Integrand size = 24, antiderivative size = 199 \[ \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}+\frac {\arctan \left (\frac {\sqrt {c+d x}}{\sqrt {3} \sqrt {c}}\right )}{2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \] Output:
-1/6*arctan(3^(1/2)*c^(1/6)*(c^(1/3)+2^(1/3)*d^(1/3)*x^(1/3))/(d*x+c)^(1/2 ))*2^(1/3)*3^(1/2)/c^(5/6)/d^(2/3)+1/6*arctan(1/3*(d*x+c)^(1/2)*3^(1/2)/c^ (1/2))*2^(1/3)*3^(1/2)/c^(5/6)/d^(2/3)-1/2*arctanh(c^(1/6)*(c^(1/3)-2^(1/3 )*d^(1/3)*x^(1/3))/(d*x+c)^(1/2))*2^(1/3)/c^(5/6)/d^(2/3)+1/6*arctanh((d*x +c)^(1/2)/c^(1/2))*2^(1/3)/c^(5/6)/d^(2/3)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx=\frac {3 x^{2/3} \sqrt {\frac {c+d x}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x}{c},-\frac {d x}{4 c}\right )}{8 c \sqrt {c+d x}} \] Input:
Integrate[1/(x^(1/3)*Sqrt[c + d*x]*(4*c + d*x)),x]
Output:
(3*x^(2/3)*Sqrt[(c + d*x)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x)/c), -1/4*( d*x)/c])/(8*c*Sqrt[c + d*x])
Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {148, 986}
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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx\) |
| \(\Big \downarrow \) 148 |
| \(\displaystyle 3 \int \frac {\sqrt [3]{x}}{\sqrt {c+d x} (4 c+d x)}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 986 |
| \(\displaystyle 3 \left (-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{3\ 2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}+\frac {\arctan \left (\frac {\sqrt {c+d x}}{\sqrt {3} \sqrt {c}}\right )}{3\ 2^{2/3} \sqrt {3} c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{9\ 2^{2/3} c^{5/6} d^{2/3}}\right )\) |
Input:
Int[1/(x^(1/3)*Sqrt[c + d*x]*(4*c + d*x)),x]
Output:
3*(-1/3*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + 2^(1/3)*d^(1/3)*x^(1/3)))/Sqrt[ c + d*x]]/(2^(2/3)*Sqrt[3]*c^(5/6)*d^(2/3)) + ArcTan[Sqrt[c + d*x]/(Sqrt[3 ]*Sqrt[c])]/(3*2^(2/3)*Sqrt[3]*c^(5/6)*d^(2/3)) - ArcTanh[(c^(1/6)*(c^(1/3 ) - 2^(1/3)*d^(1/3)*x^(1/3)))/Sqrt[c + d*x]]/(3*2^(2/3)*c^(5/6)*d^(2/3)) + ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(9*2^(2/3)*c^(5/6)*d^(2/3)))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/b))^n*(e + f*(x^k/b))^p, x], x, (b*x)^(1/k)], x]] /; FreeQ[{b, c, d, e, f, n, p}, x] && FractionQ[m] && IntegerQ[p]
Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi th[{q = Rt[d/c, 3]}, Simp[q*(ArcTanh[Sqrt[c + d*x^3]/Rt[c, 2]]/(9*2^(2/3)*b *Rt[c, 2])), x] + (-Simp[q*(ArcTanh[Rt[c, 2]*((1 - 2^(1/3)*q*x)/Sqrt[c + d* x^3])]/(3*2^(2/3)*b*Rt[c, 2])), x] + Simp[q*(ArcTan[Sqrt[c + d*x^3]/(Sqrt[3 ]*Rt[c, 2])]/(3*2^(2/3)*Sqrt[3]*b*Rt[c, 2])), x] - Simp[q*(ArcTan[Sqrt[3]*R t[c, 2]*((1 + 2^(1/3)*q*x)/Sqrt[c + d*x^3])]/(3*2^(2/3)*Sqrt[3]*b*Rt[c, 2]) ), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[4*b*c - a*d, 0] && PosQ[c]
\[\int \frac {1}{x^{\frac {1}{3}} \sqrt {x d +c}\, \left (x d +4 c \right )}d x\]
Input:
int(1/x^(1/3)/(d*x+c)^(1/2)/(d*x+4*c),x)
Output:
int(1/x^(1/3)/(d*x+c)^(1/2)/(d*x+4*c),x)
Timed out. \[ \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx=\text {Timed out} \] Input:
integrate(1/x^(1/3)/(d*x+c)^(1/2)/(d*x+4*c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx=\int \frac {1}{\sqrt [3]{x} \sqrt {c + d x} \left (4 c + d x\right )}\, dx \] Input:
integrate(1/x**(1/3)/(d*x+c)**(1/2)/(d*x+4*c),x)
Output:
Integral(1/(x**(1/3)*sqrt(c + d*x)*(4*c + d*x)), x)
\[ \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx=\int { \frac {1}{{\left (d x + 4 \, c\right )} \sqrt {d x + c} x^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/x^(1/3)/(d*x+c)^(1/2)/(d*x+4*c),x, algorithm="maxima")
Output:
integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)), x)
\[ \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx=\int { \frac {1}{{\left (d x + 4 \, c\right )} \sqrt {d x + c} x^{\frac {1}{3}}} \,d x } \] Input:
integrate(1/x^(1/3)/(d*x+c)^(1/2)/(d*x+4*c),x, algorithm="giac")
Output:
integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx=\int \frac {1}{x^{1/3}\,\left (4\,c+d\,x\right )\,\sqrt {c+d\,x}} \,d x \] Input:
int(1/(x^(1/3)*(4*c + d*x)*(c + d*x)^(1/2)),x)
Output:
int(1/(x^(1/3)*(4*c + d*x)*(c + d*x)^(1/2)), x)
\[ \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx=\int \frac {\sqrt {d x +c}}{4 x^{\frac {1}{3}} c^{2}+5 x^{\frac {4}{3}} c d +x^{\frac {7}{3}} d^{2}}d x \] Input:
int(1/x^(1/3)/(d*x+c)^(1/2)/(d*x+4*c),x)
Output:
int(sqrt(c + d*x)/(4*x**(1/3)*c**2 + 5*x**(1/3)*c*d*x + x**(1/3)*d**2*x**2 ),x)