Integrand size = 11, antiderivative size = 186 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx=\frac {3 x}{8 a \left (a+b x^{4/3}\right )^2}+\frac {15 x}{32 a^2 \left (a+b x^{4/3}\right )}-\frac {15 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{9/4} b^{3/4}}+\frac {15 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{9/4} b^{3/4}}-\frac {15 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt {a}+\sqrt {b} x^{2/3}}\right )}{64 \sqrt {2} a^{9/4} b^{3/4}} \] Output:
3/8*x/a/(a+b*x^(4/3))^2+15/32*x/a^2/(a+b*x^(4/3))-15/128*arctan(1-2^(1/2)* b^(1/4)*x^(1/3)/a^(1/4))*2^(1/2)/a^(9/4)/b^(3/4)+15/128*arctan(1+2^(1/2)*b ^(1/4)*x^(1/3)/a^(1/4))*2^(1/2)/a^(9/4)/b^(3/4)-15/128*arctanh(2^(1/2)*a^( 1/4)*b^(1/4)*x^(1/3)/(a^(1/2)+b^(1/2)*x^(2/3)))*2^(1/2)/a^(9/4)/b^(3/4)
Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx=\frac {3 \left (\frac {4 \sqrt [4]{a} x \left (9 a+5 b x^{4/3}\right )}{\left (a+b x^{4/3}\right )^2}-\frac {5 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x^{2/3}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}\right )}{b^{3/4}}-\frac {5 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt {a}+\sqrt {b} x^{2/3}}\right )}{b^{3/4}}\right )}{128 a^{9/4}} \] Input:
Integrate[(a + b*x^(4/3))^(-3),x]
Output:
(3*((4*a^(1/4)*x*(9*a + 5*b*x^(4/3)))/(a + b*x^(4/3))^2 - (5*Sqrt[2]*ArcTa n[(Sqrt[a] - Sqrt[b]*x^(2/3))/(Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3))])/b^(3/4) - (5*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3))/(Sqrt[a] + Sqrt[b]* x^(2/3))])/b^(3/4)))/(128*a^(9/4))
Time = 0.72 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.48, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {774, 819, 819, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 3 \int \frac {x^{2/3}}{\left (b x^{4/3}+a\right )^3}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle 3 \left (\frac {5 \int \frac {x^{2/3}}{\left (b x^{4/3}+a\right )^2}d\sqrt [3]{x}}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\int \frac {x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\frac {\int \frac {\sqrt {b} x^{2/3}+\sqrt {a}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\frac {\frac {\int \frac {1}{x^{2/3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt [3]{x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x^{2/3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt [3]{x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\frac {\frac {\int \frac {1}{-x^{2/3}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^{2/3}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{b} \left (x^{2/3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^{2/3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{b} \left (x^{2/3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^{2/3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt [3]{x}}{x^{2/3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}+\sqrt [4]{a}}{x^{2/3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt [3]{x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 3 \left (\frac {5 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}+\sqrt {a}+\sqrt {b} x^{2/3}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}+\sqrt {a}+\sqrt {b} x^{2/3}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{4 a}+\frac {x}{4 a \left (a+b x^{4/3}\right )}\right )}{8 a}+\frac {x}{8 a \left (a+b x^{4/3}\right )^2}\right )\) |
Input:
Int[(a + b*x^(4/3))^(-3),x]
Output:
3*(x/(8*a*(a + b*x^(4/3))^2) + (5*(x/(4*a*(a + b*x^(4/3))) + ((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x^(1/3))/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x^(1/3))/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b ]) - (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3) + Sqrt[b]*x^(2/3) ]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3 ) + Sqrt[b]*x^(2/3)]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b]))/(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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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[((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 = 0.39 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(\frac {3 x}{8 a \left (a +b \,x^{\frac {4}{3}}\right )^{2}}+\frac {\frac {15 x}{32 a \left (a +b \,x^{\frac {4}{3}}\right )}+\frac {15 \sqrt {2}\, \left (\ln \left (\frac {x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{\frac {2}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a}\) | \(150\) |
