3.12.1 \(\int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx\)

Optimal. Leaf size=241 \[ -\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt [4]{3} e} \]

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {63, 331, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt [4]{3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((6 - 3*e*x)^(1/4)*(2 + e*x)^(3/4)),x]

[Out]

(Sqrt[2]*ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)])/(3^(1/4)*e) - (Sqrt[2]*ArcTan[1 + (Sqrt[2]*(2
- e*x)^(1/4))/(2 + e*x)^(1/4)])/(3^(1/4)*e) - Log[(Sqrt[6 - 3*e*x] - Sqrt[6]*(2 - e*x)^(1/4)*(2 + e*x)^(1/4) +
Sqrt[3]*Sqrt[2 + e*x])/Sqrt[2 + e*x]]/(Sqrt[2]*3^(1/4)*e) + Log[(Sqrt[6 - 3*e*x] + Sqrt[6]*(2 - e*x)^(1/4)*(2
+ e*x)^(1/4) + Sqrt[3]*Sqrt[2 + e*x])/Sqrt[2 + e*x]]/(Sqrt[2]*3^(1/4)*e)

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 204

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

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 0.02, size = 42, normalized size = 0.17 \begin {gather*} -\frac {\sqrt {2} (6-3 e x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {1}{12} (6-3 e x)\right )}{9 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((6 - 3*e*x)^(1/4)*(2 + e*x)^(3/4)),x]

[Out]

-1/9*(Sqrt[2]*(6 - 3*e*x)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (6 - 3*e*x)/12])/e

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((6 - 3*e*x)^(1/4)*(2 + e*x)^(3/4)),x]

[Out]

Defer[IntegrateAlgebraic][1/((6 - 3*e*x)^(1/4)*(2 + e*x)^(3/4)), x]

________________________________________________________________________________________

fricas [B]  time = 1.41, size = 505, normalized size = 2.10 \begin {gather*} 2 \, \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} \frac {1}{e^{4}}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} e^{3} \frac {1}{e^{4}}^{\frac {3}{4}} - \sqrt {3} \sqrt {2} \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (e^{4} x - 2 \, e^{3}\right )} \sqrt {\frac {\sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} e \frac {1}{e^{4}}^{\frac {1}{4}} + 3 \, \sqrt {\frac {1}{3}} {\left (e^{3} x - 2 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} - \sqrt {e x + 2} \sqrt {-3 \, e x + 6}}{e x - 2}} \frac {1}{e^{4}}^{\frac {3}{4}} + e x - 2}{e x - 2}\right ) + 2 \, \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} \frac {1}{e^{4}}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} e^{3} \frac {1}{e^{4}}^{\frac {3}{4}} - \sqrt {3} \sqrt {2} \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (e^{4} x - 2 \, e^{3}\right )} \sqrt {-\frac {\sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} e \frac {1}{e^{4}}^{\frac {1}{4}} - 3 \, \sqrt {\frac {1}{3}} {\left (e^{3} x - 2 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} + \sqrt {e x + 2} \sqrt {-3 \, e x + 6}}{e x - 2}} \frac {1}{e^{4}}^{\frac {3}{4}} - e x + 2}{e x - 2}\right ) - \frac {1}{2} \, \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} \frac {1}{e^{4}}^{\frac {1}{4}} \log \left (\frac {3 \, {\left (\sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} e \frac {1}{e^{4}}^{\frac {1}{4}} + 3 \, \sqrt {\frac {1}{3}} {\left (e^{3} x - 2 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} - \sqrt {e x + 2} \sqrt {-3 \, e x + 6}\right )}}{e x - 2}\right ) + \frac {1}{2} \, \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} \frac {1}{e^{4}}^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (\sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {1}{4}} {\left (-3 \, e x + 6\right )}^{\frac {3}{4}} e \frac {1}{e^{4}}^{\frac {1}{4}} - 3 \, \sqrt {\frac {1}{3}} {\left (e^{3} x - 2 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} + \sqrt {e x + 2} \sqrt {-3 \, e x + 6}\right )}}{e x - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x, algorithm="fricas")

