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

Optimal. Leaf size=220 \[ \frac{b^5 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.170158, antiderivative size = 220, 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 \[ \frac{b^5 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{5 a b^4 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{5 a^2 b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{5 a^4 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{10 a^3 b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

-((a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x*(a + b*x))) + (10*a^3*b^2*x*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(a + b*x) + (5*a^2*b^3*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a
+ b*x) + (5*a*b^4*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (b^5*x^4*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (5*a^4*b*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 = 22.061, size = 178, normalized size = 0.81 \[ \frac{5 a^{4} b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + 5 a^{3} b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{5 a^{2} b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6} + \frac{5 a b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3} + \frac{5 b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{x} \]

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**2,x)

[Out]

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

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Mathematica [A]  time = 0.0398923, size = 79, normalized size = 0.36 \[ \frac{\sqrt{(a+b x)^2} \left (-12 a^5+60 a^4 b x \log (x)+120 a^3 b^2 x^2+60 a^2 b^3 x^3+20 a b^4 x^4+3 b^5 x^5\right )}{12 x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 76, normalized size = 0.4 \[{\frac{3\,{b}^{5}{x}^{5}+20\,a{b}^{4}{x}^{4}+60\,{a}^{2}{b}^{3}{x}^{3}+60\,{a}^{4}b\ln \left ( x \right ) x+120\,{a}^{3}{b}^{2}{x}^{2}-12\,{a}^{5}}{12\, \left ( bx+a \right ) ^{5}x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^2,x)

[Out]

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

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223044, size = 80, normalized size = 0.36 \[ \frac{3 \, b^{5} x^{5} + 20 \, a b^{4} x^{4} + 60 \, a^{2} b^{3} x^{3} + 120 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x \log \left (x\right ) - 12 \, a^{5}}{12 \, x} \]

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^2,x, algorithm="fricas")

[Out]

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

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

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**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216723, size = 123, normalized size = 0.56 \[ \frac{1}{4} \, b^{5} x^{4}{\rm sign}\left (b x + a\right ) + \frac{5}{3} \, a b^{4} x^{3}{\rm sign}\left (b x + a\right ) + 5 \, a^{2} b^{3} x^{2}{\rm sign}\left (b x + a\right ) + 10 \, a^{3} b^{2} x{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{a^{5}{\rm sign}\left (b x + a\right )}{x} \]

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^2,x, algorithm="giac")

[Out]

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