Integrand size = 26, antiderivative size = 65 \[ \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{2 \sqrt {3} \sqrt {c}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6 \sqrt {c}} \] Output:
1/6*arctan(1/3*(d*x^3+c)^(1/2)*3^(1/2)/c^(1/2))*3^(1/2)/c^(1/2)-1/6*arctan h((d*x^3+c)^(1/2)/c^(1/2))/c^(1/2)
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )-\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{6 \sqrt {c}} \] Input:
Integrate[Sqrt[c + d*x^3]/(x*(4*c + d*x^3)),x]
Output:
(Sqrt[3]*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])] - ArcTanh[Sqrt[c + d*x^ 3]/Sqrt[c]])/(6*Sqrt[c])
Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {948, 94, 73, 216, 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.
\(\displaystyle \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt {d x^3+c}}{x^3 \left (d x^3+4 c\right )}dx^3\) |
\(\Big \downarrow \) 94 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{4} \int \frac {1}{x^3 \sqrt {d x^3+c}}dx^3+\frac {3}{4} d \int \frac {1}{\sqrt {d x^3+c} \left (d x^3+4 c\right )}dx^3\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^6+3 c}d\sqrt {d x^3+c}+\frac {\int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{2 d}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {1}{\frac {x^6}{d}-\frac {c}{d}}d\sqrt {d x^3+c}}{2 d}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{2 \sqrt {c}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {3} \arctan \left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )\) |
Input:
Int[Sqrt[c + d*x^3]/(x*(4*c + d*x^3)),x]
Output:
((Sqrt[3]*ArcTan[Sqrt[c + d*x^3]/(Sqrt[3]*Sqrt[c])])/(2*Sqrt[c]) - ArcTanh [Sqrt[c + d*x^3]/Sqrt[c]]/(2*Sqrt[c]))/3
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]
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]
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])
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]
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]]
Time = 1.83 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (\frac {\sqrt {d \,x^{3}+c}\, \sqrt {3}}{3 \sqrt {c}}\right ) \sqrt {3}-\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{6 \sqrt {c}}\) | \(45\) |
default | \(\frac {\frac {2 \sqrt {d \,x^{3}+c}}{3}-\frac {2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right )}{3}}{4 c}-\frac {2 \sqrt {d \,x^{3}+c}-2 \sqrt {c}\, \sqrt {3}\, \arctan \left (\frac {\sqrt {d \,x^{3}+c}\, \sqrt {3}}{3 \sqrt {c}}\right )}{12 c}\) | \(81\) |
elliptic | \(\text {Expression too large to display}\) | \(1502\) |
Input:
int((d*x^3+c)^(1/2)/x/(d*x^3+4*c),x,method=_RETURNVERBOSE)
Output:
1/6*(arctan(1/3*(d*x^3+c)^(1/2)*3^(1/2)/c^(1/2))*3^(1/2)-arctanh((d*x^3+c) ^(1/2)/c^(1/2)))/c^(1/2)
Time = 0.13 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.22 \[ \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx=\left [-\frac {2 \, \sqrt {3} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {c}}{\sqrt {d x^{3} + c}}\right ) - \sqrt {c} \log \left (\frac {d x^{3} - 2 \, \sqrt {d x^{3} + c} \sqrt {c} + 2 \, c}{x^{3}}\right )}{12 \, c}, -\frac {\sqrt {3} \sqrt {-c} \log \left (\frac {d x^{3} - 2 \, \sqrt {3} \sqrt {d x^{3} + c} \sqrt {-c} - 2 \, c}{d x^{3} + 4 \, c}\right ) - 2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{3} + c}}\right )}{12 \, c}\right ] \] Input:
integrate((d*x^3+c)^(1/2)/x/(d*x^3+4*c),x, algorithm="fricas")
Output:
[-1/12*(2*sqrt(3)*sqrt(c)*arctan(sqrt(3)*sqrt(c)/sqrt(d*x^3 + c)) - sqrt(c )*log((d*x^3 - 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3))/c, -1/12*(sqrt(3)*sq rt(-c)*log((d*x^3 - 2*sqrt(3)*sqrt(d*x^3 + c)*sqrt(-c) - 2*c)/(d*x^3 + 4*c )) - 2*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^3 + c)))/c]
Time = 2.78 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx=\begin {cases} \frac {2 \left (\frac {d \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{\sqrt {- c}} \right )}}{12 \sqrt {- c}} + \frac {\sqrt {3} d \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {c + d x^{3}}}{3 \sqrt {c}} \right )}}{12 \sqrt {c}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\log {\left (x^{3} \right )}}{12 \sqrt {c}} & \text {otherwise} \end {cases} \] Input:
integrate((d*x**3+c)**(1/2)/x/(d*x**3+4*c),x)
Output:
Piecewise((2*(d*atan(sqrt(c + d*x**3)/sqrt(-c))/(12*sqrt(-c)) + sqrt(3)*d* atan(sqrt(3)*sqrt(c + d*x**3)/(3*sqrt(c)))/(12*sqrt(c)))/d, Ne(d, 0)), (lo g(x**3)/(12*sqrt(c)), True))
\[ \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{{\left (d x^{3} + 4 \, c\right )} x} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/x/(d*x^3+4*c),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)/((d*x^3 + 4*c)*x), x)
Time = 0.13 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right )}{6 \, \sqrt {c}} + \frac {\arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{6 \, \sqrt {-c}} \] Input:
integrate((d*x^3+c)^(1/2)/x/(d*x^3+4*c),x, algorithm="giac")
Output:
1/6*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(d*x^3 + c)/sqrt(c))/sqrt(c) + 1/6*arct an(sqrt(d*x^3 + c)/sqrt(-c))/sqrt(-c)
Time = 3.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx=\frac {\ln \left (\frac {{\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}^3\,\left (\sqrt {d\,x^3+c}+\sqrt {c}\right )}{x^6}\right )}{12\,\sqrt {c}}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}\,d\,x^3-2\,\sqrt {3}\,c+\sqrt {c}\,\sqrt {d\,x^3+c}\,6{}\mathrm {i}}{d\,x^3+4\,c}\right )\,1{}\mathrm {i}}{12\,\sqrt {c}} \] Input:
int((c + d*x^3)^(1/2)/(x*(4*c + d*x^3)),x)
Output:
log((((c + d*x^3)^(1/2) - c^(1/2))^3*((c + d*x^3)^(1/2) + c^(1/2)))/x^6)/( 12*c^(1/2)) + (3^(1/2)*log((c^(1/2)*(c + d*x^3)^(1/2)*6i - 2*3^(1/2)*c + 3 ^(1/2)*d*x^3)/(4*c + d*x^3))*1i)/(12*c^(1/2))
\[ \int \frac {\sqrt {c+d x^3}}{x \left (4 c+d x^3\right )} \, dx=\frac {\sqrt {c}\, \mathrm {log}\left (\sqrt {d \,x^{3}+c}-\sqrt {c}\right )-\sqrt {c}\, \mathrm {log}\left (\sqrt {d \,x^{3}+c}+\sqrt {c}\right )+9 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{2}}{d^{2} x^{6}+5 c d \,x^{3}+4 c^{2}}d x \right ) c d}{12 c} \] Input:
int((d*x^3+c)^(1/2)/x/(d*x^3+4*c),x)
Output:
(sqrt(c)*log(sqrt(c + d*x**3) - sqrt(c)) - sqrt(c)*log(sqrt(c + d*x**3) + sqrt(c)) + 9*int((sqrt(c + d*x**3)*x**2)/(4*c**2 + 5*c*d*x**3 + d**2*x**6) ,x)*c*d)/(12*c)