\(\int \frac {1}{\sqrt [3]{2+3 x^2} (6 d+d x^2)} \, dx\) [156]

Optimal result

Integrand size = 23, antiderivative size = 123 \[ \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\arctan \left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d} \]

[Out]

-1/8*arctanh(2^(1/6)*(2^(1/3)-(3*x^2+2)^(1/3))/x)*2^(1/6)/d+1/24*arctan(1/18*(2^(1/3)-(3*x^2+2)^(1/3))^2*2^(5/
6)/x*3^(1/2))*2^(1/6)/d*3^(1/2)+1/24*arctan(1/6*x*6^(1/2))*2^(1/6)/d*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {403} \[ \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx=\frac {\arctan \left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\arctan \left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )}{x}\right )}{4\ 2^{5/6} d} \]

[In]

Int[1/((2 + 3*x^2)^(1/3)*(6*d + d*x^2)),x]

[Out]

ArcTan[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) + ArcTan[(2^(1/3) - (2 + 3*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(
5/6)*Sqrt[3]*d) - ArcTanh[(2^(1/6)*(2^(1/3) - (2 + 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)

Rule 403

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[q*(ArcTan[
q*(x/3)]/(12*Rt[a, 3]*d)), x] + (Simp[q*(ArcTan[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)]/(12*Rt[a,
3]*d)), x] - Simp[q*(ArcTanh[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)]/(4*Sqrt[3]*Rt[a, 3]*d))
, x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && PosQ[b/a]

Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tan ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 4.50 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx=-\frac {9 x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {3 x^2}{2},-\frac {x^2}{6}\right )}{d \left (6+x^2\right ) \sqrt [3]{2+3 x^2} \left (-9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {3 x^2}{2},-\frac {x^2}{6}\right )+x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-\frac {3 x^2}{2},-\frac {x^2}{6}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-\frac {3 x^2}{2},-\frac {x^2}{6}\right )\right )\right )} \]

[In]

Integrate[1/((2 + 3*x^2)^(1/3)*(6*d + d*x^2)),x]

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, (-3*x^2)/2, -1/6*x^2])/(d*(6 + x^2)*(2 + 3*x^2)^(1/3)*(-9*AppellF1[1/2, 1/3,
1, 3/2, (-3*x^2)/2, -1/6*x^2] + x^2*(AppellF1[3/2, 1/3, 2, 5/2, (-3*x^2)/2, -1/6*x^2] + 3*AppellF1[3/2, 4/3, 1
, 5/2, (-3*x^2)/2, -1/6*x^2])))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 76.62 (sec) , antiderivative size = 1066, normalized size of antiderivative = 8.67

method	result	size
trager	\(\text {Expression too large to display}\)	\(1066\)

[In]

int(1/(3*x^2+2)^(1/3)/(d*x^2+6*d),x,method=_RETURNVERBOSE)

[Out]

-1/24*(ln(-(16*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^6*x-768*RootOf(RootOf(
_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)^2*RootOf(_Z^6+54)^5*x-(3*x^2+2)^(1/3)*RootOf(_Z^6+54)^5*x+72*RootOf
(_Z^6+54)^4*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*(3*x^2+2)^(1/3)*x-1152*RootOf(_Z^6+54)^3*
RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)^2*(3*x^2+2)^(1/3)*x-36*RootOf(_Z^6+54)^3*RootOf(RootO
f(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*x^2+72*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*R
ootOf(_Z^6+54)^3+18*RootOf(_Z^6+54)^2*(3*x^2+2)^(1/3)-432*RootOf(_Z^6+54)*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootO
f(_Z^6+54)+576*_Z^2)*(3*x^2+2)^(1/3)+54*(3*x^2+2)^(2/3))/(x^2+6))*RootOf(_Z^6+54)-24*RootOf(RootOf(_Z^6+54)^2-
24*_Z*RootOf(_Z^6+54)+576*_Z^2)*ln(-(4*RootOf(_Z^6+54)^7*x-288*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+
576*_Z^2)*RootOf(_Z^6+54)^6*x+4608*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)^2*RootOf(_Z^6+54)^
5*x-144*RootOf(_Z^6+54)^4*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*(3*x^2+2)^(1/3)*x+6912*Root
Of(_Z^6+54)^3*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)^2*(3*x^2+2)^(1/3)*x-9*x^2*RootOf(_Z^6+5
4)^4+216*RootOf(_Z^6+54)^3*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*x^2+18*RootOf(_Z^6+54)^4-4
32*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^3+2592*RootOf(_Z^6+54)*RootOf(Root
Of(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*(3*x^2+2)^(1/3)+324*(3*x^2+2)^(2/3))/(x^2+6))-24*ln(-(16*RootOf(
RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^6*x-768*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootO
f(_Z^6+54)+576*_Z^2)^2*RootOf(_Z^6+54)^5*x-(3*x^2+2)^(1/3)*RootOf(_Z^6+54)^5*x+72*RootOf(_Z^6+54)^4*RootOf(Roo
tOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*(3*x^2+2)^(1/3)*x-1152*RootOf(_Z^6+54)^3*RootOf(RootOf(_Z^6+54)
^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)^2*(3*x^2+2)^(1/3)*x-36*RootOf(_Z^6+54)^3*RootOf(RootOf(_Z^6+54)^2-24*_Z*Roo
tOf(_Z^6+54)+576*_Z^2)*x^2+72*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^3+18*Ro
otOf(_Z^6+54)^2*(3*x^2+2)^(1/3)-432*RootOf(_Z^6+54)*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*(
3*x^2+2)^(1/3)+54*(3*x^2+2)^(2/3))/(x^2+6))*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2))/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1843 vs. \(2 (90) = 180\).

