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

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

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Rubi [A]  time = 0.0570967, 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, Rules used = {367, 12, 266, 50, 63, 205} \[ \frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n}-\frac{2 a \sqrt{b (c x)^n-a}}{n}+\frac{2 \left (b (c x)^n-a\right )^{3/2}}{3 n} \]

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

Rule 367

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Dist[1/c, Subst[Int[((d*x)/c)^m*(a
+ b*x^n)^p, x], x, c*x], x] /; FreeQ[{a, b, c, d, m, n, p}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{c \left (-a+b x^n\right )^{3/2}}{x} \, dx,x,c x\right )}{c}\\ &=\operatorname{Subst}\left (\int \frac{\left (-a+b x^n\right )^{3/2}}{x} \, dx,x,c x\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-a+b x)^{3/2}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac{2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{-a+b x}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=-\frac{2 a \sqrt{-a+b (c x)^n}}{n}+\frac{2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a+b x}} \, dx,x,(c x)^n\right )}{n}\\ &=-\frac{2 a \sqrt{-a+b (c x)^n}}{n}+\frac{2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b (c x)^n}\right )}{b n}\\ &=-\frac{2 a \sqrt{-a+b (c x)^n}}{n}+\frac{2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{-a+b (c x)^n}}{\sqrt{a}}\right )}{n}\\ \end{align*}

Mathematica [A]  time = 0.0482343, 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 = 0.004, size = 65, normalized size = 0.9 \begin{align*}{\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}} \end{align*}

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. \begin{align*} \int \frac{{\left (\left (c x\right )^{n} b - a\right )}^{\frac{3}{2}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b - a)^(3/2)/x, x)

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Fricas [A]  time = 1.57029, size = 306, normalized size = 4.03 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a+b*(c*x)^n)^(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 = 44.4081, size = 97, normalized size = 1.28 \begin{align*} - \begin{cases} \left (a \sqrt{- a + b} - b \sqrt{- a + b}\right ) \log{\left (x \right )} & \text{for}\: n = 0 \\\frac{- a \left (2 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )} - 2 \sqrt{- a + b \left (c x\right )^{n}}\right ) + b \left (\begin{cases} - \sqrt{- a} \left (c x\right )^{n} & \text{for}\: b = 0 \\- \frac{2 \left (- a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right )}{n} & \text{otherwise} \end{cases} \end{align*}

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)), ((-a*(2*sqrt(a)*atan(sqrt(-a + b*(c*x)**n)/sq
rt(a)) - 2*sqrt(-a + b*(c*x)**n)) + b*Piecewise((-sqrt(-a)*(c*x)**n, Eq(b, 0)), (-2*(-a + b*(c*x)**n)**(3/2)/(
3*b), True)))/n, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\left (c x\right )^{n} b - a\right )}^{\frac{3}{2}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b - a)^(3/2)/x, x)