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

Optimal. Leaf size=221 \[ \frac{b^5 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{5 a b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{10 a^2 b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^4 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.158253, antiderivative size = 221, 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 \[ \frac{b^5 x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{5 a b^4 x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac{10 a^2 b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^5 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^4 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^4*b*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (5*a^3*b^2*x^2*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(a + b*x) + (10*a^2*b^3*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
3*(a + b*x)) + (5*a*b^4*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (b^5*
x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (a^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 = 23.0865, size = 168, normalized size = 0.76 \[ \frac{a^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + a^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{a^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6} + \frac{a^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3} + \frac{a \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{20} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5} \]

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,x)

[Out]

a**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a + b*x) + a**4*sqrt(a**2 + 2*a*b*
x + b**2*x**2) + a**3*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/6 + a**2*(a
**2 + 2*a*b*x + b**2*x**2)**(3/2)/3 + a*(5*a + 5*b*x)*(a**2 + 2*a*b*x + b**2*x**
2)**(3/2)/20 + (a**2 + 2*a*b*x + b**2*x**2)**(5/2)/5

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Mathematica [A]  time = 0.0412464, size = 74, normalized size = 0.33 \[ \frac{\sqrt{(a+b x)^2} \left (60 a^5 \log (x)+b x \left (300 a^4+300 a^3 b x+200 a^2 b^2 x^2+75 a b^3 x^3+12 b^4 x^4\right )\right )}{60 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 73, normalized size = 0.3 \[{\frac{12\,{b}^{5}{x}^{5}+75\,a{b}^{4}{x}^{4}+200\,{a}^{2}{b}^{3}{x}^{3}+300\,{a}^{3}{b}^{2}{x}^{2}+60\,{a}^{5}\ln \left ( x \right ) +300\,{a}^{4}bx}{60\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(12*b^5*x^5+75*a*b^4*x^4+200*a^2*b^3*x^3+300*a^3*b^2*x^2+
60*a^5*ln(x)+300*a^4*b*x)/(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222745, size = 72, normalized size = 0.33 \[ \frac{1}{5} \, b^{5} x^{5} + \frac{5}{4} \, a b^{4} x^{4} + \frac{10}{3} \, a^{2} b^{3} x^{3} + 5 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5} \log \left (x\right ) \]

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,x, algorithm="fricas")

[Out]

1/5*b^5*x^5 + 5/4*a*b^4*x^4 + 10/3*a^2*b^3*x^3 + 5*a^3*b^2*x^2 + 5*a^4*b*x + a^5
*log(x)

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

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,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x, x)

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GIAC/XCAS [A]  time = 0.212569, size = 122, normalized size = 0.55 \[ \frac{1}{5} \, b^{5} x^{5}{\rm sign}\left (b x + a\right ) + \frac{5}{4} \, a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + \frac{10}{3} \, a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 5 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b x{\rm sign}\left (b x + a\right ) + a^{5}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) \]

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,x, algorithm="giac")

[Out]

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