Integrand size = 11, antiderivative size = 81 \[ \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx=-\frac {3 \sqrt [3]{x}}{4 b \left (a+b x^{2/3}\right )^2}+\frac {3 \sqrt [3]{x}}{8 a b \left (a+b x^{2/3}\right )}+\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2}} \] Output:
-3/4*x^(1/3)/b/(a+b*x^(2/3))^2+3/8*x^(1/3)/a/b/(a+b*x^(2/3))+3/8*arctan(b^ (1/2)*x^(1/3)/a^(1/2))/a^(3/2)/b^(3/2)
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx=-\frac {3 \left (a-b x^{2/3}\right ) \sqrt [3]{x}}{8 a b \left (a+b x^{2/3}\right )^2}+\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a}}\right )}{8 a^{3/2} b^{3/2}} \] Input:
Integrate[(a + b*x^(2/3))^(-3),x]
Output:
(-3*(a - b*x^(2/3))*x^(1/3))/(8*a*b*(a + b*x^(2/3))^2) + (3*ArcTan[(Sqrt[b ]*x^(1/3))/Sqrt[a]])/(8*a^(3/2)*b^(3/2))
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {774, 252, 215, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 3 \int \frac {x^{2/3}}{\left (a+b x^{2/3}\right )^3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle 3 \left (\frac {\int \frac {1}{\left (a+b x^{2/3}\right )^2}d\sqrt [3]{x}}{4 b}-\frac {\sqrt [3]{x}}{4 b \left (a+b x^{2/3}\right )^2}\right )\) |
\(\Big \downarrow \) 215 |
\(\displaystyle 3 \left (\frac {\frac {\int \frac {1}{a+b x^{2/3}}d\sqrt [3]{x}}{2 a}+\frac {\sqrt [3]{x}}{2 a \left (a+b x^{2/3}\right )}}{4 b}-\frac {\sqrt [3]{x}}{4 b \left (a+b x^{2/3}\right )^2}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle 3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sqrt [3]{x}}{2 a \left (a+b x^{2/3}\right )}}{4 b}-\frac {\sqrt [3]{x}}{4 b \left (a+b x^{2/3}\right )^2}\right )\) |
Input:
Int[(a + b*x^(2/3))^(-3),x]
Output:
3*(-1/4*x^(1/3)/(b*(a + b*x^(2/3))^2) + (x^(1/3)/(2*a*(a + b*x^(2/3))) + A rcTan[(Sqrt[b]*x^(1/3))/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]))/(4*b))
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Time = 0.39 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(\frac {\frac {3 x}{8 a}-\frac {3 x^{\frac {1}{3}}}{8 b}}{\left (a +b \,x^{\frac {2}{3}}\right )^{2}}+\frac {3 \arctan \left (\frac {b \,x^{\frac {1}{3}}}{\sqrt {a b}}\right )}{8 a b \sqrt {a b}}\) | \(52\) |
default | \(\text {Expression too large to display}\) | \(1273\) |
Input:
int(1/(a+b*x^(2/3))^3,x,method=_RETURNVERBOSE)
Output:
3*(1/8*x/a-1/8*x^(1/3)/b)/(a+b*x^(2/3))^2+3/8/a/b/(a*b)^(1/2)*arctan(b*x^( 1/3)/(a*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (57) = 114\).
Time = 0.08 (sec) , antiderivative size = 356, normalized size of antiderivative = 4.40 \[ \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx=\left [-\frac {3 \, {\left (6 \, a^{2} b^{5} x^{3} - 6 \, a^{5} b^{2} x + {\left (b^{6} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{6}\right )} \sqrt {-a b} \log \left (\frac {b^{3} x^{2} + 2 \, \sqrt {-a b} a b x - a^{3} - 2 \, {\left (\sqrt {-a b} b^{2} x - a^{2} b\right )} x^{\frac {2}{3}} - 2 \, {\left (a b^{2} x + \sqrt {-a b} a^{2}\right )} x^{\frac {1}{3}}}{b^{3} x^{2} + a^{3}}\right ) - 2 \, {\left (a b^{6} x^{3} - 5 \, a^{4} b^{3} x\right )} x^{\frac {2}{3}} - 2 \, {\left (5 \, a^{3} b^{4} x^{2} - a^{6} b\right )} x^{\frac {1}{3}}\right )}}{16 \, {\left (a^{2} b^{8} x^{4} + 2 \, a^{5} b^{5} x^{2} + a^{8} b^{2}\right )}}, -\frac {3 \, {\left (3 \, a^{2} b^{5} x^{3} - 3 \, a^{5} b^{2} x - {\left (b^{6} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{6}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x^{\frac {1}{3}}}{a}\right ) - {\left (a b^{6} x^{3} - 5 \, a^{4} b^{3} x\right )} x^{\frac {2}{3}} - {\left (5 \, a^{3} b^{4} x^{2} - a^{6} b\right )} x^{\frac {1}{3}}\right )}}{8 \, {\left (a^{2} b^{8} x^{4} + 2 \, a^{5} b^{5} x^{2} + a^{8} b^{2}\right )}}\right ] \] Input:
integrate(1/(a+b*x^(2/3))^3,x, algorithm="fricas")
Output:
[-3/16*(6*a^2*b^5*x^3 - 6*a^5*b^2*x + (b^6*x^4 + 2*a^3*b^3*x^2 + a^6)*sqrt (-a*b)*log((b^3*x^2 + 2*sqrt(-a*b)*a*b*x - a^3 - 2*(sqrt(-a*b)*b^2*x - a^2 *b)*x^(2/3) - 2*(a*b^2*x + sqrt(-a*b)*a^2)*x^(1/3))/(b^3*x^2 + a^3)) - 2*( a*b^6*x^3 - 5*a^4*b^3*x)*x^(2/3) - 2*(5*a^3*b^4*x^2 - a^6*b)*x^(1/3))/(a^2 *b^8*x^4 + 2*a^5*b^5*x^2 + a^8*b^2), -3/8*(3*a^2*b^5*x^3 - 3*a^5*b^2*x - ( b^6*x^4 + 2*a^3*b^3*x^2 + a^6)*sqrt(a*b)*arctan(sqrt(a*b)*x^(1/3)/a) - (a* b^6*x^3 - 5*a^4*b^3*x)*x^(2/3) - (5*a^3*b^4*x^2 - a^6*b)*x^(1/3))/(a^2*b^8 *x^4 + 2*a^5*b^5*x^2 + a^8*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (70) = 140\).