Time = 192.78 (sec) , antiderivative size = 1843, normalized size of antiderivative = 14.98 \[ \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/48*(1/864)^(1/6)*(sqrt(-3) + 1)*(-1/d^6)^(1/6)*log(-1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 - 92*d^4*x^3 - 36*d^4*x +
sqrt(-3)*(7*d^4*x^5 - 92*d^4*x^3 - 36*d^4*x))*(-1/d^6)^(2/3) + 4*(10*x^3 + 3*sqrt(1/6)*(d^3*x^4 - 24*d^3*x^2 +
12*d^3)*sqrt(-1/d^6) - 36*x)*(3*x^2 + 2)^(2/3) - 2*(432*(1/864)^(5/6)*(5*d^5*x^4 - 36*d^5*x^2 - 12*d^5 - sqrt
(-3)*(5*d^5*x^4 - 36*d^5*x^2 - 12*d^5))*(-1/d^6)^(5/6) - (1/4)^(1/3)*(d^2*x^5 - 52*d^2*x^3 + 36*d^2*x - sqrt(-
3)*(d^2*x^5 - 52*d^2*x^3 + 36*d^2*x))*(-1/d^6)^(1/3))*(3*x^2 + 2)^(1/3) - (1/864)^(1/6)*(d*x^6 - 210*d*x^4 + 2
52*d*x^2 + sqrt(-3)*(d*x^6 - 210*d*x^4 + 252*d*x^2 + 72*d) + 72*d)*(-1/d^6)^(1/6))/(x^6 + 18*x^4 + 108*x^2 + 2
16)) - 1/48*(1/864)^(1/6)*(sqrt(-3) + 1)*(-1/d^6)^(1/6)*log(-1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 - 92*d^4*x^3 - 36*d
^4*x + sqrt(-3)*(7*d^4*x^5 - 92*d^4*x^3 - 36*d^4*x))*(-1/d^6)^(2/3) + 4*(10*x^3 - 3*sqrt(1/6)*(d^3*x^4 - 24*d^
3*x^2 + 12*d^3)*sqrt(-1/d^6) - 36*x)*(3*x^2 + 2)^(2/3) + 2*(432*(1/864)^(5/6)*(5*d^5*x^4 - 36*d^5*x^2 - 12*d^5
- sqrt(-3)*(5*d^5*x^4 - 36*d^5*x^2 - 12*d^5))*(-1/d^6)^(5/6) + (1/4)^(1/3)*(d^2*x^5 - 52*d^2*x^3 + 36*d^2*x -
sqrt(-3)*(d^2*x^5 - 52*d^2*x^3 + 36*d^2*x))*(-1/d^6)^(1/3))*(3*x^2 + 2)^(1/3) + (1/864)^(1/6)*(d*x^6 - 210*d*
x^4 + 252*d*x^2 + sqrt(-3)*(d*x^6 - 210*d*x^4 + 252*d*x^2 + 72*d) + 72*d)*(-1/d^6)^(1/6))/(x^6 + 18*x^4 + 108*
x^2 + 216)) - 1/48*(1/864)^(1/6)*(sqrt(-3) - 1)*(-1/d^6)^(1/6)*log(-1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 - 92*d^4*x^3
- 36*d^4*x - sqrt(-3)*(7*d^4*x^5 - 92*d^4*x^3 - 36*d^4*x))*(-1/d^6)^(2/3) + 4*(10*x^3 + 3*sqrt(1/6)*(d^3*x^4
- 24*d^3*x^2 + 12*d^3)*sqrt(-1/d^6) - 36*x)*(3*x^2 + 2)^(2/3) - 2*(432*(1/864)^(5/6)*(5*d^5*x^4 - 36*d^5*x^2 -
12*d^5 + sqrt(-3)*(5*d^5*x^4 - 36*d^5*x^2 - 12*d^5))*(-1/d^6)^(5/6) - (1/4)^(1/3)*(d^2*x^5 - 52*d^2*x^3 + 36*
d^2*x + sqrt(-3)*(d^2*x^5 - 52*d^2*x^3 + 36*d^2*x))*(-1/d^6)^(1/3))*(3*x^2 + 2)^(1/3) - (1/864)^(1/6)*(d*x^6 -
210*d*x^4 + 252*d*x^2 - sqrt(-3)*(d*x^6 - 210*d*x^4 + 252*d*x^2 + 72*d) + 72*d)*(-1/d^6)^(1/6))/(x^6 + 18*x^4
+ 108*x^2 + 216)) + 1/48*(1/864)^(1/6)*(sqrt(-3) - 1)*(-1/d^6)^(1/6)*log(-1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 - 92*
