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

Optimal. Leaf size=143 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{3 \sqrt{c}}\right )}{6 c^{5/6} d^{2/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.462651, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {130, 486, 444, 63, 206, 2138, 2145, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} \sqrt [3]{x}\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{3 \sqrt{c}}\right )}{6 c^{5/6} d^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 130

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + (b*x^k)/e)^m*(c + (d*x^k)/e)^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 486

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[d/c, 3]}, Dist[(d*q)/(4*
b), Int[x^2/((8*c - d*x^3)*Sqrt[c + d*x^3]), x], x] + (-Dist[q^2/(12*b), Int[(1 + q*x)/((2 - q*x)*Sqrt[c + d*x
^3]), x], x] + Dist[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 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[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 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

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

Rule 2138

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(-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 2145

Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbo
l] :> Dist[-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] /;
FreeQ[{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]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{x} (8 c-d x) \sqrt{c+d x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{2 \sqrt [3]{c} d^{2/3}-2 d x-\frac{d^{4/3} x^2}{\sqrt [3]{c}}}{\left (4+\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}\right ) \sqrt{c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 c d}+\frac{\operatorname{Subst}\left (\int \frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (2-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 c^{2/3} \sqrt [3]{d}}-\frac{\left (3 \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}} \, dx,x,\sqrt [3]{x}\right )}{4 \sqrt [3]{c}}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{9-c x^2} \, dx,x,\frac{\left (1+\frac{\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{\sqrt{c+d x}}\right )}{2 \sqrt [3]{c} d^{2/3}}-\frac{\sqrt [3]{d} \operatorname{Subst}\left (\int \frac{1}{(8 c-d x) \sqrt{c+d x}} \, dx,x,x\right )}{4 \sqrt [3]{c}}+\frac{d^{4/3} \operatorname{Subst}\left (\int \frac{1}{-\frac{2 d^2}{c}-6 d^2 x^2} \, dx,x,\frac{1+\frac{\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}}{\sqrt{c+d x}}\right )}{c^{4/3}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} \left (1+\frac{\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )}{\sqrt{c+d x}}\right )}{2 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \left (1+\frac{\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{3 \sqrt{c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{9 c-x^2} \, dx,x,\sqrt{c+d x}\right )}{2 \sqrt [3]{c} d^{2/3}}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} \left (1+\frac{\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )}{\sqrt{c+d x}}\right )}{2 \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \left (1+\frac{\sqrt [3]{d} \sqrt [3]{x}}{\sqrt [3]{c}}\right )^2}{3 \sqrt{c+d x}}\right )}{6 c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{3 \sqrt{c}}\right )}{6 c^{5/6} d^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0372369, size = 61, normalized size = 0.43 \[ \frac{3 x^{2/3} \sqrt{\frac{c+d x}{c}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},\frac{d x}{8 c}\right )}{16 c \sqrt{c+d x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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])

________________________________________________________________________________________

Maple [F]  time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-dx+8\,c}{\frac{1}{\sqrt [3]{x}}}{\frac{1}{\sqrt{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{\sqrt{d x + c}{\left (d x - 8 \, c\right )} x^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- 8 c \sqrt [3]{x} \sqrt{c + d x} + d x^{\frac{4}{3}} \sqrt{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{\sqrt{d x + c}{\left (d x - 8 \, c\right )} x^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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