Integrand size = 26, antiderivative size = 290 \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {c x \left (-b+a x^3\right )^{2/3}}{3 a}-\frac {b c \arctan \left (\frac {\frac {x}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{a}}}{x}\right )}{3 \sqrt {3} a^{4/3}}+\frac {d \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{b}}-\frac {b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{9 a^{4/3}}-\frac {d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}}+\frac {b c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{18 a^{4/3}} \]
1/3*c*x*(a*x^3-b)^(2/3)/a-1/9*b*c*arctan((1/3*x*3^(1/2)+2/3*(a*x^3-b)^(1/3 )*3^(1/2)/a^(1/3))/x)*3^(1/2)/a^(4/3)-1/3*d*arctan(-1/3*3^(1/2)+2/3*(a*x^3 -b)^(1/3)*3^(1/2)/b^(1/3))*3^(1/2)/b^(1/3)+1/3*d*ln(b^(1/3)+(a*x^3-b)^(1/3 ))/b^(1/3)-1/9*b*c*ln(-a^(1/3)*x+(a*x^3-b)^(1/3))/a^(4/3)-1/6*d*ln(b^(2/3) -b^(1/3)*(a*x^3-b)^(1/3)+(a*x^3-b)^(2/3))/b^(1/3)+1/18*b*c*ln(a^(2/3)*x^2+ a^(1/3)*x*(a*x^3-b)^(1/3)+(a*x^3-b)^(2/3))/a^(4/3)
Time = 5.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.94 \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {1}{18} \left (\frac {6 c x \left (-b+a x^3\right )^{2/3}}{a}+\frac {6 \sqrt {3} d \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {2 \sqrt {3} b c \arctan \left (\frac {1+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{a} x}}{\sqrt {3}}\right )}{a^{4/3}}+\frac {6 d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{\sqrt [3]{b}}-\frac {2 b c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{a^{4/3}}-\frac {3 d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{\sqrt [3]{b}}+\frac {b c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{a^{4/3}}\right ) \]
((6*c*x*(-b + a*x^3)^(2/3))/a + (6*Sqrt[3]*d*ArcTan[(1 - (2*(-b + a*x^3)^( 1/3))/b^(1/3))/Sqrt[3]])/b^(1/3) - (2*Sqrt[3]*b*c*ArcTan[(1 + (2*(-b + a*x ^3)^(1/3))/(a^(1/3)*x))/Sqrt[3]])/a^(4/3) + (6*d*Log[b^(1/3) + (-b + a*x^3 )^(1/3)])/b^(1/3) - (2*b*c*Log[-(a^(1/3)*x) + (-b + a*x^3)^(1/3)])/a^(4/3) - (3*d*Log[b^(2/3) - b^(1/3)*(-b + a*x^3)^(1/3) + (-b + a*x^3)^(2/3)])/b^ (1/3) + (b*c*Log[a^(2/3)*x^2 + a^(1/3)*x*(-b + a*x^3)^(1/3) + (-b + a*x^3) ^(2/3)])/a^(4/3))/18
Time = 0.39 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.66, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2383, 2009}
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.
\(\displaystyle \int \frac {c x^4-d}{x \sqrt [3]{a x^3-b}} \, dx\) |
\(\Big \downarrow \) 2383 |
\(\displaystyle \int \left (\frac {c x^3}{\sqrt [3]{a x^3-b}}-\frac {d}{x \sqrt [3]{a x^3-b}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b c \arctan \left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3}}-\frac {b c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{6 a^{4/3}}+\frac {d \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {c x \left (a x^3-b\right )^{2/3}}{3 a}+\frac {d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}}\) |
(c*x*(-b + a*x^3)^(2/3))/(3*a) + (b*c*ArcTan[(1 + (2*a^(1/3)*x)/(-b + a*x^ 3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*a^(4/3)) + (d*ArcTan[(b^(1/3) - 2*(-b + a*x ^3)^(1/3))/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*b^(1/3)) - (d*Log[x])/(2*b^(1/3)) + (d*Log[b^(1/3) + (-b + a*x^3)^(1/3)])/(2*b^(1/3)) - (b*c*Log[-(a^(1/3)*x ) + (-b + a*x^3)^(1/3)])/(6*a^(4/3))
3.29.41.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n , p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IGtQ[m, 0]
\[\int \frac {c \,x^{4}-d}{x \left (a \,x^{3}-b \right )^{\frac {1}{3}}}d x\]
Timed out. \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 1.61 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.29 \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=- \frac {c x^{4} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 \sqrt [3]{b} \Gamma \left (\frac {7}{3}\right )} + \frac {d \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac {4}{3}\right )} \]
-c*x**4*exp(2*I*pi/3)*gamma(4/3)*hyper((1/3, 4/3), (7/3,), a*x**3/b)/(3*b* *(1/3)*gamma(7/3)) + d*gamma(1/3)*hyper((1/3, 1/3), (4/3,), b*exp_polar(2* I*pi)/(a*x**3))/(3*a**(1/3)*x*gamma(4/3))
Time = 0.30 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.86 \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=-\frac {1}{18} \, {\left (\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {b \log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{a^{\frac {4}{3}}} + \frac {2 \, b \log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{a^{\frac {4}{3}}} - \frac {6 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} b}{{\left (a^{2} - \frac {{\left (a x^{3} - b\right )} a}{x^{3}}\right )} x^{2}}\right )} c - \frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} + \frac {\log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{b^{\frac {1}{3}}}\right )} d \]
-1/18*(2*sqrt(3)*b*arctan(1/3*sqrt(3)*(a^(1/3) + 2*(a*x^3 - b)^(1/3)/x)/a^ (1/3))/a^(4/3) - b*log(a^(2/3) + (a*x^3 - b)^(1/3)*a^(1/3)/x + (a*x^3 - b) ^(2/3)/x^2)/a^(4/3) + 2*b*log(-a^(1/3) + (a*x^3 - b)^(1/3)/x)/a^(4/3) - 6* (a*x^3 - b)^(2/3)*b/((a^2 - (a*x^3 - b)*a/x^3)*x^2))*c - 1/6*(2*sqrt(3)*ar ctan(1/3*sqrt(3)*(2*(a*x^3 - b)^(1/3) - b^(1/3))/b^(1/3))/b^(1/3) + log((a *x^3 - b)^(2/3) - (a*x^3 - b)^(1/3)*b^(1/3) + b^(2/3))/b^(1/3) - 2*log((a* x^3 - b)^(1/3) + b^(1/3))/b^(1/3))*d
\[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=\int { \frac {c x^{4} - d}{{\left (a x^{3} - b\right )}^{\frac {1}{3}} x} \,d x } \]
Timed out. \[ \int \frac {-d+c x^4}{x \sqrt [3]{-b+a x^3}} \, dx=-\int \frac {d-c\,x^4}{x\,{\left (a\,x^3-b\right )}^{1/3}} \,d x \]