| default | \(\text {Expression too large to display}\) | \(1585\) |
Input:
int(1/(a+b*x^(4/3))^3,x,method=_RETURNVERBOSE)
Output:
3/8*x/a/(a+b*x^(4/3))^2+15/8/a*(1/4*x/a/(a+b*x^(4/3))+1/32/a/b/(a/b)^(1/4) *2^(1/2)*(ln((x^(2/3)-(a/b)^(1/4)*x^(1/3)*2^(1/2)+(a/b)^(1/2))/(x^(2/3)+(a /b)^(1/4)*x^(1/3)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/ 3)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/3)-1)))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.83 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx=-\frac {3 \, {\left (12 \, a^{2} b^{3} x^{5} - 36 \, a^{5} x - 5 \, {\left (a^{2} b^{6} x^{8} + 2 \, a^{5} b^{3} x^{4} + a^{8}\right )} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (a^{7} b^{2} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {3}{4}} + x^{\frac {1}{3}}\right ) + 5 \, {\left (i \, a^{2} b^{6} x^{8} + 2 i \, a^{5} b^{3} x^{4} + i \, a^{8}\right )} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (i \, a^{7} b^{2} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {3}{4}} + x^{\frac {1}{3}}\right ) + 5 \, {\left (-i \, a^{2} b^{6} x^{8} - 2 i \, a^{5} b^{3} x^{4} - i \, a^{8}\right )} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (-i \, a^{7} b^{2} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {3}{4}} + x^{\frac {1}{3}}\right ) + 5 \, {\left (a^{2} b^{6} x^{8} + 2 \, a^{5} b^{3} x^{4} + a^{8}\right )} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {1}{4}} \log \left (-a^{7} b^{2} \left (-\frac {1}{a^{9} b^{3}}\right )^{\frac {3}{4}} + x^{\frac {1}{3}}\right ) - 4 \, {\left (5 \, b^{5} x^{7} + 17 \, a^{3} b^{2} x^{3}\right )} x^{\frac {2}{3}} + 4 \, {\left (a b^{4} x^{6} + 13 \, a^{4} b x^{2}\right )} x^{\frac {1}{3}}\right )}}{128 \, {\left (a^{2} b^{6} x^{8} + 2 \, a^{5} b^{3} x^{4} + a^{8}\right )}} \] Input:
integrate(1/(a+b*x^(4/3))^3,x, algorithm="fricas")
Output:
-3/128*(12*a^2*b^3*x^5 - 36*a^5*x - 5*(a^2*b^6*x^8 + 2*a^5*b^3*x^4 + a^8)* (-1/(a^9*b^3))^(1/4)*log(a^7*b^2*(-1/(a^9*b^3))^(3/4) + x^(1/3)) + 5*(I*a^ 2*b^6*x^8 + 2*I*a^5*b^3*x^4 + I*a^8)*(-1/(a^9*b^3))^(1/4)*log(I*a^7*b^2*(- 1/(a^9*b^3))^(3/4) + x^(1/3)) + 5*(-I*a^2*b^6*x^8 - 2*I*a^5*b^3*x^4 - I*a^ 8)*(-1/(a^9*b^3))^(1/4)*log(-I*a^7*b^2*(-1/(a^9*b^3))^(3/4) + x^(1/3)) + 5 *(a^2*b^6*x^8 + 2*a^5*b^3*x^4 + a^8)*(-1/(a^9*b^3))^(1/4)*log(-a^7*b^2*(-1 /(a^9*b^3))^(3/4) + x^(1/3)) - 4*(5*b^5*x^7 + 17*a^3*b^2*x^3)*x^(2/3) + 4* (a*b^4*x^6 + 13*a^4*b*x^2)*x^(1/3))/(a^2*b^6*x^8 + 2*a^5*b^3*x^4 + a^8)
Timed out. \[ \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*x**(4/3))**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx=\frac {3 \, {\left (5 \, b x^{\frac {7}{3}} + 9 \, a x\right )}}{32 \, {\left (a^{2} b^{2} x^{\frac {8}{3}} + 2 \, a^{3} b x^{\frac {4}{3}} + a^{4}\right )}} + \frac {15 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} x^{\frac {1}{3}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} x^{\frac {1}{3}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{\frac {1}{3}} + \sqrt {b} x^{\frac {2}{3}} + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{\frac {1}{3}} + \sqrt {b} x^{\frac {2}{3}} + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{256 \, a^{2}} \] Input:
integrate(1/(a+b*x^(4/3))^3,x, algorithm="maxima")
Output:
3/32*(5*b*x^(7/3) + 9*a*x)/(a^2*b^2*x^(8/3) + 2*a^3*b*x^(4/3) + a^4) + 15/ 256*(2*sqrt(2)*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(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*a rctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*x^(1/3))/sqrt(sqrt (a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4 )*b^(1/4)*x^(1/3) + sqrt(b)*x^(2/3) + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2) *log(-sqrt(2)*a^(1/4)*b^(1/4)*x^(1/3) + sqrt(b)*x^(2/3) + sqrt(a))/(a^(1/4 )*b^(3/4)))/a^2
