Integrand size = 19, antiderivative size = 99 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {(b c-a d) x}{a b \sqrt [3]{a+b x^3}}+\frac {d \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3}}-\frac {d \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 b^{4/3}} \]
(-a*d+b*c)*x/a/b/(b*x^3+a)^(1/3)-1/2*d*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(4 /3)+1/3*d*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(4/3)*3^(1 /2)
Time = 0.53 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.52 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {\frac {6 \sqrt [3]{b} (b c-a d) x}{a \sqrt [3]{a+b x^3}}+2 \sqrt {3} d \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 d \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+d \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{6 b^{4/3}} \]
((6*b^(1/3)*(b*c - a*d)*x)/(a*(a + b*x^3)^(1/3)) + 2*Sqrt[3]*d*ArcTan[(Sqr t[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 2*d*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)] + d*Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + ( a + b*x^3)^(2/3)])/(6*b^(4/3))
Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {910, 769}
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.
\(\displaystyle \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx\) |
\(\Big \downarrow \) 910 |
\(\displaystyle \frac {d \int \frac {1}{\sqrt [3]{b x^3+a}}dx}{b}+\frac {x (b c-a d)}{a b \sqrt [3]{a+b x^3}}\) |
\(\Big \downarrow \) 769 |
\(\displaystyle \frac {d \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{b}+\frac {x (b c-a d)}{a b \sqrt [3]{a+b x^3}}\) |
((b*c - a*d)*x)/(a*b*(a + b*x^3)^(1/3)) + (d*(ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*b^(1/3)) - Log[-(b^(1/3)*x) + (a + b*x^ 3)^(1/3)]/(2*b^(1/3))))/b
3.1.59.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
Time = 3.98 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.63
method | result | size |
pseudoelliptic | \(\frac {6 b^{\frac {4}{3}} c x -2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a d \left (b \,x^{3}+a \right )^{\frac {1}{3}}-6 a d x \,b^{\frac {1}{3}}-2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a d \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a d \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{6 b^{\frac {4}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} a}\) | \(161\) |
1/6*(6*b^(4/3)*c*x-2*3^(1/2)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/ 3))/b^(1/3)/x)*a*d*(b*x^3+a)^(1/3)-6*a*d*x*b^(1/3)-2*ln((-b^(1/3)*x+(b*x^3 +a)^(1/3))/x)*a*d*(b*x^3+a)^(1/3)+ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)* x+(b*x^3+a)^(2/3))/x^2)*a*d*(b*x^3+a)^(1/3))/b^(4/3)/(b*x^3+a)^(1/3)/a
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (81) = 162\).
Time = 0.31 (sec) , antiderivative size = 488, normalized size of antiderivative = 4.93 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d x^{3} + a^{2} b d\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b^{2} c - a b d\right )} x - 2 \, {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} d x^{3} + a^{2} b d\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) - 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b^{2} c - a b d\right )} x + 2 \, {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (a b^{3} x^{3} + a^{2} b^{2}\right )}}\right ] \]
[1/6*(3*sqrt(1/3)*(a*b^2*d*x^3 + a^2*b*d)*sqrt((-b)^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3)*((-b)^(1/3)*b*x^3 - (b*x ^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b)^(2/3)*x)*sqrt((-b)^(1/3)/b) + 2*a) + 6*(b*x^3 + a)^(2/3)*(b^2*c - a*b*d)*x - 2*(a*b*d*x^3 + a^2*d)*(- b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + (a*b*d*x^3 + a^2*d)*( -b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2))/(a*b^3*x^3 + a^2*b^2), -1/6*(6*sqrt(1/3)*(a*b^2*d*x^3 + a^ 2*b*d)*sqrt(-(-b)^(1/3)/b)*arctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a) ^(1/3))*sqrt(-(-b)^(1/3)/b)/x) - 6*(b*x^3 + a)^(2/3)*(b^2*c - a*b*d)*x + 2 *(a*b*d*x^3 + a^2*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - (a*b*d*x^3 + a^2*d)*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*( -b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2))/(a*b^3*x^3 + a^2*b^2)]
Result contains complex when optimal does not.
Time = 2.83 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.72 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {c x \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {4}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {4}{3}} \Gamma \left (\frac {7}{3}\right )} \]
c*x*gamma(1/3)/(3*a**(4/3)*(1 + b*x**3/a)**(1/3)*gamma(4/3)) + d*x**4*gamm a(4/3)*hyper((4/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(4/3)*gam ma(7/3))
Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.35 \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=-\frac {1}{6} \, d {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} + \frac {6 \, x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} b} - \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}}\right )} + \frac {c x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} a} \]
-1/6*d*(2*sqrt(3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^( 1/3))/b^(4/3) + 6*x/((b*x^3 + a)^(1/3)*b) - log(b^(2/3) + (b*x^3 + a)^(1/3 )*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(4/3) + 2*log(-b^(1/3) + (b*x^3 + a )^(1/3)/x)/b^(4/3)) + c*x/((b*x^3 + a)^(1/3)*a)
\[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\int { \frac {d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac {4}{3}}} \,d x } \]
Timed out. \[ \int \frac {c+d x^3}{\left (a+b x^3\right )^{4/3}} \, dx=\int \frac {d\,x^3+c}{{\left (b\,x^3+a\right )}^{4/3}} \,d x \]