Integrand size = 19, antiderivative size = 333 \[ \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx=-\frac {d x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^2 \sqrt [3]{a+b x^3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b c^3-a d^3}}-\frac {\arctan \left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b c^3-a d^3}}+\frac {\log \left (c^3+d^3 x^3\right )}{3 \sqrt [3]{b c^3-a d^3}}-\frac {\log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b c^3-a d^3}}-\frac {\log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b c^3-a d^3}} \]
-1/2*d*x^2*(1+b*x^3/a)^(1/3)*AppellF1(2/3,1/3,1,5/3,-b*x^3/a,-d^3*x^3/c^3) /c^2/(b*x^3+a)^(1/3)+1/3*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(1/3)-1/2*ln((-a*d ^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/(-a*d^3+b*c^3)^(1/3)-1/2*ln((-a*d^3+b *c^3)^(1/3)+d*(b*x^3+a)^(1/3))/(-a*d^3+b*c^3)^(1/3)+1/3*arctan(1/3*(1+2*(- a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(1/2))/(-a*d^3+b*c^3)^(1/3)*3^(1 /2)-1/3*arctan(1/3*(1-2*d*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)^(1/3))*3^(1/2))/( -a*d^3+b*c^3)^(1/3)*3^(1/2)
\[ \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx=\int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx \]
Time = 0.59 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2581, 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.
\(\displaystyle \int \frac {1}{\sqrt [3]{a+b x^3} (c+d x)} \, dx\) |
\(\Big \downarrow \) 2581 |
\(\displaystyle \int \left (-\frac {c d x}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )}+\frac {d^2 x^2}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )}+\frac {c^2}{\sqrt [3]{a+b x^3} \left (c^3+d^3 x^3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d x^2 \sqrt [3]{\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{2 c^2 \sqrt [3]{a+b x^3}}+\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b c^3-a d^3}}-\frac {\arctan \left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b c^3-a d^3}}+\frac {\log \left (c^3+d^3 x^3\right )}{3 \sqrt [3]{b c^3-a d^3}}-\frac {\log \left (\frac {x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b c^3-a d^3}}-\frac {\log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b c^3-a d^3}}\) |
-1/2*(d*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(c^2*(a + b*x^3)^(1/3)) + ArcTan[(1 + (2*(b*c^3 - a*d^ 3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*(b*c^3 - a*d^3)^(1/3) ) - ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]]/(S qrt[3]*(b*c^3 - a*d^3)^(1/3)) + Log[c^3 + d^3*x^3]/(3*(b*c^3 - a*d^3)^(1/3 )) - Log[((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)]/(2*(b*c^3 - a*d^ 3)^(1/3)) - Log[(b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)]/(2*(b*c^3 - a *d^3)^(1/3))
3.1.33.3.1 Defintions of rubi rules used
Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d ^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ[q, 0 ] && RationalQ[p] && EqQ[Denominator[p], 3]
\[\int \frac {1}{\left (d x +c \right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x\]
Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx=\int \frac {1}{\sqrt [3]{a + b x^{3}} \left (c + d x\right )}\, dx \]
\[ \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \]
\[ \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{a+b x^3}} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \]