3.5.64 \(\int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx\) [464]

3.5.64.1 Optimal result

Integrand size = 25, antiderivative size = 143 \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{2 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{6 c^{5/6} d^{2/3}} \]

output

1/6*arctanh(1/3*(c^(1/3)+d^(1/3)*x^(1/3))^2/c^(1/6)/(d*x+c)^(1/2))/c^(5/6) 
/d^(2/3)-1/6*arctanh(1/3*(d*x+c)^(1/2)/c^(1/2))/c^(5/6)/d^(2/3)-1/6*arctan 
(c^(1/6)*(c^(1/3)+d^(1/3)*x^(1/3))*3^(1/2)/(d*x+c)^(1/2))/c^(5/6)/d^(2/3)* 
3^(1/2)
 

3.5.64.2 Mathematica [C] (verified)

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 = 0.43 \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\frac {3 x^{2/3} \sqrt {\frac {c+d x}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x}{c},\frac {d x}{8 c}\right )}{16 c \sqrt {c+d x}} \]

input

Integrate[1/(x^(1/3)*(8*c - d*x)*Sqrt[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), (d*x)/ 
(8*c)])/(16*c*Sqrt[c + d*x])
 

3.5.64.3 Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {148, 988, 946, 73, 219, 2563, 219, 2570, 218}

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.

\(\displaystyle \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 148

\(\displaystyle 3 \int \frac {\sqrt [3]{x}}{(8 c-d x) \sqrt {c+d x}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 988

\(\displaystyle 3 \left (-\frac {\int \frac {-\frac {x^{2/3} d^{4/3}}{\sqrt [3]{c}}-2 \sqrt [3]{x} d+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^{2/3}}{c^{2/3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}+4\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c d}+\frac {\int \frac {\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} \sqrt [3]{x}\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{d} \int \frac {x^{2/3}}{(8 c-d x) \sqrt {c+d x}}d\sqrt [3]{x}}{4 \sqrt [3]{c}}\right )\)

\(\Big \downarrow \) 946

\(\displaystyle 3 \left (-\frac {\int \frac {-\frac {x^{2/3} d^{4/3}}{\sqrt [3]{c}}-2 \sqrt [3]{x} d+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^{2/3}}{c^{2/3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}+4\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c d}+\frac {\int \frac {\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} \sqrt [3]{x}\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{d} \int \frac {1}{(8 c-d x) \sqrt {c+d x}}dx}{12 \sqrt [3]{c}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle 3 \left (-\frac {\int \frac {-\frac {x^{2/3} d^{4/3}}{\sqrt [3]{c}}-2 \sqrt [3]{x} d+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^{2/3}}{c^{2/3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}+4\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c d}+\frac {\int \frac {\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} \sqrt [3]{x}\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c^{2/3} \sqrt [3]{d}}-\frac {\int \frac {1}{9 c-\sqrt [3]{c+d x}}d\sqrt {c+d x}}{6 \sqrt [3]{c} d^{2/3}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 3 \left (-\frac {\int \frac {-\frac {x^{2/3} d^{4/3}}{\sqrt [3]{c}}-2 \sqrt [3]{x} d+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^{2/3}}{c^{2/3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}+4\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c d}+\frac {\int \frac {\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}}{\left (2 \sqrt [3]{c}-\sqrt [3]{d} \sqrt [3]{x}\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c^{2/3} \sqrt [3]{d}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )\)

\(\Big \downarrow \) 2563

\(\displaystyle 3 \left (\frac {\int \frac {1}{9-c x^{2/3}}d\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{c^{2/3} \sqrt {c+d x}}}{6 \sqrt [3]{c} d^{2/3}}-\frac {\int \frac {-\frac {x^{2/3} d^{4/3}}{\sqrt [3]{c}}-2 \sqrt [3]{x} d+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^{2/3}}{c^{2/3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}+4\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c d}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 3 \left (-\frac {\int \frac {-\frac {x^{2/3} d^{4/3}}{\sqrt [3]{c}}-2 \sqrt [3]{x} d+2 \sqrt [3]{c} d^{2/3}}{\left (\frac {d^{2/3} x^{2/3}}{c^{2/3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}+4\right ) \sqrt {c+d x}}d\sqrt [3]{x}}{12 c d}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )\)

