Integrand size = 11, antiderivative size = 173 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx=\frac {x}{a^2}+\frac {3 b x}{4 a^2 \left (b+a x^{4/3}\right )}+\frac {21 b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{8 \sqrt {2} a^{11/4}}-\frac {21 b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{8 \sqrt {2} a^{11/4}}+\frac {21 b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt {b}+\sqrt {a} x^{2/3}}\right )}{8 \sqrt {2} a^{11/4}} \] Output:
x/a^2+3/4*b*x/a^2/(b+a*x^(4/3))-21/16*b^(3/4)*arctan(-1+2^(1/2)*a^(1/4)*x^ (1/3)/b^(1/4))*2^(1/2)/a^(11/4)-21/16*b^(3/4)*arctan(1+2^(1/2)*a^(1/4)*x^( 1/3)/b^(1/4))*2^(1/2)/a^(11/4)+21/16*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^(11/4)
Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx=\frac {\frac {4 a^{3/4} x \left (7 b+4 a x^{4/3}\right )}{b+a x^{4/3}}-21 \sqrt {2} b^{3/4} \arctan \left (\frac {-\sqrt {b}+\sqrt {a} x^{2/3}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}\right )+21 \sqrt {2} b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt {b}+\sqrt {a} x^{2/3}}\right )}{16 a^{11/4}} \] Input:
Integrate[(a + b/x^(4/3))^(-2),x]
Output:
((4*a^(3/4)*x*(7*b + 4*a*x^(4/3)))/(b + a*x^(4/3)) - 21*Sqrt[2]*b^(3/4)*Ar cTan[(-Sqrt[b] + Sqrt[a]*x^(2/3))/(Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3))] + 21* Sqrt[2]*b^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3))/(Sqrt[b] + Sqrt[ a]*x^(2/3))])/(16*a^(11/4))
Time = 0.73 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.55, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {774, 795, 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.
| \(\displaystyle \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 3 \int \frac {x^{2/3}}{\left (a+\frac {b}{x^{4/3}}\right )^2}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle 3 \int \frac {x^{10/3}}{\left (a x^{4/3}+b\right )^2}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 3 \left (\frac {7 \int \frac {x^2}{a x^{4/3}+b}d\sqrt [3]{x}}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \int \frac {x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\int \frac {\sqrt {a} x^{2/3}+\sqrt {b}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\int \frac {1}{x^{2/3}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}d\sqrt [3]{x}}{2 \sqrt {a}}+\frac {\int \frac {1}{x^{2/3}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}d\sqrt [3]{x}}{2 \sqrt {a}}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\int \frac {1}{-x^{2/3}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^{2/3}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{a x^{4/3}+b}d\sqrt [3]{x}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{a} \left (x^{2/3}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}+\sqrt [4]{b}\right )}{\sqrt [4]{a} \left (x^{2/3}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{a} \left (x^{2/3}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}+\sqrt [4]{b}\right )}{\sqrt [4]{a} \left (x^{2/3}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} \sqrt [3]{x}}{x^{2/3}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt {a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}+\sqrt [4]{b}}{x^{2/3}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}d\sqrt [3]{x}}{2 \sqrt {a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 3 \left (\frac {7 \left (\frac {x}{3 a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}+\sqrt {a} x^{2/3}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}+\sqrt {a} x^{2/3}+\sqrt {b}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}\right )}{4 a}-\frac {x^{7/3}}{4 a \left (a x^{4/3}+b\right )}\right )\) |
Input:
Int[(a + b/x^(4/3))^(-2),x]
Output:
