3.3.53 \(\int \frac {(c \sqrt {a+b x^2})^{3/2}}{x} \, dx\) [253]

3.3.53.1 Optimal result

Integrand size = 21, antiderivative size = 117 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\frac {2}{3} \left (c \sqrt {a+b x^2}\right )^{3/2}+\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \arctan \left (\sqrt [4]{1+\frac {b x^2}{a}}\right )}{\left (1+\frac {b x^2}{a}\right )^{3/4}}-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \text {arctanh}\left (\sqrt [4]{1+\frac {b x^2}{a}}\right )}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \]

output

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

3.3.53.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.82 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (2 \left (a+b x^2\right )^{3/4}+3 a^{3/4} \arctan \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )-3 a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right )}{3 \left (a+b x^2\right )^{3/4}} \]

input

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

output

((c*Sqrt[a + b*x^2])^(3/2)*(2*(a + b*x^2)^(3/4) + 3*a^(3/4)*ArcTan[(a + b* 
x^2)^(1/4)/a^(1/4)] - 3*a^(3/4)*ArcTanh[(a + b*x^2)^(1/4)/a^(1/4)]))/(3*(a 
 + b*x^2)^(3/4))
 

3.3.53.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.81, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2045, 243, 60, 73, 25, 27, 827, 216, 219}

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.

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

\(\Big \downarrow \) 2045

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \int \frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x}dx}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \int \frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^2}dx^2}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\int \frac {1}{x^2 \sqrt [4]{\frac {b x^2}{a}+1}}dx^2+\frac {4}{3} \left (\frac {b x^2}{a}+1\right )^{3/4}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {4 a \int -\frac {b x^4}{a \left (1-x^8\right )}d\sqrt [4]{\frac {b x^2}{a}+1}}{b}+\frac {4}{3} \left (\frac {b x^2}{a}+1\right )^{3/4}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {4}{3} \left (\frac {b x^2}{a}+1\right )^{3/4}-\frac {4 a \int \frac {b x^4}{a \left (1-x^8\right )}d\sqrt [4]{\frac {b x^2}{a}+1}}{b}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {4}{3} \left (\frac {b x^2}{a}+1\right )^{3/4}-4 \int \frac {x^4}{1-x^8}d\sqrt [4]{\frac {b x^2}{a}+1}\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {4}{3} \left (\frac {b x^2}{a}+1\right )^{3/4}-4 \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{\frac {b x^2}{a}+1}-\frac {1}{2} \int \frac {1}{x^4+1}d\sqrt [4]{\frac {b x^2}{a}+1}\right )\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {4}{3} \left (\frac {b x^2}{a}+1\right )^{3/4}-4 \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{\frac {b x^2}{a}+1}-\frac {1}{2} \arctan \left (\sqrt [4]{\frac {b x^2}{a}+1}\right )\right )\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {4}{3} \left (\frac {b x^2}{a}+1\right )^{3/4}-4 \left (\frac {1}{2} \text {arctanh}\left (\sqrt [4]{\frac {b x^2}{a}+1}\right )-\frac {1}{2} \arctan \left (\sqrt [4]{\frac {b x^2}{a}+1}\right )\right )\right )}{2 \left (\frac {b x^2}{a}+1\right )^{3/4}}\)

input

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

output

((c*Sqrt[a + b*x^2])^(3/2)*((4*(1 + (b*x^2)/a)^(3/4))/3 - 4*(-1/2*ArcTan[( 
1 + (b*x^2)/a)^(1/4)] + ArcTanh[(1 + (b*x^2)/a)^(1/4)]/2)))/(2*(1 + (b*x^2 
)/a)^(3/4))
 

3.3.53.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 2045

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si 
mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)]   Int[u*(1 + b*(x^n/a))^(p*q) 
, x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]
 

3.3.53.4 Maple [F]

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

input

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

output

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

3.3.53.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.3.53.6 Sympy [F]

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

input

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

output

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

3.3.53.7 Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=\frac {3 \, a c^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} c}}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}} + \frac {\log \left (\frac {\sqrt {\sqrt {b x^{2} + a} c} - \left (a c^{2}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {b x^{2} + a} c} + \left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )} + 4 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} c^{2}}{6 \, c^{2}} \]

input

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

output

1/6*(3*a*c^4*(2*arctan(sqrt(sqrt(b*x^2 + a)*c)/(a*c^2)^(1/4))/(a*c^2)^(1/4 
) + log((sqrt(sqrt(b*x^2 + a)*c) - (a*c^2)^(1/4))/(sqrt(sqrt(b*x^2 + a)*c) 
 + (a*c^2)^(1/4)))/(a*c^2)^(1/4)) + 4*(sqrt(b*x^2 + a)*c)^(3/2)*c^2)/c^2
 

3.3.53.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.62 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x} \, dx=-\frac {1}{12} \, {\left (6 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + 6 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - 3 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right ) + 3 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right ) - 8 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}}\right )} c^{\frac {3}{2}} \]

input

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

output

-1/12*(6*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b* 
x^2 + a)^(1/4))/(-a)^(1/4)) + 6*sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sq 
rt(2)*(-a)^(1/4) - 2*(b*x^2 + a)^(1/4))/(-a)^(1/4)) - 3*sqrt(2)*(-a)^(3/4) 
*log(sqrt(2)*(b*x^2 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^2 + a) + sqrt(-a)) + 
3*sqrt(2)*(-a)^(3/4)*log(-sqrt(2)*(b*x^2 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^ 
2 + a) + sqrt(-a)) - 8*(b*x^2 + a)^(3/4))*c^(3/2)
 

3.3.53.9 Mupad [F(-1)]

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

input

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

output

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