Integrand size = 11, antiderivative size = 195 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx=\frac {x}{a^3}-\frac {3 b^2 x}{8 a^3 \left (b+a x^{4/3}\right )^2}+\frac {57 b x}{32 a^3 \left (b+a x^{4/3}\right )}+\frac {231 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{64 \sqrt {2} a^{15/4}}-\frac {231 b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{64 \sqrt {2} a^{15/4}}+\frac {231 b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt {b}+\sqrt {a} x^{2/3}}\right )}{64 \sqrt {2} a^{15/4}} \] Output:
x/a^3-3/8*b^2*x/a^3/(b+a*x^(4/3))^2+57/32*b*x/a^3/(b+a*x^(4/3))-231/128*b^ (3/4)*arctan(-1+2^(1/2)*a^(1/4)*x^(1/3)/b^(1/4))*2^(1/2)/a^(15/4)-231/128* b^(3/4)*arctan(1+2^(1/2)*a^(1/4)*x^(1/3)/b^(1/4))*2^(1/2)/a^(15/4)+231/128 *b^(3/4)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x^(1/3)/(b^(1/2)+a^(1/2)*x^(2/3)) )*2^(1/2)/a^(15/4)
Time = 0.39 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx=\frac {\frac {4 a^{3/4} x \left (77 b^2+121 a b x^{4/3}+32 a^2 x^{8/3}\right )}{\left (b+a x^{4/3}\right )^2}-231 \sqrt {2} b^{3/4} \arctan \left (\frac {-\sqrt {b}+\sqrt {a} x^{2/3}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}\right )+231 \sqrt {2} b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt {b}+\sqrt {a} x^{2/3}}\right )}{128 a^{15/4}} \] Input:
Integrate[(a + b/x^(4/3))^(-3),x]
Output:
((4*a^(3/4)*x*(77*b^2 + 121*a*b*x^(4/3) + 32*a^2*x^(8/3)))/(b + a*x^(4/3)) ^2 - 231*Sqrt[2]*b^(3/4)*ArcTan[(-Sqrt[b] + Sqrt[a]*x^(2/3))/(Sqrt[2]*a^(1 /4)*b^(1/4)*x^(1/3))] + 231*Sqrt[2]*b^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/ 4)*x^(1/3))/(Sqrt[b] + Sqrt[a]*x^(2/3))])/(128*a^(15/4))
Time = 0.77 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.53, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.182, Rules used = {774, 795, 817, 817, 843, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 3 \int \frac {x^{2/3}}{\left (a+\frac {b}{x^{4/3}}\right )^3}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle 3 \int \frac {x^{14/3}}{\left (a x^{4/3}+b\right )^3}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 3 \left (\frac {11 \int \frac {x^{10/3}}{\left (a x^{4/3}+b\right )^2}d\sqrt [3]{x}}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \int \frac {x^2}{a x^{4/3}+b}d\sqrt [3]{x}}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \int \frac {x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\int \frac {\sqrt {a} x^{2/3}+\sqrt {b}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\int \frac {1}{x^{2/3}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}d\sqrt [3]{x}}{2 \sqrt {a}}+\frac {\int \frac {1}{x^{2/3}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}d\sqrt [3]{x}}{2 \sqrt {a}}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\int \frac {1}{-x^{2/3}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^{2/3}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{a} \left (x^{2/3}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}+\sqrt [4]{b}\right )}{\sqrt [4]{a} \left (x^{2/3}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{a} \left (x^{2/3}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}+\sqrt [4]{b}\right )}{\sqrt [4]{a} \left (x^{2/3}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} \sqrt [3]{x}}{x^{2/3}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt {a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}+\sqrt [4]{b}}{x^{2/3}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}d\sqrt [3]{x}}{2 \sqrt {a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 3 \left (\frac {11 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}+\sqrt {a} x^{2/3}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}+\sqrt {a} x^{2/3}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )}{8 a}-\frac {x^{11/3}}{8 a \left (a x^{4/3}+b\right )^2}\right )\) |