\(\Big \downarrow \) 2570

\(\displaystyle 3 \left (\frac {d^{4/3} \int \frac {1}{-6 x^{2/3} d^2-\frac {2 d^2}{c}}d\frac {\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c} \sqrt {c+d x}}}{3 c^{4/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle 3 \left (-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt {c+d x}}\right )}{6 \sqrt {3} c^{5/6} d^{2/3}}+\frac {\text {arctanh}\left (\frac {\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt {c+d x}}\right )}{18 c^{5/6} d^{2/3}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{3 \sqrt {c}}\right )}{18 c^{5/6} d^{2/3}}\right )\)

input

Int[1/(x^(1/3)*(8*c - d*x)*Sqrt[c + d*x]),x]
 

output

3*(-1/6*ArcTan[(Sqrt[3]*c^(1/6)*(c^(1/3) + d^(1/3)*x^(1/3)))/Sqrt[c + d*x] 
]/(Sqrt[3]*c^(5/6)*d^(2/3)) + ArcTanh[(c^(1/3) + d^(1/3)*x^(1/3))^2/(3*c^( 
1/6)*Sqrt[c + d*x])]/(18*c^(5/6)*d^(2/3)) - ArcTanh[Sqrt[c + d*x]/(3*Sqrt[ 
c])]/(18*c^(5/6)*d^(2/3)))
 

3.5.64.3.1 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 148

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]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

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 946

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[(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] && EqQ[m - n 
+ 1, 0]
 

rule 988

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> Wi 
th[{q = Rt[d/c, 3]}, Simp[d*(q/(4*b))   Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x 
^3]), x], x] + (-Simp[q^2/(12*b)   Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x^3] 
), x], x] + Simp[1/(12*b*c)   Int[(2*c*q^2 - 2*d*x - d*q*x^2)/((4 + 2*q*x + 
 q^2*x^2)*Sqrt[c + d*x^3]), x], x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && EqQ[8*b*c + a*d, 0]
 

rule 2563

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> Simp[-2*(e/d)   Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ 
Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & 
& EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
 

rule 2570

Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)* 
Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Simp[-2*g*h   Subst[Int[1/(2*e*h - 
 (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8 
*a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]
 

3.5.64.4 Maple [F]

\[\int \frac {1}{x^{\frac {1}{3}} \left (-d x +8 c \right ) \sqrt {d x +c}}d x\]

input

int(1/x^(1/3)/(-d*x+8*c)/(d*x+c)^(1/2),x)
 

output

int(1/x^(1/3)/(-d*x+8*c)/(d*x+c)^(1/2),x)
 

3.5.64.5 Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\text {Timed out} \]

input

integrate(1/x^(1/3)/(-d*x+8*c)/(d*x+c)^(1/2),x, algorithm="fricas")
 

output

Timed out
 

3.5.64.6 Sympy [F]

\[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=- \int \frac {1}{- 8 c \sqrt [3]{x} \sqrt {c + d x} + d x^{\frac {4}{3}} \sqrt {c + d x}}\, dx \]

input

integrate(1/x**(1/3)/(-d*x+8*c)/(d*x+c)**(1/2),x)
 

output

-Integral(1/(-8*c*x**(1/3)*sqrt(c + d*x) + d*x**(4/3)*sqrt(c + d*x)), x)
 

3.5.64.7 Maxima [F]

\[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\int { -\frac {1}{\sqrt {d x + c} {\left (d x - 8 \, c\right )} x^{\frac {1}{3}}} \,d x } \]

input

integrate(1/x^(1/3)/(-d*x+8*c)/(d*x+c)^(1/2),x, algorithm="maxima")
 

output

-integrate(1/(sqrt(d*x + c)*(d*x - 8*c)*x^(1/3)), x)
 

3.5.64.8 Giac [F]

\[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\int { -\frac {1}{\sqrt {d x + c} {\left (d x - 8 \, c\right )} x^{\frac {1}{3}}} \,d x } \]

input

integrate(1/x^(1/3)/(-d*x+8*c)/(d*x+c)^(1/2),x, algorithm="giac")
 

output

integrate(-1/(sqrt(d*x + c)*(d*x - 8*c)*x^(1/3)), x)
 

3.5.64.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{x} (8 c-d x) \sqrt {c+d x}} \, dx=\int \frac {1}{x^{1/3}\,\left (8\,c-d\,x\right )\,\sqrt {c+d\,x}} \,d x \]

input

int(1/(x^(1/3)*(8*c - d*x)*(c + d*x)^(1/2)),x)
 

output

int(1/(x^(1/3)*(8*c - d*x)*(c + d*x)^(1/2)), x)