3.2492 \(\int \frac{\left (a+b x^n\right )^{5/2}}{x} \, dx\)

Optimal. Leaf size=85 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n}+\frac{2 a^2 \sqrt{a+b x^n}}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n} \]

[Out]

(2*a^2*Sqrt[a + b*x^n])/n + (2*a*(a + b*x^n)^(3/2))/(3*n) + (2*(a + b*x^n)^(5/2)
)/(5*n) - (2*a^(5/2)*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/n

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Rubi [A]  time = 0.115577, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n}+\frac{2 a^2 \sqrt{a+b x^n}}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(5/2)/x,x]

[Out]

(2*a^2*Sqrt[a + b*x^n])/n + (2*a*(a + b*x^n)^(3/2))/(3*n) + (2*(a + b*x^n)^(5/2)
)/(5*n) - (2*a^(5/2)*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/n

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Rubi in Sympy [A]  time = 12.8996, size = 73, normalized size = 0.86 \[ - \frac{2 a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 a^{2} \sqrt{a + b x^{n}}}{n} + \frac{2 a \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 n} + \frac{2 \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**(5/2)/x,x)

[Out]

-2*a**(5/2)*atanh(sqrt(a + b*x**n)/sqrt(a))/n + 2*a**2*sqrt(a + b*x**n)/n + 2*a*
(a + b*x**n)**(3/2)/(3*n) + 2*(a + b*x**n)**(5/2)/(5*n)

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Mathematica [A]  time = 0.0926866, size = 69, normalized size = 0.81 \[ \frac{2 \sqrt{a+b x^n} \left (23 a^2+11 a b x^n+3 b^2 x^{2 n}\right )-30 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{15 n} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x^n]*(23*a^2 + 11*a*b*x^n + 3*b^2*x^(2*n)) - 30*a^(5/2)*ArcTanh[Sq
rt[a + b*x^n]/Sqrt[a]])/(15*n)

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Maple [A]  time = 0.006, size = 62, normalized size = 0.7 \[{\frac{1}{n} \left ({\frac{2}{5} \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}}+2\,\sqrt{a+b{x}^{n}}{a}^{2}-2\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(5/2)/x,x)

[Out]

1/n*(2/5*(a+b*x^n)^(5/2)+2/3*(a+b*x^n)^(3/2)*a+2*(a+b*x^n)^(1/2)*a^2-2*a^(5/2)*a
rctanh((a+b*x^n)^(1/2)/a^(1/2)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230742, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{5}{2}} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \,{\left (3 \, b^{2} x^{2 \, n} + 11 \, a b x^{n} + 23 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, n}, -\frac{2 \,{\left (15 \, \sqrt{-a} a^{2} \arctan \left (\frac{\sqrt{b x^{n} + a}}{\sqrt{-a}}\right ) -{\left (3 \, b^{2} x^{2 \, n} + 11 \, a b x^{n} + 23 \, a^{2}\right )} \sqrt{b x^{n} + a}\right )}}{15 \, n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2)/x,x, algorithm="fricas")

[Out]

[1/15*(15*a^(5/2)*log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/x^n) + 2*(3*b^2*
x^(2*n) + 11*a*b*x^n + 23*a^2)*sqrt(b*x^n + a))/n, -2/15*(15*sqrt(-a)*a^2*arctan
(sqrt(b*x^n + a)/sqrt(-a)) - (3*b^2*x^(2*n) + 11*a*b*x^n + 23*a^2)*sqrt(b*x^n +
a))/n]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**n)**(5/2)/x,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2)/x,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(5/2)/x, x)