Input:
Int[(a + b/x^(4/3))^(-3),x]
Output:
3*(-1/8*x^(11/3)/(a*(b + a*x^(4/3))^2) + (11*(-1/4*x^(7/3)/(a*(b + a*x^(4/ 3))) + (7*(x/(3*a) - (b*((-(ArcTan[1 - (Sqrt[2]*a^(1/4)*x^(1/3))/b^(1/4)]/ (Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*a^(1/4)*x^(1/3))/b^(1/4)] /(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) - (-1/2*Log[Sqrt[b] - Sqrt[2]*a^(1 /4)*b^(1/4)*x^(1/3) + Sqrt[a]*x^(2/3)]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqr t[b] + Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3) + Sqrt[a]*x^(2/3)]/(2*Sqrt[2]*a^(1/ 4)*b^(1/4)))/(2*Sqrt[a])))/a))/(4*a)))/(8*a))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.36 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {x}{a^{3}}-\frac {3 b \left (\frac {-\frac {19 a \,x^{\frac {7}{3}}}{32}-\frac {15 b x}{32}}{\left (b +a \,x^{\frac {4}{3}}\right )^{2}}+\frac {77 \sqrt {2}\, \left (\ln \left (\frac {x^{\frac {2}{3}}-\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {b}{a}}}{x^{\frac {2}{3}}+\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {b}{a}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{a^{3}}\) | \(144\) |
| default | \(\frac {x}{a^{3}}-\frac {3 b \left (\frac {-\frac {19 a \,x^{\frac {7}{3}}}{32}-\frac {15 b x}{32}}{\left (b +a \,x^{\frac {4}{3}}\right )^{2}}+\frac {77 \sqrt {2}\, \left (\ln \left (\frac {x^{\frac {2}{3}}-\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {b}{a}}}{x^{\frac {2}{3}}+\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {b}{a}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{a^{3}}\) | \(144\) |
Input:
int(1/(a+b/x^(4/3))^3,x,method=_RETURNVERBOSE)
Output:
x/a^3-3/a^3*b*((-19/32*a*x^(7/3)-15/32*b*x)/(b+a*x^(4/3))^2+77/256/a/(b/a) ^(1/4)*2^(1/2)*(ln((x^(2/3)-(b/a)^(1/4)*x^(1/3)*2^(1/2)+(b/a)^(1/2))/(x^(2 /3)+(b/a)^(1/4)*x^(1/3)*2^(1/2)+(b/a)^(1/2)))+2*arctan(2^(1/2)/(b/a)^(1/4) *x^(1/3)+1)+2*arctan(2^(1/2)/(b/a)^(1/4)*x^(1/3)-1)))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.87 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx=\frac {128 \, a^{6} x^{9} + 580 \, a^{3} b^{3} x^{5} + 308 \, b^{6} x - 231 \, {\left (a^{9} x^{8} + 2 \, a^{6} b^{3} x^{4} + a^{3} b^{6}\right )} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} \log \left (12326391 \, a^{11} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {3}{4}} + 12326391 \, b^{2} x^{\frac {1}{3}}\right ) - 231 \, {\left (-i \, a^{9} x^{8} - 2 i \, a^{6} b^{3} x^{4} - i \, a^{3} b^{6}\right )} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} \log \left (12326391 i \, a^{11} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {3}{4}} + 12326391 \, b^{2} x^{\frac {1}{3}}\right ) - 231 \, {\left (i \, a^{9} x^{8} + 2 i \, a^{6} b^{3} x^{4} + i \, a^{3} b^{6}\right )} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-12326391 i \, a^{11} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {3}{4}} + 12326391 \, b^{2} x^{\frac {1}{3}}\right ) + 231 \, {\left (a^{9} x^{8} + 2 \, a^{6} b^{3} x^{4} + a^{3} b^{6}\right )} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-12326391 \, a^{11} \left (-\frac {b^{3}}{a^{15}}\right )^{\frac {3}{4}} + 12326391 \, b^{2} x^{\frac {1}{3}}\right ) + 12 \, {\left (19 \, a^{5} b x^{7} + 7 \, a^{2} b^{4} x^{3}\right )} x^{\frac {2}{3}} - 12 \, {\left (23 \, a^{4} b^{2} x^{6} + 11 \, a b^{5} x^{2}\right )} x^{\frac {1}{3}}}{128 \, {\left (a^{9} x^{8} + 2 \, a^{6} b^{3} x^{4} + a^{3} b^{6}\right )}} \] Input:
integrate(1/(a+b/x^(4/3))^3,x, algorithm="fricas")
Output:
1/128*(128*a^6*x^9 + 580*a^3*b^3*x^5 + 308*b^6*x - 231*(a^9*x^8 + 2*a^6*b^ 3*x^4 + a^3*b^6)*(-b^3/a^15)^(1/4)*log(12326391*a^11*(-b^3/a^15)^(3/4) + 1 2326391*b^2*x^(1/3)) - 231*(-I*a^9*x^8 - 2*I*a^6*b^3*x^4 - I*a^3*b^6)*(-b^ 3/a^15)^(1/4)*log(12326391*I*a^11*(-b^3/a^15)^(3/4) + 12326391*b^2*x^(1/3) ) - 231*(I*a^9*x^8 + 2*I*a^6*b^3*x^4 + I*a^3*b^6)*(-b^3/a^15)^(1/4)*log(-1 2326391*I*a^11*(-b^3/a^15)^(3/4) + 12326391*b^2*x^(1/3)) + 231*(a^9*x^8 + 2*a^6*b^3*x^4 + a^3*b^6)*(-b^3/a^15)^(1/4)*log(-12326391*a^11*(-b^3/a^15)^ (3/4) + 12326391*b^2*x^(1/3)) + 12*(19*a^5*b*x^7 + 7*a^2*b^4*x^3)*x^(2/3) - 12*(23*a^4*b^2*x^6 + 11*a*b^5*x^2)*x^(1/3))/(a^9*x^8 + 2*a^6*b^3*x^4 + a ^3*b^6)
Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b/x**(4/3))**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx=\frac {32 \, a^{2} + \frac {121 \, a b}{x^{\frac {4}{3}}} + \frac {77 \, b^{2}}{x^{\frac {8}{3}}}}{32 \, {\left (\frac {a^{5}}{x} + \frac {2 \, a^{4} b}{x^{\frac {7}{3}}} + \frac {a^{3} b^{2}}{x^{\frac {11}{3}}}\right )}} + \frac {231 \, {\left (\frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + \frac {2 \, \sqrt {b}}{x^{\frac {1}{3}}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - \frac {2 \, \sqrt {b}}{x^{\frac {1}{3}}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\frac {\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}}{x^{\frac {1}{3}}} + \sqrt {a} + \frac {\sqrt {b}}{x^{\frac {2}{3}}}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (-\frac {\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}}{x^{\frac {1}{3}}} + \sqrt {a} + \frac {\sqrt {b}}{x^{\frac {2}{3}}}\right )}{a^{\frac {3}{4}}}\right )}}{256 \, a^{3}} \] Input:
integrate(1/(a+b/x^(4/3))^3,x, algorithm="maxima")
Output:
1/32*(32*a^2 + 121*a*b/x^(4/3) + 77*b^2/x^(8/3))/(a^5/x + 2*a^4*b/x^(7/3) + a^3*b^2/x^(11/3)) + 231/256*(2*sqrt(2)*b*arctan(1/2*sqrt(2)*(sqrt(2)*a^( 1/4)*b^(1/4) + 2*sqrt(b)/x^(1/3))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqr t(a)*sqrt(b))) + 2*sqrt(2)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)/x^(1/3))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b)) ) + sqrt(2)*b^(3/4)*log(sqrt(2)*a^(1/4)*b^(1/4)/x^(1/3) + sqrt(a) + sqrt(b )/x^(2/3))/a^(3/4) - sqrt(2)*b^(3/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)/x^(1/3) + sqrt(a) + sqrt(b)/x^(2/3))/a^(3/4))/a^3
Time = 0.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx=\frac {x}{a^{3}} - \frac {231 \, \sqrt {2} \left (a^{3} b\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{a}\right )^{\frac {1}{4}} + 2 \, x^{\frac {1}{3}}\right )}}{2 \, \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{128 \, a^{6}} - \frac {231 \, \sqrt {2} \left (a^{3} b\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{a}\right )^{\frac {1}{4}} - 2 \, x^{\frac {1}{3}}\right )}}{2 \, \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{128 \, a^{6}} + \frac {231 \, \sqrt {2} \left (a^{3} b\right )^{\frac {3}{4}} \log \left (\sqrt {2} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{4}} + x^{\frac {2}{3}} + \sqrt {\frac {b}{a}}\right )}{256 \, a^{6}} - \frac {231 \, \sqrt {2} \left (a^{3} b\right )^{\frac {3}{4}} \log \left (-\sqrt {2} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{4}} + x^{\frac {2}{3}} + \sqrt {\frac {b}{a}}\right )}{256 \, a^{6}} + \frac {3 \, {\left (19 \, a b x^{\frac {7}{3}} + 15 \, b^{2} x\right )}}{32 \, {\left (a x^{\frac {4}{3}} + b\right )}^{2} a^{3}} \] Input:
