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

Optimal. Leaf size=196 \[ \frac{3 a^2 b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a b+b^2 x^n\right )}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]

[Out]

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

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Rubi [A]  time = 0.139127, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{3 a^2 b^2 x^n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{n \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a b+b^2 x^n\right )}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.0133, size = 143, normalized size = 0.73 \[ \frac{2 a^{3} b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}} \log{\left (x \right )}}{2 a b + 2 b^{2} x^{n}} + \frac{a^{2} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{n} + \frac{\left (2 a^{2} + 2 a b x^{n}\right ) \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{4 n} + \frac{\left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac{3}{2}}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2)/x,x)

[Out]

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

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Mathematica [A]  time = 0.0640078, size = 66, normalized size = 0.34 \[ \frac{\sqrt{\left (a+b x^n\right )^2} \left (6 a^3 n \log (x)+b x^n \left (18 a^2+9 a b x^n+2 b^2 x^{2 n}\right )\right )}{6 n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^n)^2]*(b*x^n*(18*a^2 + 9*a*b*x^n + 2*b^2*x^(2*n)) + 6*a^3*n*Log[x
]))/(6*n*(a + b*x^n))

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Maple [A]  time = 0.031, size = 127, normalized size = 0.7 \[{\frac{{a}^{3}\ln \left ( x \right ) }{a+b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{b}^{3} \left ({x}^{n} \right ) ^{3}}{ \left ( 3\,a+3\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{3\,a{b}^{2} \left ({x}^{n} \right ) ^{2}}{ \left ( 2\,a+2\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+3\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{a}^{2}b{x}^{n}}{ \left ( a+b{x}^{n} \right ) n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2)/x,x)

[Out]

((a+b*x^n)^2)^(1/2)/(a+b*x^n)*a^3*ln(x)+1/3*((a+b*x^n)^2)^(1/2)/(a+b*x^n)*b^3/n*
(x^n)^3+3/2*((a+b*x^n)^2)^(1/2)/(a+b*x^n)*a*b^2/n*(x^n)^2+3*((a+b*x^n)^2)^(1/2)/
(a+b*x^n)*a^2*b/n*x^n

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Maxima [A]  time = 0.753302, size = 58, normalized size = 0.3 \[ a^{3} \log \left (x\right ) + \frac{2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)/x,x, algorithm="maxima")

[Out]

a^3*log(x) + 1/6*(2*b^3*x^(3*n) + 9*a*b^2*x^(2*n) + 18*a^2*b*x^n)/n

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Fricas [A]  time = 0.271802, size = 59, normalized size = 0.3 \[ \frac{6 \, a^{3} n \log \left (x\right ) + 2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)/x,x, algorithm="fricas")

[Out]

1/6*(6*a^3*n*log(x) + 2*b^3*x^(3*n) + 9*a*b^2*x^(2*n) + 18*a^2*b*x^n)/n

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2)/x,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)/x,x, algorithm="giac")

[Out]

integrate((b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2)/x, x)