d^4*x^3 - 36*d^4*x - sqrt(-3)*(7*d^4*x^5 - 92*d^4*x^3 - 36*d^4*x))*(-1/d^6)^(2/3) + 4*(10*x^3 - 3*sqrt(1/6)*(d
^3*x^4 - 24*d^3*x^2 + 12*d^3)*sqrt(-1/d^6) - 36*x)*(3*x^2 + 2)^(2/3) + 2*(432*(1/864)^(5/6)*(5*d^5*x^4 - 36*d^
5*x^2 - 12*d^5 + sqrt(-3)*(5*d^5*x^4 - 36*d^5*x^2 - 12*d^5))*(-1/d^6)^(5/6) + (1/4)^(1/3)*(d^2*x^5 - 52*d^2*x^
3 + 36*d^2*x + sqrt(-3)*(d^2*x^5 - 52*d^2*x^3 + 36*d^2*x))*(-1/d^6)^(1/3))*(3*x^2 + 2)^(1/3) + (1/864)^(1/6)*(
d*x^6 - 210*d*x^4 + 252*d*x^2 - sqrt(-3)*(d*x^6 - 210*d*x^4 + 252*d*x^2 + 72*d) + 72*d)*(-1/d^6)^(1/6))/(x^6 +
18*x^4 + 108*x^2 + 216)) - 1/24*(1/864)^(1/6)*(-1/d^6)^(1/6)*log(1/4*(4*(1/4)^(2/3)*(7*d^4*x^5 - 92*d^4*x^3 -
36*d^4*x)*(-1/d^6)^(2/3) - 2*(10*x^3 + 3*sqrt(1/6)*(d^3*x^4 - 24*d^3*x^2 + 12*d^3)*sqrt(-1/d^6) - 36*x)*(3*x^
2 + 2)^(2/3) - 2*(432*(1/864)^(5/6)*(5*d^5*x^4 - 36*d^5*x^2 - 12*d^5)*(-1/d^6)^(5/6) - (1/4)^(1/3)*(d^2*x^5 -
52*d^2*x^3 + 36*d^2*x)*(-1/d^6)^(1/3))*(3*x^2 + 2)^(1/3) - (1/864)^(1/6)*(d*x^6 - 210*d*x^4 + 252*d*x^2 + 72*d
)*(-1/d^6)^(1/6))/(x^6 + 18*x^4 + 108*x^2 + 216)) + 1/24*(1/864)^(1/6)*(-1/d^6)^(1/6)*log(1/4*(4*(1/4)^(2/3)*(
7*d^4*x^5 - 92*d^4*x^3 - 36*d^4*x)*(-1/d^6)^(2/3) - 2*(10*x^3 - 3*sqrt(1/6)*(d^3*x^4 - 24*d^3*x^2 + 12*d^3)*sq
rt(-1/d^6) - 36*x)*(3*x^2 + 2)^(2/3) + 2*(432*(1/864)^(5/6)*(5*d^5*x^4 - 36*d^5*x^2 - 12*d^5)*(-1/d^6)^(5/6) +
(1/4)^(1/3)*(d^2*x^5 - 52*d^2*x^3 + 36*d^2*x)*(-1/d^6)^(1/3))*(3*x^2 + 2)^(1/3) + (1/864)^(1/6)*(d*x^6 - 210*
d*x^4 + 252*d*x^2 + 72*d)*(-1/d^6)^(1/6))/(x^6 + 18*x^4 + 108*x^2 + 216))

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx=\frac {\int \frac {1}{x^{2} \sqrt [3]{3 x^{2} + 2} + 6 \sqrt [3]{3 x^{2} + 2}}\, dx}{d} \]

[In]

integrate(1/(3*x**2+2)**(1/3)/(d*x**2+6*d),x)

[Out]

Integral(1/(x**2*(3*x**2 + 2)**(1/3) + 6*(3*x**2 + 2)**(1/3)), x)/d

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx=\int { \frac {1}{{\left (d x^{2} + 6 \, d\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(1/((d*x^2 + 6*d)*(3*x^2 + 2)^(1/3)), x)

Giac [F]

\[ \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx=\int { \frac {1}{{\left (d x^{2} + 6 \, d\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integrate(1/((d*x^2 + 6*d)*(3*x^2 + 2)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx=\int \frac {1}{{\left (3\,x^2+2\right )}^{1/3}\,\left (d\,x^2+6\,d\right )} \,d x \]

[In]

int(1/((3*x^2 + 2)^(1/3)*(6*d + d*x^2)),x)

[Out]

int(1/((3*x^2 + 2)^(1/3)*(6*d + d*x^2)), x)