \left (x\right ) - 300 \, a b^{4} x^{4} - 300 \, a^{2} b^{3} x^{3} - 200 \, a^{3} b^{2} x^{2} - 75 \, a^{4} b x - 12 \, a^{5}}{60 \, x^{5}} \]

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^6,x, algorithm="fricas")

[Out]

1/60*(60*b^5*x^5*log(x) - 300*a*b^4*x^4 - 300*a^2*b^3*x^3 - 200*a^3*b^2*x^2 - 75
*a^4*b*x - 12*a^5)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{6}}\, dx \]

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**6,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.210237, size = 126, normalized size = 0.57 \[ b^{5}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{300 \, a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 300 \, a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 200 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 75 \, a^{4} b x{\rm sign}\left (b x + a\right ) + 12 \, a^{5}{\rm sign}\left (b x + a\right )}{60 \, x^{5}} \]

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^6,x, algorithm="giac")

[Out]

b^5*ln(abs(x))*sign(b*x + a) - 1/60*(300*a*b^4*x^4*sign(b*x + a) + 300*a^2*b^3*x
^3*sign(b*x + a) + 200*a^3*b^2*x^2*sign(b*x + a) + 75*a^4*b*x*sign(b*x + a) + 12
*a^5*sign(b*x + a))/x^5