Integrand size = 15, antiderivative size = 110 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{3 a x^3}-\frac {b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3}}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}} \] Output:
-1/3*(b*x^3+a)^(2/3)/a/x^3-1/9*b*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))*3^ (1/2)/a^(1/3))*3^(1/2)/a^(4/3)+1/6*b*ln(x)/a^(4/3)-1/6*b*ln(a^(1/3)-(b*x^3 +a)^(1/3))/a^(4/3)
Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx=-\frac {6 \sqrt [3]{a} \left (a+b x^3\right )^{2/3}+2 \sqrt {3} b x^3 \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b x^3 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-b x^3 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{18 a^{4/3} x^3} \] Input:
Integrate[1/(x^4*(a + b*x^3)^(1/3)),x]
Output:
-1/18*(6*a^(1/3)*(a + b*x^3)^(2/3) + 2*Sqrt[3]*b*x^3*ArcTan[(1 + (2*(a + b *x^3)^(1/3))/a^(1/3))/Sqrt[3]] + 2*b*x^3*Log[-a^(1/3) + (a + b*x^3)^(1/3)] - b*x^3*Log[a^(2/3) + a^(1/3)*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(a^ (4/3)*x^3)
Time = 0.35 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 52, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{x^6 \sqrt [3]{b x^3+a}}dx^3\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{3} \left (-\frac {b \int \frac {1}{x^3 \sqrt [3]{b x^3+a}}dx^3}{3 a}-\frac {\left (a+b x^3\right )^{2/3}}{a x^3}\right )\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {1}{3} \left (-\frac {b \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {\left (a+b x^3\right )^{2/3}}{a x^3}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (-\frac {b \left (\frac {3}{2} \int \frac {1}{x^6+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {\left (a+b x^3\right )^{2/3}}{a x^3}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (-\frac {b \left (-\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {\left (a+b x^3\right )^{2/3}}{a x^3}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (-\frac {b \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^3\right )}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {\left (a+b x^3\right )^{2/3}}{a x^3}\right )\) |
Input:
Int[1/(x^4*(a + b*x^3)^(1/3)),x]
Output:
(-((a + b*x^3)^(2/3)/(a*x^3)) - (b*((Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1 /3))/a^(1/3))/Sqrt[3]])/a^(1/3) - Log[x^3]/(2*a^(1/3)) + (3*Log[a^(1/3) - (a + b*x^3)^(1/3)])/(2*a^(1/3))))/(3*a))/3
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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]
Time = 0.67 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(\frac {-2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b \,x^{3}-2 \ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b \,x^{3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b \,x^{3}-6 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{\frac {1}{3}}}{18 a^{\frac {4}{3}} x^{3}}\) | \(111\) |
Input:
int(1/x^4/(b*x^3+a)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/18*(-2*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))*3^(1/2)/a^(1/3))*3^(1/2)*b *x^3-2*ln((b*x^3+a)^(1/3)-a^(1/3))*b*x^3+ln((b*x^3+a)^(2/3)+a^(1/3)*(b*x^3 +a)^(1/3)+a^(2/3))*b*x^3-6*(b*x^3+a)^(2/3)*a^(1/3))/a^(4/3)/x^3
Time = 0.09 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.13 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x^{3} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x^{3} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x^{3}}\right ) + \left (-a\right )^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{18 \, a^{2} x^{3}}, -\frac {6 \, \sqrt {\frac {1}{3}} a b x^{3} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - \left (-a\right )^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{18 \, a^{2} x^{3}}\right ] \] Input:
integrate(1/x^4/(b*x^3+a)^(1/3),x, algorithm="fricas")
Output:
[1/18*(3*sqrt(1/3)*a*b*x^3*sqrt((-a)^(1/3)/a)*log((2*b*x^3 - 3*sqrt(1/3)*( 2*(b*x^3 + a)^(2/3)*(-a)^(2/3) - (b*x^3 + a)^(1/3)*a + (-a)^(1/3)*a)*sqrt( (-a)^(1/3)/a) - 3*(b*x^3 + a)^(1/3)*(-a)^(2/3) + 3*a)/x^3) + (-a)^(2/3)*b* x^3*log((b*x^3 + a)^(2/3) - (b*x^3 + a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) - 2 *(-a)^(2/3)*b*x^3*log((b*x^3 + a)^(1/3) + (-a)^(1/3)) - 6*(b*x^3 + a)^(2/3 )*a)/(a^2*x^3), -1/18*(6*sqrt(1/3)*a*b*x^3*sqrt(-(-a)^(1/3)/a)*arctan(sqrt (1/3)*(2*(b*x^3 + a)^(1/3) - (-a)^(1/3))*sqrt(-(-a)^(1/3)/a)) - (-a)^(2/3) *b*x^3*log((b*x^3 + a)^(2/3) - (b*x^3 + a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) + 2*(-a)^(2/3)*b*x^3*log((b*x^3 + a)^(1/3) + (-a)^(1/3)) + 6*(b*x^3 + a)^( 2/3)*a)/(a^2*x^3)]
Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.35 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx=- \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 \sqrt [3]{b} x^{4} \Gamma \left (\frac {7}{3}\right )} \] Input:
integrate(1/x**4/(b*x**3+a)**(1/3),x)
Output:
-gamma(4/3)*hyper((1/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**3))/(3*b**(1 /3)*x**4*gamma(7/3))
Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {4}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{3 \, {\left ({\left (b x^{3} + a\right )} a - a^{2}\right )}} + \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, a^{\frac {4}{3}}} - \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {4}{3}}} \] Input:
integrate(1/x^4/(b*x^3+a)^(1/3),x, algorithm="maxima")
Output:
-1/9*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1/3)) /a^(4/3) - 1/3*(b*x^3 + a)^(2/3)*b/((b*x^3 + a)*a - a^2) + 1/18*b*log((b*x ^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) - 1/9*b*log(( b*x^3 + a)^(1/3) - a^(1/3))/a^(4/3)
Time = 0.41 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx=-\frac {1}{18} \, b {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {2 \, \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {4}{3}}} + \frac {6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{a b x^{3}}\right )} \] Input:
integrate(1/x^4/(b*x^3+a)^(1/3),x, algorithm="giac")
Output:
-1/18*b*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + a^(1/3))/a^(1 /3))/a^(4/3) - log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*a^(1/3) + a^(2/3) )/a^(4/3) + 2*log(abs((b*x^3 + a)^(1/3) - a^(1/3)))/a^(4/3) + 6*(b*x^3 + a )^(2/3)/(a*b*x^3))
Time = 0.56 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx=-\frac {b\,\ln \left ({\left (b\,x^3+a\right )}^{1/3}-a^{1/3}\right )}{9\,a^{4/3}}-\frac {{\left (b\,x^3+a\right )}^{2/3}}{3\,a\,x^3}+\frac {\ln \left (\frac {{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}}-\frac {b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9\,a^2}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{18\,a^{4/3}}+\frac {\ln \left (\frac {{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}}-\frac {b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9\,a^2}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{18\,a^{4/3}} \] Input:
int(1/(x^4*(a + b*x^3)^(1/3)),x)
Output:
(log((b - 3^(1/2)*b*1i)^2/(36*a^(5/3)) - (b^2*(a + b*x^3)^(1/3))/(9*a^2))* (b - 3^(1/2)*b*1i))/(18*a^(4/3)) - (a + b*x^3)^(2/3)/(3*a*x^3) - (b*log((a + b*x^3)^(1/3) - a^(1/3)))/(9*a^(4/3)) + (log((b + 3^(1/2)*b*1i)^2/(36*a^ (5/3)) - (b^2*(a + b*x^3)^(1/3))/(9*a^2))*(b + 3^(1/2)*b*1i))/(18*a^(4/3))
\[ \int \frac {1}{x^4 \sqrt [3]{a+b x^3}} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{4}}d x \] Input:
int(1/x^4/(b*x^3+a)^(1/3),x)
Output:
int(1/((a + b*x**3)**(1/3)*x**4),x)