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

Optimal. Leaf size=222 \[ \frac{b^5 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{10 a^3 b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.163054, antiderivative size = 222, 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 \[ \frac{b^5 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{10 a^2 b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{10 a^3 b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (5*a^4*b*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(x*(a + b*x)) + (10*a^2*b^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a
+ b*x) + (5*a*b^4*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(a + b*x)) + (b^5*x^3*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (10*a^3*b^2*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 = 23.2245, size = 187, normalized size = 0.84 \[ \frac{10 a^{3} b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + 10 a^{2} b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{5 a b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3} + \frac{10 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3} - \frac{5 b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{2 x} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{2 x^{2}} \]

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**3,x)

[Out]

10*a**3*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a + b*x) + 10*a**2*b**2*sq
rt(a**2 + 2*a*b*x + b**2*x**2) + 5*a*b**2*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b*
*2*x**2)/3 + 10*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/3 - 5*b*(a + b*x)*(a**2
+ 2*a*b*x + b**2*x**2)**(3/2)/(2*x) - (a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(2*x*
*2)

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Mathematica [A]  time = 0.0377474, size = 79, normalized size = 0.36 \[ \frac{\sqrt{(a+b x)^2} \left (-3 a^5-30 a^4 b x+60 a^3 b^2 x^2 \log (x)+60 a^2 b^3 x^3+15 a b^4 x^4+2 b^5 x^5\right )}{6 x^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 76, normalized size = 0.3 \[{\frac{2\,{b}^{5}{x}^{5}+15\,a{b}^{4}{x}^{4}+60\,{a}^{3}{b}^{2}\ln \left ( x \right ){x}^{2}+60\,{a}^{2}{b}^{3}{x}^{3}-30\,{a}^{4}bx-3\,{a}^{5}}{6\, \left ( bx+a \right ) ^{5}{x}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^3,x)

[Out]

1/6*((b*x+a)^2)^(5/2)*(2*b^5*x^5+15*a*b^4*x^4+60*a^3*b^2*ln(x)*x^2+60*a^2*b^3*x^
3-30*a^4*b*x-3*a^5)/(b*x+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^2 + 2*a*b*x + a^2)^(5/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220854, size = 80, normalized size = 0.36 \[ \frac{2 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 60 \, a^{3} b^{2} x^{2} \log \left (x\right ) - 30 \, a^{4} b x - 3 \, a^{5}}{6 \, x^{2}} \]

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^3,x, algorithm="fricas")

[Out]

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

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

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**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216525, size = 123, normalized size = 0.55 \[ \frac{1}{3} \, b^{5} x^{3}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, a b^{4} x^{2}{\rm sign}\left (b x + a\right ) + 10 \, a^{2} b^{3} x{\rm sign}\left (b x + a\right ) + 10 \, a^{3} b^{2}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{10 \, a^{4} b x{\rm sign}\left (b x + a\right ) + a^{5}{\rm sign}\left (b x + a\right )}{2 \, x^{2}} \]

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^3,x, algorithm="giac")

[Out]

1/3*b^5*x^3*sign(b*x + a) + 5/2*a*b^4*x^2*sign(b*x + a) + 10*a^2*b^3*x*sign(b*x
+ a) + 10*a^3*b^2*ln(abs(x))*sign(b*x + a) - 1/2*(10*a^4*b*x*sign(b*x + a) + a^5
*sign(b*x + a))/x^2