3.463 \(\int \frac{1}{\sqrt [3]{x} \sqrt{c+d x} (4 c+d x)} \, dx\)

Optimal. Leaf size=199 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.100251, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {130, 484} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(1/3)*Sqrt[c + d*x]*(4*c + d*x)),x]

[Out]

-(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])]/(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]]/(2^(2/3)*c^(5/6)*d^(2/3)) + ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(3*2^(2/3)*
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 484

Int[(x_)/(((a_) + (b_.)*(x_)^3)*Sqrt[(c_) + (d_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[d/c, 3]}, Simp[(q*ArcTan
h[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]*Rt[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]

Rubi steps

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

Mathematica [C]  time = 0.0368889, size = 61, normalized size = 0.31 \[ \frac{3 x^{2/3} \sqrt{\frac{c+d x}{c}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )}{8 c \sqrt{c+d x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^(1/3)*Sqrt[c + d*x]*(4*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)/(4*c)])/(8*c*Sqrt[c + d*x])

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+4\,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+4*c)/(d*x+c)^(1/2),x)

[Out]

int(1/x^(1/3)/(d*x+4*c)/(d*x+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + 4 \, c\right )} \sqrt{d x + c} 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+4*c)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)), x)

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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+4*c)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/3)/(d*x+4*c)/(d*x+c)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + 4 \, c\right )} \sqrt{d x + c} 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+4*c)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/((d*x + 4*c)*sqrt(d*x + c)*x^(1/3)), x)