Time = 7.45 (sec) , antiderivative size = 666, normalized size of antiderivative = 8.22 \[ \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {x}{a^{3}} & \text {for}\: b = 0 \\- \frac {1}{b^{3} x} & \text {for}\: a = 0 \\\frac {3 a^{2} \log {\left (\sqrt [3]{x} - \sqrt {- \frac {a}{b}} \right )}}{16 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 32 a^{2} b^{3} x^{\frac {2}{3}} \sqrt {- \frac {a}{b}} + 16 a b^{4} x^{\frac {4}{3}} \sqrt {- \frac {a}{b}}} - \frac {3 a^{2} \log {\left (\sqrt [3]{x} + \sqrt {- \frac {a}{b}} \right )}}{16 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 32 a^{2} b^{3} x^{\frac {2}{3}} \sqrt {- \frac {a}{b}} + 16 a b^{4} x^{\frac {4}{3}} \sqrt {- \frac {a}{b}}} + \frac {6 a b x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} - \sqrt {- \frac {a}{b}} \right )}}{16 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 32 a^{2} b^{3} x^{\frac {2}{3}} \sqrt {- \frac {a}{b}} + 16 a b^{4} x^{\frac {4}{3}} \sqrt {- \frac {a}{b}}} - \frac {6 a b x^{\frac {2}{3}} \log {\left (\sqrt [3]{x} + \sqrt {- \frac {a}{b}} \right )}}{16 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 32 a^{2} b^{3} x^{\frac {2}{3}} \sqrt {- \frac {a}{b}} + 16 a b^{4} x^{\frac {4}{3}} \sqrt {- \frac {a}{b}}} - \frac {6 a b \sqrt [3]{x} \sqrt {- \frac {a}{b}}}{16 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 32 a^{2} b^{3} x^{\frac {2}{3}} \sqrt {- \frac {a}{b}} + 16 a b^{4} x^{\frac {4}{3}} \sqrt {- \frac {a}{b}}} + \frac {3 b^{2} x^{\frac {4}{3}} \log {\left (\sqrt [3]{x} - \sqrt {- \frac {a}{b}} \right )}}{16 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 32 a^{2} b^{3} x^{\frac {2}{3}} \sqrt {- \frac {a}{b}} + 16 a b^{4} x^{\frac {4}{3}} \sqrt {- \frac {a}{b}}} - \frac {3 b^{2} x^{\frac {4}{3}} \log {\left (\sqrt [3]{x} + \sqrt {- \frac {a}{b}} \right )}}{16 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 32 a^{2} b^{3} x^{\frac {2}{3}} \sqrt {- \frac {a}{b}} + 16 a b^{4} x^{\frac {4}{3}} \sqrt {- \frac {a}{b}}} + \frac {6 b^{2} x \sqrt {- \frac {a}{b}}}{16 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 32 a^{2} b^{3} x^{\frac {2}{3}} \sqrt {- \frac {a}{b}} + 16 a b^{4} x^{\frac {4}{3}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b*x**(2/3))**3,x)
Output:
Piecewise((zoo/x, Eq(a, 0) & Eq(b, 0)), (x/a**3, Eq(b, 0)), (-1/(b**3*x), Eq(a, 0)), (3*a**2*log(x**(1/3) - sqrt(-a/b))/(16*a**3*b**2*sqrt(-a/b) + 3 2*a**2*b**3*x**(2/3)*sqrt(-a/b) + 16*a*b**4*x**(4/3)*sqrt(-a/b)) - 3*a**2* log(x**(1/3) + sqrt(-a/b))/(16*a**3*b**2*sqrt(-a/b) + 32*a**2*b**3*x**(2/3 )*sqrt(-a/b) + 16*a*b**4*x**(4/3)*sqrt(-a/b)) + 6*a*b*x**(2/3)*log(x**(1/3 ) - sqrt(-a/b))/(16*a**3*b**2*sqrt(-a/b) + 32*a**2*b**3*x**(2/3)*sqrt(-a/b ) + 16*a*b**4*x**(4/3)*sqrt(-a/b)) - 6*a*b*x**(2/3)*log(x**(1/3) + sqrt(-a /b))/(16*a**3*b**2*sqrt(-a/b) + 32*a**2*b**3*x**(2/3)*sqrt(-a/b) + 