Time = 0.13 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx=\frac {3 \, {\left (5 \, b x^{\frac {7}{3}} + 9 \, a x\right )}}{32 \, {\left (b x^{\frac {4}{3}} + a\right )}^{2} a^{2}} + \frac {15 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, x^{\frac {1}{3}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {15 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, x^{\frac {1}{3}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} - \frac {15 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x^{\frac {2}{3}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} + \frac {15 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x^{\frac {2}{3}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} \] Input:
integrate(1/(a+b*x^(4/3))^3,x, algorithm="giac")
Output:
3/32*(5*b*x^(7/3) + 9*a*x)/((b*x^(4/3) + a)^2*a^2) + 15/128*sqrt(2)*(a*b^3 )^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*x^(1/3))/(a/b)^(1/4))/ (a^3*b^3) + 15/128*sqrt(2)*(a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b )^(1/4) - 2*x^(1/3))/(a/b)^(1/4))/(a^3*b^3) - 15/256*sqrt(2)*(a*b^3)^(3/4) *log(sqrt(2)*x^(1/3)*(a/b)^(1/4) + x^(2/3) + sqrt(a/b))/(a^3*b^3) + 15/256 *sqrt(2)*(a*b^3)^(3/4)*log(-sqrt(2)*x^(1/3)*(a/b)^(1/4) + x^(2/3) + sqrt(a /b))/(a^3*b^3)
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx=\frac {\frac {27\,x}{32\,a}+\frac {15\,b\,x^{7/3}}{32\,a^2}}{a^2+b^2\,x^{8/3}+2\,a\,b\,x^{4/3}}+\frac {15\,\mathrm {atan}\left (\frac {b^{1/4}\,x^{1/3}}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{9/4}\,b^{3/4}}-\frac {15\,\mathrm {atanh}\left (\frac {b^{1/4}\,x^{1/3}}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{9/4}\,b^{3/4}} \] Input:
int(1/(a + b*x^(4/3))^3,x)
Output:
((27*x)/(32*a) + (15*b*x^(7/3))/(32*a^2))/(a^2 + b^2*x^(8/3) + 2*a*b*x^(4/ 3)) + (15*atan((b^(1/4)*x^(1/3))/(-a)^(1/4)))/(64*(-a)^(9/4)*b^(3/4)) - (1 5*atanh((b^(1/4)*x^(1/3))/(-a)^(1/4)))/(64*(-a)^(9/4)*b^(3/4))
Time = 0.22 (sec) , antiderivative size = 498, normalized size of antiderivative = 2.68 \[ \int \frac {1}{\left (a+b x^{4/3}\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(a+b*x^(4/3))^3,x)
Output:
(3*( - 10*x**(2/3)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt( 2) - 2*x**(1/3)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*x**2 - 20*x**(1 /3)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*x**(1/3) *sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*x - 10*b**(1/4)*a**(3/4)*sqrt(2 )*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*x**(1/3)*sqrt(b))/(b**(1/4)*a**(1/4) *sqrt(2)))*a**2 + 10*x**(2/3)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a** (1/4)*sqrt(2) + 2*x**(1/3)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*x**2 + 20*x**(1/3)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*x**(1/3)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*x + 10*b**(1/4)*a**( 3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*x**(1/3)*sqrt(b))/(b**(1/ 4)*a**(1/4)*sqrt(2)))*a**2 + 5*x**(2/3)*b**(1/4)*a**(3/4)*sqrt(2)*log( - x **(1/3)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + x**(2/3)*sqrt(b))*b**2*x**2 - 5*x**(2/3)*b**(1/4)*a**(3/4)*sqrt(2)*log(x**(1/3)*b**(1/4)*a**(1/4)*sqrt (2) + sqrt(a) + x**(2/3)*sqrt(b))*b**2*x**2 + 10*x**(1/3)*b**(1/4)*a**(3/4 )*sqrt(2)*log( - x**(1/3)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + x**(2/3)*s qrt(b))*a*b*x - 10*x**(1/3)*b**(1/4)*a**(3/4)*sqrt(2)*log(x**(1/3)*b**(1/4 )*a**(1/4)*sqrt(2) + sqrt(a) + x**(2/3)*sqrt(b))*a*b*x + 5*b**(1/4)*a**(3/ 4)*sqrt(2)*log( - x**(1/3)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + x**(2/3)* sqrt(b))*a**2 - 5*b**(1/4)*a**(3/4)*sqrt(2)*log(x**(1/3)*b**(1/4)*a**(1/4) *sqrt(2) + sqrt(a) + x**(2/3)*sqrt(b))*a**2 + 40*x**(1/3)*a*b**2*x**2 +...