3.30.72 \(\int \frac {\sqrt {a+b (c x^3)^{3/2}}}{x} \, dx\) [2972]

3.30.72.1 Optimal result

Integrand size = 21, antiderivative size = 55 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x} \, dx=\frac {4}{9} \sqrt {a+b \left (c x^3\right )^{3/2}}-\frac {4}{9} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right ) \]

output

-4/9*arctanh((a+b*(c*x^3)^(3/2))^(1/2)/a^(1/2))*a^(1/2)+4/9*(a+b*(c*x^3)^( 
3/2))^(1/2)
 

3.30.72.2 Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x} \, dx=\frac {4}{9} \left (\sqrt {a+b \left (c x^3\right )^{3/2}}-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right )\right ) \]

input

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

output

(4*(Sqrt[a + b*(c*x^3)^(3/2)] - Sqrt[a]*ArcTanh[Sqrt[a + b*(c*x^3)^(3/2)]/ 
Sqrt[a]]))/9
 

3.30.72.3 Rubi [A] (warning: unable to verify)

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {893, 798, 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.

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

\(\Big \downarrow \) 893

\(\displaystyle \int \frac {\sqrt {a+b c^{3/2} x^{9/2}}}{x}dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {2}{9} \int \frac {\sqrt {b c^{3/2} x^{9/2}+a}}{x^{9/2}}dx^{9/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2}{9} \left (a \int \frac {1}{x^{9/2} \sqrt {b c^{3/2} x^{9/2}+a}}dx^{9/2}+2 \sqrt {a+b c^{3/2} x^{9/2}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2}{9} \left (\frac {2 a \int \frac {1}{\frac {x^9}{b c^{3/2}}-\frac {a}{b c^{3/2}}}d\sqrt {b c^{3/2} x^{9/2}+a}}{b c^{3/2}}+2 \sqrt {a+b c^{3/2} x^{9/2}}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2}{9} \left (2 \sqrt {a+b c^{3/2} x^{9/2}}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{\sqrt {a}}\right )\right )\)

input

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

output

(2*(2*Sqrt[a + b*c^(3/2)*x^(9/2)] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*c^(3/2)*x 
^(9/2)]/Sqrt[a]]))/9
 

3.30.72.3.1 Defintions of rubi rules used

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 893

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x 
], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, 
d, m, p, q}, x] && FractionQ[n]
 

3.30.72.4 Maple [F]

\[\int \frac {\sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{x}d x\]

input

int((a+b*(c*x^3)^(3/2))^(1/2)/x,x)
 

output

int((a+b*(c*x^3)^(3/2))^(1/2)/x,x)
 

3.30.72.5 Fricas [F(-1)]

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

input

integrate((a+b*(c*x^3)^(3/2))^(1/2)/x,x, algorithm="fricas")
 

output

Timed out
 

3.30.72.6 Sympy [F]

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

input

integrate((a+b*(c*x**3)**(3/2))**(1/2)/x,x)
 

output

Integral(sqrt(a + b*(c*x**3)**(3/2))/x, x)
 

3.30.72.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x} \, dx=\frac {2}{9} \, \sqrt {a} \log \left (\frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} - \sqrt {a}}{\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} + \sqrt {a}}\right ) + \frac {4}{9} \, \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} \]

input

integrate((a+b*(c*x^3)^(3/2))^(1/2)/x,x, algorithm="maxima")
 

output

2/9*sqrt(a)*log((sqrt((c*x^3)^(3/2)*b + a) - sqrt(a))/(sqrt((c*x^3)^(3/2)* 
b + a) + sqrt(a))) + 4/9*sqrt((c*x^3)^(3/2)*b + a)
 

3.30.72.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x} \, dx=\frac {4 \, {\left (\frac {a c \arctan \left (\frac {\sqrt {\sqrt {c x} b c^{5} x^{4} + a c^{4}}}{\sqrt {-a} c^{2}}\right )}{\sqrt {-a}} + \frac {\sqrt {\sqrt {c x} b c^{5} x^{4} + a c^{4}}}{c}\right )} {\left | c \right |}^{2}}{9 \, c^{3}} \]

input

integrate((a+b*(c*x^3)^(3/2))^(1/2)/x,x, algorithm="giac")
 

output

4/9*(a*c*arctan(sqrt(sqrt(c*x)*b*c^5*x^4 + a*c^4)/(sqrt(-a)*c^2))/sqrt(-a) 
 + sqrt(sqrt(c*x)*b*c^5*x^4 + a*c^4)/c)*abs(c)^2/c^3
 

3.30.72.9 Mupad [F(-1)]

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

input

int((a + b*(c*x^3)^(3/2))^(1/2)/x,x)
 

output

int((a + b*(c*x^3)^(3/2))^(1/2)/x, x)