16*a*b* *4*x**(4/3)*sqrt(-a/b)) - 6*a*b*x**(1/3)*sqrt(-a/b)/(16*a**3*b**2*sqrt(-a/ b) + 32*a**2*b**3*x**(2/3)*sqrt(-a/b) + 16*a*b**4*x**(4/3)*sqrt(-a/b)) + 3 *b**2*x**(4/3)*log(x**(1/3) - sqrt(-a/b))/(16*a**3*b**2*sqrt(-a/b) + 32*a* *2*b**3*x**(2/3)*sqrt(-a/b) + 16*a*b**4*x**(4/3)*sqrt(-a/b)) - 3*b**2*x**( 4/3)*log(x**(1/3) + sqrt(-a/b))/(16*a**3*b**2*sqrt(-a/b) + 32*a**2*b**3*x* *(2/3)*sqrt(-a/b) + 16*a*b**4*x**(4/3)*sqrt(-a/b)) + 6*b**2*x*sqrt(-a/b)/( 16*a**3*b**2*sqrt(-a/b) + 32*a**2*b**3*x**(2/3)*sqrt(-a/b) + 16*a*b**4*x** (4/3)*sqrt(-a/b)), True))
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx=\frac {3 \, {\left (b x - a x^{\frac {1}{3}}\right )}}{8 \, {\left (a b^{3} x^{\frac {4}{3}} + 2 \, a^{2} b^{2} x^{\frac {2}{3}} + a^{3} b\right )}} + \frac {3 \, \arctan \left (\frac {b x^{\frac {1}{3}}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b} \] Input:
integrate(1/(a+b*x^(2/3))^3,x, algorithm="maxima")
Output:
3/8*(b*x - a*x^(1/3))/(a*b^3*x^(4/3) + 2*a^2*b^2*x^(2/3) + a^3*b) + 3/8*ar ctan(b*x^(1/3)/sqrt(a*b))/(sqrt(a*b)*a*b)
Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx=\frac {3 \, \arctan \left (\frac {b x^{\frac {1}{3}}}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b} + \frac {3 \, {\left (b x - a x^{\frac {1}{3}}\right )}}{8 \, {\left (b x^{\frac {2}{3}} + a\right )}^{2} a b} \] Input:
integrate(1/(a+b*x^(2/3))^3,x, algorithm="giac")
Output:
3/8*arctan(b*x^(1/3)/sqrt(a*b))/(sqrt(a*b)*a*b) + 3/8*(b*x - a*x^(1/3))/(( b*x^(2/3) + a)^2*a*b)
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx=\frac {\frac {3\,x}{8\,a}-\frac {3\,x^{1/3}}{8\,b}}{a^2+b^2\,x^{4/3}+2\,a\,b\,x^{2/3}}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{1/3}}{\sqrt {a}}\right )}{8\,a^{3/2}\,b^{3/2}} \] Input:
int(1/(a + b*x^(2/3))^3,x)
Output:
((3*x)/(8*a) - (3*x^(1/3))/(8*b))/(a^2 + b^2*x^(4/3) + 2*a*b*x^(2/3)) + (3 *atan((b^(1/2)*x^(1/3))/a^(1/2)))/(8*a^(3/2)*b^(3/2))
Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\left (a+b x^{2/3}\right )^3} \, dx=\frac {\frac {3 x^{\frac {2}{3}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} b}{\sqrt {b}\, \sqrt {a}}\right ) a b}{4}+\frac {3 x^{\frac {4}{3}} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} b}{\sqrt {b}\, \sqrt {a}}\right ) b^{2}}{8}+\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {x^{\frac {1}{3}} b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2}}{8}-\frac {3 x^{\frac {1}{3}} a^{2} b}{8}+\frac {3 a \,b^{2} x}{8}}{a^{2} b^{2} \left (2 x^{\frac {2}{3}} a b +x^{\frac {4}{3}} b^{2}+a^{2}\right )} \] Input:
int(1/(a+b*x^(2/3))^3,x)
Output:
(3*(2*x**(2/3)*sqrt(b)*sqrt(a)*atan((x**(1/3)*b)/(sqrt(b)*sqrt(a)))*a*b + x**(1/3)*sqrt(b)*sqrt(a)*atan((x**(1/3)*b)/(sqrt(b)*sqrt(a)))*b**2*x + sqr t(b)*sqrt(a)*atan((x**(1/3)*b)/(sqrt(b)*sqrt(a)))*a**2 - x**(1/3)*a**2*b + a*b**2*x))/(8*a**2*b**2*(2*x**(2/3)*a*b + x**(1/3)*b**2*x + a**2))