[Out]

2*sqrt(2)*(1/3)^(1/4)*(e^(-4))^(1/4)*arctan(-(sqrt(2)*(1/3)^(3/4)*(e*x + 2)^(1/4)*(-3*e*x + 6)^(3/4)*e^3*(e^(-
4))^(3/4) - sqrt(3)*sqrt(2)*(1/3)^(3/4)*(e^4*x - 2*e^3)*sqrt((sqrt(2)*(1/3)^(1/4)*(e*x + 2)^(1/4)*(-3*e*x + 6)
^(3/4)*e*(e^(-4))^(1/4) + 3*sqrt(1/3)*(e^3*x - 2*e^2)*sqrt(e^(-4)) - sqrt(e*x + 2)*sqrt(-3*e*x + 6))/(e*x - 2)
)*(e^(-4))^(3/4) + e*x - 2)/(e*x - 2)) + 2*sqrt(2)*(1/3)^(1/4)*(e^(-4))^(1/4)*arctan(-(sqrt(2)*(1/3)^(3/4)*(e*
x + 2)^(1/4)*(-3*e*x + 6)^(3/4)*e^3*(e^(-4))^(3/4) - sqrt(3)*sqrt(2)*(1/3)^(3/4)*(e^4*x - 2*e^3)*sqrt(-(sqrt(2
)*(1/3)^(1/4)*(e*x + 2)^(1/4)*(-3*e*x + 6)^(3/4)*e*(e^(-4))^(1/4) - 3*sqrt(1/3)*(e^3*x - 2*e^2)*sqrt(e^(-4)) +
sqrt(e*x + 2)*sqrt(-3*e*x + 6))/(e*x - 2))*(e^(-4))^(3/4) - e*x + 2)/(e*x - 2)) - 1/2*sqrt(2)*(1/3)^(1/4)*(e^
(-4))^(1/4)*log(3*(sqrt(2)*(1/3)^(1/4)*(e*x + 2)^(1/4)*(-3*e*x + 6)^(3/4)*e*(e^(-4))^(1/4) + 3*sqrt(1/3)*(e^3*
x - 2*e^2)*sqrt(e^(-4)) - sqrt(e*x + 2)*sqrt(-3*e*x + 6))/(e*x - 2)) + 1/2*sqrt(2)*(1/3)^(1/4)*(e^(-4))^(1/4)*
log(-3*(sqrt(2)*(1/3)^(1/4)*(e*x + 2)^(1/4)*(-3*e*x + 6)^(3/4)*e*(e^(-4))^(1/4) - 3*sqrt(1/3)*(e^3*x - 2*e^2)*
sqrt(e^(-4)) + sqrt(e*x + 2)*sqrt(-3*e*x + 6))/(e*x - 2))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (e x + 2\right )}^{\frac {3}{4}} {\left (-3 \, e x + 6\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((e*x + 2)^(3/4)*(-3*e*x + 6)^(1/4)), x)

________________________________________________________________________________________

maple [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (-3 e x +6\right )^{\frac {1}{4}} \left (e x +2\right )^{\frac {3}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x)

[Out]

int(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (e x + 2\right )}^{\frac {3}{4}} {\left (-3 \, e x + 6\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*e*x+6)^(1/4)/(e*x+2)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((e*x + 2)^(3/4)*(-3*e*x + 6)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (e\,x+2\right )}^{3/4}\,{\left (6-3\,e\,x\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e*x + 2)^(3/4)*(6 - 3*e*x)^(1/4)),x)

[Out]

int(1/((e*x + 2)^(3/4)*(6 - 3*e*x)^(1/4)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {3^{\frac {3}{4}} \int \frac {1}{\sqrt [4]{- e x + 2} \left (e x + 2\right )^{\frac {3}{4}}}\, dx}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*e*x+6)**(1/4)/(e*x+2)**(3/4),x)

[Out]

3**(3/4)*Integral(1/((-e*x + 2)**(1/4)*(e*x + 2)**(3/4)), x)/3

________________________________________________________________________________________