integrate(1/(a+b/x^(4/3))^3,x, algorithm="giac")
Output:
x/a^3 - 231/128*sqrt(2)*(a^3*b)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/a)^(1 /4) + 2*x^(1/3))/(b/a)^(1/4))/a^6 - 231/128*sqrt(2)*(a^3*b)^(3/4)*arctan(- 1/2*sqrt(2)*(sqrt(2)*(b/a)^(1/4) - 2*x^(1/3))/(b/a)^(1/4))/a^6 + 231/256*s qrt(2)*(a^3*b)^(3/4)*log(sqrt(2)*x^(1/3)*(b/a)^(1/4) + x^(2/3) + sqrt(b/a) )/a^6 - 231/256*sqrt(2)*(a^3*b)^(3/4)*log(-sqrt(2)*x^(1/3)*(b/a)^(1/4) + x ^(2/3) + sqrt(b/a))/a^6 + 3/32*(19*a*b*x^(7/3) + 15*b^2*x)/((a*x^(4/3) + b )^2*a^3)
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx=\frac {\frac {45\,b^2\,x}{32}+\frac {57\,a\,b\,x^{7/3}}{32}}{a^3\,b^2+a^5\,x^{8/3}+2\,a^4\,b\,x^{4/3}}+\frac {x}{a^3}+\frac {231\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {a^{1/4}\,x^{1/3}}{{\left (-b\right )}^{1/4}}\right )}{64\,a^{15/4}}+\frac {{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {a^{1/4}\,x^{1/3}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,231{}\mathrm {i}}{64\,a^{15/4}} \] Input:
int(1/(a + b/x^(4/3))^3,x)
Output:
((45*b^2*x)/32 + (57*a*b*x^(7/3))/32)/(a^3*b^2 + a^5*x^(8/3) + 2*a^4*b*x^( 4/3)) + x/a^3 + (231*(-b)^(3/4)*atan((a^(1/4)*x^(1/3))/(-b)^(1/4)))/(64*a^ (15/4)) + ((-b)^(3/4)*atan((a^(1/4)*x^(1/3)*1i)/(-b)^(1/4))*231i)/(64*a^(1 5/4))
Time = 0.18 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(a+b/x^(4/3))^3,x)
Output:
(462*x**(2/3)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*x**(1/3)*sqrt(a))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*x**2 + 924*x**(1/3)* b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*x**(1/3)*sqr t(a))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*x + 462*b**(3/4)*a**(1/4)*sqrt(2)*a tan((b**(1/4)*a**(1/4)*sqrt(2) - 2*x**(1/3)*sqrt(a))/(b**(1/4)*a**(1/4)*sq rt(2)))*b**2 - 462*x**(2/3)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1 /4)*sqrt(2) + 2*x**(1/3)*sqrt(a))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*x**2 - 924*x**(1/3)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*x**(1/3)*sqrt(a))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*x - 462*b**(3/4)*a**( 1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*x**(1/3)*sqrt(a))/(b**(1/ 4)*a**(1/4)*sqrt(2)))*b**2 - 231*x**(2/3)*b**(3/4)*a**(1/4)*sqrt(2)*log( - x**(1/3)*b**(1/4)*a**(1/4)*sqrt(2) + x**(2/3)*sqrt(a) + sqrt(b))*a**2*x** 2 + 231*x**(2/3)*b**(3/4)*a**(1/4)*sqrt(2)*log(x**(1/3)*b**(1/4)*a**(1/4)* sqrt(2) + x**(2/3)*sqrt(a) + sqrt(b))*a**2*x**2 - 462*x**(1/3)*b**(3/4)*a* *(1/4)*sqrt(2)*log( - x**(1/3)*b**(1/4)*a**(1/4)*sqrt(2) + x**(2/3)*sqrt(a ) + sqrt(b))*a*b*x + 462*x**(1/3)*b**(3/4)*a**(1/4)*sqrt(2)*log(x**(1/3)*b **(1/4)*a**(1/4)*sqrt(2) + x**(2/3)*sqrt(a) + sqrt(b))*a*b*x - 231*b**(3/4 )*a**(1/4)*sqrt(2)*log( - x**(1/3)*b**(1/4)*a**(1/4)*sqrt(2) + x**(2/3)*sq rt(a) + sqrt(b))*b**2 + 231*b**(3/4)*a**(1/4)*sqrt(2)*log(x**(1/3)*b**(1/4 )*a**(1/4)*sqrt(2) + x**(2/3)*sqrt(a) + sqrt(b))*b**2 + 256*x**(2/3)*a*...