\(\int \frac {\sqrt {c+d x^3}}{x (a+b x^3)} \, dx\) [534]

Optimal result

Integrand size = 24, antiderivative size = 85 \[ \int \frac {\sqrt {c+d x^3}}{x \left (a+b x^3\right )} \, dx=-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a}+\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a \sqrt {b}} \] Output:

-2/3*c^(1/2)*arctanh((d*x^3+c)^(1/2)/c^(1/2))/a+2/3*(-a*d+b*c)^(1/2)*arcta 
nh(b^(1/2)*(d*x^3+c)^(1/2)/(-a*d+b*c)^(1/2))/a/b^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x^3}}{x \left (a+b x^3\right )} \, dx=\frac {2 \left (\frac {\sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{\sqrt {b}}-\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )\right )}{3 a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {948, 94, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 94

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{3} \left (\frac {2 c \int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{a d}-\frac {2 (b c-a d) \int \frac {1}{\frac {b x^6}{d}+a-\frac {b c}{d}}d\sqrt {d x^3+c}}{a d}\right )\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 94

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[(e + f*x)^(p - 1)/(a + b*x), x], 
x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[(e + f*x)^(p - 1)/(c + d*x), x], x] 
 /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]
 

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 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {-\frac {2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3}+\frac {2 \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 \sqrt {\left (a d -b c \right ) b}}}{a}\)	\(71\)
default	\(\frac {\frac {2 \sqrt {d \,x^{3}+c}}{3}-\frac {2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3}}{a}-\frac {2 \left (\sqrt {d \,x^{3}+c}-\frac {\left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 a}\)	\(98\)
elliptic	\(\text {Expression too large to display}\)	\(1543\)

Input:

int((d*x^3+c)^(1/2)/x/(b*x^3+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 377, normalized size of antiderivative = 4.44 \[ \int \frac {\sqrt {c+d x^3}}{x \left (a+b x^3\right )} \, dx=\left [\frac {\sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right ) + \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right )}{3 \, a}, \frac {2 \, \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right )}{3 \, a}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) + \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{3} + a}\right )}{3 \, a}, \frac {2 \, {\left (\sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{3} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{3} + c}}\right )\right )}}{3 \, a}\right ] \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (73) = 146\).

Time = 5.38 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.94 \[ \int \frac {\sqrt {c+d x^3}}{x \left (a+b x^3\right )} \, dx=\begin {cases} \frac {2 \left (\frac {c d \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{\sqrt {- c}} \right )}}{3 a \sqrt {- c}} + \frac {d \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{3 a b \sqrt {\frac {a d - b c}{b}}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (- \frac {2 b \left (\begin {cases} \frac {\frac {a}{2 b} + x^{3}}{a} & \text {for}\: b = 0 \\- \frac {\log {\left (a - 2 b \left (\frac {a}{2 b} + x^{3}\right ) \right )}}{2 b} & \text {otherwise} \end {cases}\right )}{3 a} - \frac {2 b \left (\begin {cases} \frac {\frac {a}{2 b} + x^{3}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + 2 b \left (\frac {a}{2 b} + x^{3}\right ) \right )}}{2 b} & \text {otherwise} \end {cases}\right )}{3 a}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

integrate(sqrt(d*x^3 + c)/((b*x^3 + a)*x), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x^3}}{x \left (a+b x^3\right )} \, dx=-\frac {2 \, {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} a} + \frac {2 \, c \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a \sqrt {-c}} \] Input:

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

Output:

-2/3*(b*c - a*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2 
*c + a*b*d)*a) + 2/3*c*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(a*sqrt(-c))
 

Mupad [B] (verification not implemented)

Time = 6.70 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {c+d x^3}}{x \left (a+b x^3\right )} \, dx=\frac {\sqrt {c}\,\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )}{3\,a}+\frac {\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\sqrt {a\,d-b\,c}\,1{}\mathrm {i}}{3\,a\,\sqrt {b}} \] Input:

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

Output:

(c^(1/2)*log((((c + d*x^3)^(1/2) - c^(1/2))^3*((c + d*x^3)^(1/2) + c^(1/2) 
))/x^6))/(3*a) + (log((2*b*c - a*d + b^(1/2)*(c + d*x^3)^(1/2)*(a*d - b*c) 
^(1/2)*2i + b*d*x^3)/(a + b*x^3))*(a*d - b*c)^(1/2)*1i)/(3*a*b^(1/2))
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x^3}}{x \left (a+b x^3\right )} \, dx=\int \frac {\sqrt {d \,x^{3}+c}}{b \,x^{4}+a x}d x \] Input:

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

Output:

int(sqrt(c + d*x**3)/(a*x + b*x**4),x)