3*(-1/4*x^(7/3)/(a*(b + a*x^(4/3))) + (7*(x/(3*a) - (b*((-(ArcTan[1 - (Sqr t[2]*a^(1/4)*x^(1/3))/b^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sq rt[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)]/(Sqr t[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[b] + Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3) + Sq rt[a]*x^(2/3)]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/a))/(4*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 = 0.37 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {x}{a^{2}}-\frac {3 b \left (-\frac {x}{4 \left (b +a \,x^{\frac {4}{3}}\right )}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {x^{\frac {2}{3}}-\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {b}{a}}}{x^{\frac {2}{3}}+\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {b}{a}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{a^{2}}\) | \(135\) |
| default | \(\frac {x}{a^{2}}-\frac {3 b \left (-\frac {x}{4 \left (b +a \,x^{\frac {4}{3}}\right )}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {x^{\frac {2}{3}}-\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {b}{a}}}{x^{\frac {2}{3}}+\left (\frac {b}{a}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {b}{a}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{a^{2}}\) | \(135\) |
Input:
int(1/(a+b/x^(4/3))^2,x,method=_RETURNVERBOSE)
Output:
1/a^2*x-3*b/a^2*(-1/4*x/(b+a*x^(4/3))+7/32/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 = 264, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx=\frac {16 \, a^{3} x^{5} + 12 \, a^{2} b x^{\frac {11}{3}} - 12 \, a b^{2} x^{\frac {7}{3}} + 28 \, b^{3} x - 21 \, {\left (a^{5} x^{4} + a^{2} b^{3}\right )} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (9261 \, a^{8} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {3}{4}} + 9261 \, b^{2} x^{\frac {1}{3}}\right ) - 21 \, {\left (-i \, a^{5} x^{4} - i \, a^{2} b^{3}\right )} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (9261 i \, a^{8} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {3}{4}} + 9261 \, b^{2} x^{\frac {1}{3}}\right ) - 21 \, {\left (i \, a^{5} x^{4} + i \, a^{2} b^{3}\right )} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-9261 i \, a^{8} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {3}{4}} + 9261 \, b^{2} x^{\frac {1}{3}}\right ) + 21 \, {\left (a^{5} x^{4} + a^{2} b^{3}\right )} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-9261 \, a^{8} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {3}{4}} + 9261 \, b^{2} x^{\frac {1}{3}}\right )}{16 \, {\left (a^{5} x^{4} + a^{2} b^{3}\right )}} \] Input:
integrate(1/(a+b/x^(4/3))^2,x, algorithm="fricas")
Output:
1/16*(16*a^3*x^5 + 12*a^2*b*x^(11/3) - 12*a*b^2*x^(7/3) + 28*b^3*x - 21*(a ^5*x^4 + a^2*b^3)*(-b^3/a^11)^(1/4)*log(9261*a^8*(-b^3/a^11)^(3/4) + 9261* b^2*x^(1/3)) - 21*(-I*a^5*x^4 - I*a^2*b^3)*(-b^3/a^11)^(1/4)*log(9261*I*a^ 8*(-b^3/a^11)^(3/4) + 9261*b^2*x^(1/3)) - 21*(I*a^5*x^4 + I*a^2*b^3)*(-b^3 /a^11)^(1/4)*log(-9261*I*a^8*(-b^3/a^11)^(3/4) + 9261*b^2*x^(1/3)) + 21*(a ^5*x^4 + a^2*b^3)*(-b^3/a^11)^(1/4)*log(-9261*a^8*(-b^3/a^11)^(3/4) + 9261 *b^2*x^(1/3)))/(a^5*x^4 + a^2*b^3)
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (163) = 326\).
