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

Optimal result

Integrand size = 17, antiderivative size = 70 \[ \int \frac {\left (a+b (c x)^n\right )^{3/2}}{x} \, dx=\frac {2 a \sqrt {a+b (c x)^n}}{n}+\frac {2 \left (a+b (c x)^n\right )^{3/2}}{3 n}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{n} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b (c x)^n\right )^{3/2}}{x} \, dx=\frac {2 \sqrt {a+b (c x)^n} \left (4 a+b (c x)^n\right )-6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{3 n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {891, 27, 798, 60, 60, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 891

\(\displaystyle \frac {\int \frac {\left (b (c x)^n+a\right )^{3/2}}{x}d(c x)}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {\left (a+b (c x)^n\right )^{3/2}}{c x}d(c x)\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\int \frac {\left (b (c x)^n+a\right )^{3/2}}{c x}d(c x)^n}{n}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {a \int \frac {\sqrt {b (c x)^n+a}}{c x}d(c x)^n+\frac {2}{3} \left (a+b (c x)^n\right )^{3/2}}{n}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {a \left (a \int \frac {1}{c x \sqrt {b (c x)^n+a}}d(c x)^n+2 \sqrt {a+b (c x)^n}\right )+\frac {2}{3} \left (a+b (c x)^n\right )^{3/2}}{n}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {a \left (\frac {2 a \int \frac {1}{\frac {c^2 x^2}{b}-\frac {a}{b}}d\sqrt {b (c x)^n+a}}{b}+2 \sqrt {a+b (c x)^n}\right )+\frac {2}{3} \left (a+b (c x)^n\right )^{3/2}}{n}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {a \left (2 \sqrt {a+b (c x)^n}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+b (c x)^n\right )^{3/2}}{n}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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 891

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> 
Simp[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]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {\frac {2 \left (a +b \left (c x \right )^{n}\right )^{\frac {3}{2}}}{3}+2 \sqrt {a +b \left (c x \right )^{n}}\, a -2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\)	\(54\)
default	\(\frac {\frac {2 \left (a +b \left (c x \right )^{n}\right )^{\frac {3}{2}}}{3}+2 \sqrt {a +b \left (c x \right )^{n}}\, a -2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\)	\(54\)
risch	\(\frac {2 \left (b \,{\mathrm e}^{n \ln \left (c x \right )}+4 a \right ) \sqrt {a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}{3 n}-\frac {2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}{\sqrt {a}}\right )}{n}\)	\(59\)

Input:

int((a+b*(c*x)^n)^(3/2)/x,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b (c x)^n\right )^{3/2}}{x} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b + a} \sqrt {a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, {\left (\left (c x\right )^{n} b + 4 \, a\right )} \sqrt {\left (c x\right )^{n} b + a}}{3 \, n}, \frac {2 \, {\left (3 \, \sqrt {-a} a \arctan \left (\frac {\sqrt {-a}}{\sqrt {\left (c x\right )^{n} b + a}}\right ) + {\left (\left (c x\right )^{n} b + 4 \, a\right )} \sqrt {\left (c x\right )^{n} b + a}\right )}}{3 \, n}\right ] \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 16.32 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b (c x)^n\right )^{3/2}}{x} \, dx=\begin {cases} \frac {\begin {cases} \frac {2 a^{2} \operatorname {atan}{\left (\frac {\sqrt {a + b \left (c x\right )^{n}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 a \sqrt {a + b \left (c x\right )^{n}} + \frac {2 \left (a + b \left (c x\right )^{n}\right )^{\frac {3}{2}}}{3} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \log {\left (\left (c x\right )^{n} \right )} & \text {otherwise} \end {cases}}{n} & \text {for}\: n \neq 0 \\\left (a \sqrt {a + b} + b \sqrt {a + b}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \] Input:

integrate((a+b*(c*x)**n)**(3/2)/x,x)
 

Output:

Piecewise((Piecewise((2*a**2*atan(sqrt(a + b*(c*x)**n)/sqrt(-a))/sqrt(-a) 
+ 2*a*sqrt(a + b*(c*x)**n) + 2*(a + b*(c*x)**n)**(3/2)/3, Ne(b, 0)), (a**( 
3/2)*log((c*x)**n), True))/n, Ne(n, 0)), ((a*sqrt(a + b) + b*sqrt(a + b))* 
log(x), True))
 

Maxima [F]

\[ \int \frac {\left (a+b (c x)^n\right )^{3/2}}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b + a\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:

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

Output:

integrate(((c*x)^n*b + a)^(3/2)/x, x)
 

Giac [F]

\[ \int \frac {\left (a+b (c x)^n\right )^{3/2}}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b + a\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:

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

Output:

integrate(((c*x)^n*b + a)^(3/2)/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b (c x)^n\right )^{3/2}}{x} \, dx=\int \frac {{\left (a+b\,{\left (c\,x\right )}^n\right )}^{3/2}}{x} \,d x \] Input:

int((a + b*(c*x)^n)^(3/2)/x,x)
 

Output:

int((a + b*(c*x)^n)^(3/2)/x, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\left (a+b (c x)^n\right )^{3/2}}{x} \, dx=\frac {2 x^{n} c^{n} \sqrt {x^{n} c^{n} b +a}\, b +8 \sqrt {x^{n} c^{n} b +a}\, a +3 \left (\int \frac {\sqrt {x^{n} c^{n} b +a}}{x^{n} c^{n} b x +a x}d x \right ) a^{2} n}{3 n} \] Input:

int((a+b*(c*x)^n)^(3/2)/x,x)
 

Output:

(2*x**n*c**n*sqrt(x**n*c**n*b + a)*b + 8*sqrt(x**n*c**n*b + a)*a + 3*int(s 
qrt(x**n*c**n*b + a)/(x**n*c**n*b*x + a*x),x)*a**2*n)/(3*n)