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

Optimal. Leaf size=93 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n}+\frac{2 a^2 \sqrt{a+b (c x)^n}}{n}+\frac{2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n} \]

[Out]

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

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Rubi [A]  time = 0.170548, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n}+\frac{2 a^2 \sqrt{a+b (c x)^n}}{n}+\frac{2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*(c*x)^n)^(5/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.18671, size = 80, normalized size = 0.86 \[ - \frac{2 a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 a^{2} \sqrt{a + b \left (c x\right )^{n}}}{n} + \frac{2 a \left (a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}{3 n} + \frac{2 \left (a + b \left (c x\right )^{n}\right )^{\frac{5}{2}}}{5 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0969101, size = 77, normalized size = 0.83 \[ \frac{2 \sqrt{a+b (c x)^n} \left (23 a^2+11 a b (c x)^n+3 b^2 (c x)^{2 n}\right )-30 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{15 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 70, normalized size = 0.8 \[{\frac{1}{n} \left ({\frac{2}{5} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{3}{2}}}}+2\,\sqrt{a+b \left ( cx \right ) ^{n}}{a}^{2}-2\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x)^n)^(5/2)/x,x)

[Out]

1/n*(2/5*(a+b*(c*x)^n)^(5/2)+2/3*(a+b*(c*x)^n)^(3/2)*a+2*(a+b*(c*x)^n)^(1/2)*a^2
-2*a^(5/2)*arctanh((a+b*(c*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(((c*x)^n*b + a)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284281, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{5}{2}} \log \left (\frac{\left (c x\right )^{n} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} \sqrt{a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \,{\left (11 \, \left (c x\right )^{n} a b + 3 \, \left (c x\right )^{2 \, n} b^{2} + 23 \, a^{2}\right )} \sqrt{\left (c x\right )^{n} b + a}}{15 \, n}, -\frac{2 \,{\left (15 \, \sqrt{-a} a^{2} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b + a}}{\sqrt{-a}}\right ) -{\left (11 \, \left (c x\right )^{n} a b + 3 \, \left (c x\right )^{2 \, n} b^{2} + 23 \, a^{2}\right )} \sqrt{\left (c x\right )^{n} b + a}\right )}}{15 \, n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 108.802, size = 189, normalized size = 2.03 \[ \begin{cases} \frac{- 2 a^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b \left (c x\right )^{n} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b \left (c x\right )^{n} \wedge - a < 0 \end{cases}\right ) + 2 a^{2} \sqrt{a + b \left (c x\right )^{n}} + \frac{2 a \left (a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}{3} + \frac{2 \left (a + b \left (c x\right )^{n}\right )^{\frac{5}{2}}}{5}}{n} & \text{for}\: n \neq 0 \\\left (a^{2} \sqrt{a + b} + 2 a b \sqrt{a + b} + b^{2} \sqrt{a + b}\right ) \log{\left (c x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((-2*a**3*Piecewise((-atan(sqrt(a + b*(c*x)**n)/sqrt(-a))/sqrt(-a), -a
> 0), (acoth(sqrt(a + b*(c*x)**n)/sqrt(a))/sqrt(a), (-a < 0) & (a < a + b*(c*x)
**n)), (atanh(sqrt(a + b*(c*x)**n)/sqrt(a))/sqrt(a), (-a < 0) & (a > a + b*(c*x)
**n))) + 2*a**2*sqrt(a + b*(c*x)**n) + 2*a*(a + b*(c*x)**n)**(3/2)/3 + 2*(a + b*
(c*x)**n)**(5/2)/5)/n, Ne(n, 0)), ((a**2*sqrt(a + b) + 2*a*b*sqrt(a + b) + b**2*
sqrt(a + b))*log(c*x), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x)^n*b + a)^(5/2)/x, x)