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

Optimal. Leaf size=101 \[ -\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n}+\frac{2 a^2 \sqrt{b (c x)^n-a}}{n}-\frac{2 a \left (b (c x)^n-a\right )^{3/2}}{3 n}+\frac{2 \left (b (c x)^n-a\right )^{5/2}}{5 n} \]

[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)*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/n

_______________________________________________________________________________________

Rubi [A]  time = 0.178208, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n}+\frac{2 a^2 \sqrt{b (c x)^n-a}}{n}-\frac{2 a \left (b (c x)^n-a\right )^{3/2}}{3 n}+\frac{2 \left (b (c x)^n-a\right )^{5/2}}{5 n} \]

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)*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/n

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.07787, size = 80, normalized size = 0.79 \[ - \frac{2 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{n} + \frac{2 a^{2} \sqrt{- a + b \left (c x\right )^{n}}}{n} - \frac{2 a \left (- a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}{3 n} + \frac{2 \left (- a + b \left (c x\right )^{n}\right )^{\frac{5}{2}}}{5 n} \]

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)*atan(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)

_______________________________________________________________________________________

Mathematica [A]  time = 0.102974, size = 81, normalized size = 0.8 \[ \frac{2 \sqrt{b (c x)^n-a} \left (23 a^2-11 a b (c x)^n+3 b^2 (c x)^{2 n}\right )-30 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{15 n} \]

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)*ArcTan[Sqrt[-a + b*(c*x)^n]/Sqrt[a]])/(15*n)

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 86, normalized size = 0.9 \[ -{\frac{2\,a}{3\,n} \left ( -a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{3}{2}}}}+{\frac{2}{5\,n} \left ( -a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{5}{2}}}}-2\,{\frac{{a}^{5/2}}{n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{{a}^{2}\sqrt{-a+b \left ( cx \right ) ^{n}}}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-a+b*(c*x)^n)^(5/2)/x,x)

[Out]

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Fricas [A]  time = 0.285436, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{-a} a^{2} \log \left (\frac{\left (c x\right )^{n} b - 2 \, \sqrt{\left (c x\right )^{n} b - a} \sqrt{-a} - 2 \, a}{\left (c x\right )^{n}}\right ) - 2 \,{\left (11 \, \left (c x\right )^{n} a b - 3 \, \left (c x\right )^{2 \, n} b^{2} - 23 \, a^{2}\right )} \sqrt{\left (c x\right )^{n} b - a}}{15 \, n}, -\frac{2 \,{\left (15 \, a^{\frac{5}{2}} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b - a}}{\sqrt{a}}\right ) +{\left (11 \, \left (c x\right )^{n} a b - 3 \, \left (c x\right )^{2 \, n} b^{2} - 23 \, a^{2}\right )} \sqrt{\left (c x\right )^{n} b - a}\right )}}{15 \, n}\right ] \]

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*sqrt(-a)*a^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*a^(5/2)*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]

_______________________________________________________________________________________

Sympy [A]  time = 106.39, size = 192, normalized size = 1.9 \[ \begin{cases} \frac{- 2 a^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a < - a + b \left (c x\right )^{n} \wedge a < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > - a + b \left (c x\right )^{n} \wedge a < 0 \end{cases}\right ) + 2 a^{2} \sqrt{- a + b \left (c x\right )^{n}} - \frac{2 a \left (- a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}{3} + \frac{2 \left (- a + b \left (c x\right )^{n}\right )^{\frac{5}{2}}}{5}}{n} & \text{for}\: n \neq 0 \\\left (a^{2} \sqrt{- a + b} - 2 a b \sqrt{- a + b} + b^{2} \sqrt{- a + b}\right ) \log{\left (c x \right )} & \text{otherwise} \end{cases} \]

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))

_______________________________________________________________________________________

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)