Time = 29.95 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.68 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx=\begin {cases} \tilde {\infty } x^{\frac {11}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {11}{3}}}{11 b^{2}} & \text {for}\: a = 0 \\\frac {x}{a^{2}} & \text {for}\: b = 0 \\\frac {16 a^{2} x^{\frac {7}{3}} \sqrt [4]{- \frac {b}{a}}}{16 a^{4} x^{\frac {4}{3}} \sqrt [4]{- \frac {b}{a}} + 16 a^{3} b \sqrt [4]{- \frac {b}{a}}} - \frac {21 a b x^{\frac {4}{3}} \log {\left (\sqrt [3]{x} - \sqrt [4]{- \frac {b}{a}} \right )}}{16 a^{4} x^{\frac {4}{3}} \sqrt [4]{- \frac {b}{a}} + 16 a^{3} b \sqrt [4]{- \frac {b}{a}}} + \frac {21 a b x^{\frac {4}{3}} \log {\left (\sqrt [3]{x} + \sqrt [4]{- \frac {b}{a}} \right )}}{16 a^{4} x^{\frac {4}{3}} \sqrt [4]{- \frac {b}{a}} + 16 a^{3} b \sqrt [4]{- \frac {b}{a}}} - \frac {42 a b x^{\frac {4}{3}} \operatorname {atan}{\left (\frac {\sqrt [3]{x}}{\sqrt [4]{- \frac {b}{a}}} \right )}}{16 a^{4} x^{\frac {4}{3}} \sqrt [4]{- \frac {b}{a}} + 16 a^{3} b \sqrt [4]{- \frac {b}{a}}} + \frac {28 a b x \sqrt [4]{- \frac {b}{a}}}{16 a^{4} x^{\frac {4}{3}} \sqrt [4]{- \frac {b}{a}} + 16 a^{3} b \sqrt [4]{- \frac {b}{a}}} - \frac {21 b^{2} \log {\left (\sqrt [3]{x} - \sqrt [4]{- \frac {b}{a}} \right )}}{16 a^{4} x^{\frac {4}{3}} \sqrt [4]{- \frac {b}{a}} + 16 a^{3} b \sqrt [4]{- \frac {b}{a}}} + \frac {21 b^{2} \log {\left (\sqrt [3]{x} + \sqrt [4]{- \frac {b}{a}} \right )}}{16 a^{4} x^{\frac {4}{3}} \sqrt [4]{- \frac {b}{a}} + 16 a^{3} b \sqrt [4]{- \frac {b}{a}}} - \frac {42 b^{2} \operatorname {atan}{\left (\frac {\sqrt [3]{x}}{\sqrt [4]{- \frac {b}{a}}} \right )}}{16 a^{4} x^{\frac {4}{3}} \sqrt [4]{- \frac {b}{a}} + 16 a^{3} b \sqrt [4]{- \frac {b}{a}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b/x**(4/3))**2,x)
Output:
Piecewise((zoo*x**(11/3), Eq(a, 0) & Eq(b, 0)), (3*x**(11/3)/(11*b**2), Eq (a, 0)), (x/a**2, Eq(b, 0)), (16*a**2*x**(7/3)*(-b/a)**(1/4)/(16*a**4*x**( 4/3)*(-b/a)**(1/4) + 16*a**3*b*(-b/a)**(1/4)) - 21*a*b*x**(4/3)*log(x**(1/ 3) - (-b/a)**(1/4))/(16*a**4*x**(4/3)*(-b/a)**(1/4) + 16*a**3*b*(-b/a)**(1 /4)) + 21*a*b*x**(4/3)*log(x**(1/3) + (-b/a)**(1/4))/(16*a**4*x**(4/3)*(-b /a)**(1/4) + 16*a**3*b*(-b/a)**(1/4)) - 42*a*b*x**(4/3)*atan(x**(1/3)/(-b/ a)**(1/4))/(16*a**4*x**(4/3)*(-b/a)**(1/4) + 16*a**3*b*(-b/a)**(1/4)) + 28 *a*b*x*(-b/a)**(1/4)/(16*a**4*x**(4/3)*(-b/a)**(1/4) + 16*a**3*b*(-b/a)**( 1/4)) - 21*b**2*log(x**(1/3) - (-b/a)**(1/4))/(16*a**4*x**(4/3)*(-b/a)**(1 /4) + 16*a**3*b*(-b/a)**(1/4)) + 21*b**2*log(x**(1/3) + (-b/a)**(1/4))/(16 *a**4*x**(4/3)*(-b/a)**(1/4) + 16*a**3*b*(-b/a)**(1/4)) - 42*b**2*atan(x** (1/3)/(-b/a)**(1/4))/(16*a**4*x**(4/3)*(-b/a)**(1/4) + 16*a**3*b*(-b/a)**( 1/4)), True))
Time = 0.12 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx=\frac {4 \, a + \frac {7 \, b}{x^{\frac {4}{3}}}}{4 \, {\left (\frac {a^{3}}{x} + \frac {a^{2} b}{x^{\frac {7}{3}}}\right )}} + \frac {21 \, {\left (\frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + \frac {2 \, \sqrt {b}}{x^{\frac {1}{3}}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - \frac {2 \, \sqrt {b}}{x^{\frac {1}{3}}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\frac {\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}}{x^{\frac {1}{3}}} + \sqrt {a} + \frac {\sqrt {b}}{x^{\frac {2}{3}}}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (-\frac {\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}}{x^{\frac {1}{3}}} + \sqrt {a} + \frac {\sqrt {b}}{x^{\frac {2}{3}}}\right )}{a^{\frac {3}{4}}}\right )}}{32 \, a^{2}} \] Input:
integrate(1/(a+b/x^(4/3))^2,x, algorithm="maxima")
Output:
1/4*(4*a + 7*b/x^(4/3))/(a^3/x + a^2*b/x^(7/3)) + 21/32*(2*sqrt(2)*b*arcta n(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)/x^(1/3))/sqrt(sqrt(a)*s qrt(b)))/(sqrt(a)*sqrt(sqrt(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)))/(sqr t(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^2
