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

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

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Rubi [A]  time = 0.10, antiderivative size = 199, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 {gather*}

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*}

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Mathematica [C]  time = 0.05, size = 61, normalized size = 0.31 \begin {gather*} \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}} \end {gather*}

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), -1/4*(d*x)/c])/(8*c*Sqrt[c + d*x])

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IntegrateAlgebraic [F]  time = 30.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x} \sqrt {c+d x} (4 c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (d x + 4 \, c\right )} \sqrt {d x + c} x^{\frac {1}{3}}}\,{d x} \end {gather*}

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)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d x +4 c \right ) \sqrt {d x +c}\, x^{\frac {1}{3}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (d x + 4 \, c\right )} \sqrt {d x + c} x^{\frac {1}{3}}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^{1/3}\,\left (4\,c+d\,x\right )\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x} \sqrt {c + d x} \left (4 c + d x\right )}\, dx \end {gather*}

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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