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

Optimal. Leaf size=76 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n}-\frac{2 a \sqrt{b (c x)^n-a}}{n}+\frac{2 \left (b (c x)^n-a\right )^{3/2}}{3 n} \]

[Out]

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

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Rubi [A]  time = 0.139962, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n}-\frac{2 a \sqrt{b (c x)^n-a}}{n}+\frac{2 \left (b (c x)^n-a\right )^{3/2}}{3 n} \]

Antiderivative was successfully verified.

[In]  Int[(-a + b*(c*x)^n)^(3/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.48878, size = 60, normalized size = 0.79 \[ \frac{2 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{n} - \frac{2 a \sqrt{- a + b \left (c x\right )^{n}}}{n} + \frac{2 \left (- a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0666771, size = 66, normalized size = 0.87 \[ \frac{6 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )-2 \left (4 a-b (c x)^n\right ) \sqrt{b (c x)^n-a}}{3 n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(4*a - b*(c*x)^n)*Sqrt[-a + b*(c*x)^n] + 6*a^(3/2)*ArcTan[Sqrt[-a + b*(c*x)^
n]/Sqrt[a]])/(3*n)

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Maple [A]  time = 0.005, size = 65, normalized size = 0.9 \[{\frac{2}{3\,n} \left ( -a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{3/2}}{n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }-2\,{\frac{a\sqrt{-a+b \left ( cx \right ) ^{n}}}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-a+b*(c*x)^n)^(3/2)/x,x)

[Out]

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

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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)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28255, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{-a} a \log \left (\frac{\left (c x\right )^{n} b + 2 \, \sqrt{\left (c x\right )^{n} b - a} \sqrt{-a} - 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt{\left (c x\right )^{n} b - a}{\left (\left (c x\right )^{n} b - 4 \, a\right )}}{3 \, n}, \frac{2 \,{\left (3 \, a^{\frac{3}{2}} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b - a}}{\sqrt{a}}\right ) + \sqrt{\left (c x\right )^{n} b - a}{\left (\left (c x\right )^{n} b - 4 \, a\right )}\right )}}{3 \, n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*sqrt(-a)*a*log(((c*x)^n*b + 2*sqrt((c*x)^n*b - a)*sqrt(-a) - 2*a)/(c*x)^
n) + 2*sqrt((c*x)^n*b - a)*((c*x)^n*b - 4*a))/n, 2/3*(3*a^(3/2)*arctan(sqrt((c*x
)^n*b - a)/sqrt(a)) + sqrt((c*x)^n*b - a)*((c*x)^n*b - 4*a))/n]

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Sympy [A]  time = 14.1454, size = 158, normalized size = 2.08 \[ - \begin{cases} \left (a \sqrt{- a + b} - b \sqrt{- a + b}\right ) \log{\left (x \right )} & \text{for}\: n = 0 \\\frac{- 2 a^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a < - a + b \left (c x\right )^{n} \wedge a < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > - a + b \left (c x\right )^{n} \wedge a < 0 \end{cases}\right ) + 2 a \sqrt{- a + b \left (c x\right )^{n}} - \frac{2 \left (- a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}{3}}{n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Piecewise(((a*sqrt(-a + b) - b*sqrt(-a + b))*log(x), Eq(n, 0)), ((-2*a**2*Piece
wise((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*sqrt(
-a + b*(c*x)**n) - 2*(-a + b*(c*x)**n)**(3/2)/3)/n, True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((c*x)^n*b - a)^(3/2)/x, x)