Time = 0.13 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx=\frac {x}{a^{2}} + \frac {3 \, b x}{4 \, {\left (a x^{\frac {4}{3}} + b\right )} a^{2}} - \frac {21 \, \sqrt {2} \left (a^{3} b\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{a}\right )^{\frac {1}{4}} + 2 \, x^{\frac {1}{3}}\right )}}{2 \, \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{16 \, a^{5}} - \frac {21 \, \sqrt {2} \left (a^{3} b\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{a}\right )^{\frac {1}{4}} - 2 \, x^{\frac {1}{3}}\right )}}{2 \, \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{16 \, a^{5}} + \frac {21 \, \sqrt {2} \left (a^{3} b\right )^{\frac {3}{4}} \log \left (\sqrt {2} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{4}} + x^{\frac {2}{3}} + \sqrt {\frac {b}{a}}\right )}{32 \, a^{5}} - \frac {21 \, \sqrt {2} \left (a^{3} b\right )^{\frac {3}{4}} \log \left (-\sqrt {2} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{4}} + x^{\frac {2}{3}} + \sqrt {\frac {b}{a}}\right )}{32 \, a^{5}} \] Input:
integrate(1/(a+b/x^(4/3))^2,x, algorithm="giac")
Output:
x/a^2 + 3/4*b*x/((a*x^(4/3) + b)*a^2) - 21/16*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^5 - 21/16*sq rt(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^5 + 21/32*sqrt(2)*(a^3*b)^(3/4)*log(sqrt(2)*x^(1/3)*(b/a)^( 1/4) + x^(2/3) + sqrt(b/a))/a^5 - 21/32*sqrt(2)*(a^3*b)^(3/4)*log(-sqrt(2) *x^(1/3)*(b/a)^(1/4) + x^(2/3) + sqrt(b/a))/a^5
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx=\frac {x}{a^2}+\frac {3\,b\,x}{4\,\left (a^2\,b+a^3\,x^{4/3}\right )}+\frac {21\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {a^{1/4}\,x^{1/3}}{{\left (-b\right )}^{1/4}}\right )}{8\,a^{11/4}}+\frac {{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {a^{1/4}\,x^{1/3}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,21{}\mathrm {i}}{8\,a^{11/4}} \] Input:
int(1/(a + b/x^(4/3))^2,x)
Output:
x/a^2 + (3*b*x)/(4*(a^2*b + a^3*x^(4/3))) + (21*(-b)^(3/4)*atan((a^(1/4)*x ^(1/3))/(-b)^(1/4)))/(8*a^(11/4)) + ((-b)^(3/4)*atan((a^(1/4)*x^(1/3)*1i)/ (-b)^(1/4))*21i)/(8*a^(11/4))
Time = 0.19 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (a+\frac {b}{x^{4/3}}\right )^2} \, dx=\frac {42 x^{\frac {4}{3}} b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 x^{\frac {1}{3}} \sqrt {a}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+42 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 x^{\frac {1}{3}} \sqrt {a}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-42 x^{\frac {4}{3}} b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 x^{\frac {1}{3}} \sqrt {a}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-42 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 x^{\frac {1}{3}} \sqrt {a}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-21 x^{\frac {4}{3}} b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+x^{\frac {2}{3}} \sqrt {a}+\sqrt {b}\right )+21 x^{\frac {4}{3}} b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+x^{\frac {2}{3}} \sqrt {a}+\sqrt {b}\right )-21 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+x^{\frac {2}{3}} \sqrt {a}+\sqrt {b}\right )+21 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+x^{\frac {2}{3}} \sqrt {a}+\sqrt {b}\right )+32 x^{\frac {7}{3}} a^{2}+56 a b x}{32 a^{3} \left (x^{\frac {4}{3}} a +b \right )} \] Input:
int(1/(a+b/x^(4/3))^2,x)
Output:
(42*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*x + 42*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 - 42*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*x - 42*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 - 21*x**(1/3)*b**(3/4)*a**(1/4)*s qrt(2)*log( - x**(1/3)*b**(1/4)*a**(1/4)*sqrt(2) + x**(2/3)*sqrt(a) + sqrt (b))*a*x + 21*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*x - 21*b**(3/4)*a**(1/4)*sqr t(2)*log( - x**(1/3)*b**(1/4)*a**(1/4)*sqrt(2) + x**(2/3)*sqrt(a) + sqrt(b ))*b + 21*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 + 32*x**(1/3)*a**2*x**2 + 56*a*b*x)/(32*a **3*(x**(1/3)*a*x + b))