Integrand size = 24, antiderivative size = 200 \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d}-\frac {(3 b d e-2 b c f-a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt {3} b^{4/3} d^{5/3}}-\frac {(3 b d e-2 b c f-a d f) \log (c+d x)}{6 b^{4/3} d^{5/3}}-\frac {(3 b d e-2 b c f-a d f) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3} d^{5/3}} \]
f*(b*x+a)^(2/3)*(d*x+c)^(1/3)/b/d-1/6*(-a*d*f-2*b*c*f+3*b*d*e)*ln(d*x+c)/b ^(4/3)/d^(5/3)-1/2*(-a*d*f-2*b*c*f+3*b*d*e)*ln(-1+d^(1/3)*(b*x+a)^(1/3)/b^ (1/3)/(d*x+c)^(1/3))/b^(4/3)/d^(5/3)-1/3*(-a*d*f-2*b*c*f+3*b*d*e)*arctan(1 /3*3^(1/2)+2/3*d^(1/3)*(b*x+a)^(1/3)/b^(1/3)/(d*x+c)^(1/3)*3^(1/2))/b^(4/3 )/d^(5/3)*3^(1/2)
Time = 0.42 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.17 \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\frac {6 \sqrt [3]{b} d^{2/3} f (a+b x)^{2/3} \sqrt [3]{c+d x}+2 \sqrt {3} (3 b d e-2 b c f-a d f) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )+2 (-3 b d e+2 b c f+a d f) \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )+(3 b d e-2 b c f-a d f) \log \left (d^{2/3}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {b^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{6 b^{4/3} d^{5/3}} \]
(6*b^(1/3)*d^(2/3)*f*(a + b*x)^(2/3)*(c + d*x)^(1/3) + 2*Sqrt[3]*(3*b*d*e - 2*b*c*f - a*d*f)*ArcTan[(1 + (2*b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b *x)^(1/3)))/Sqrt[3]] + 2*(-3*b*d*e + 2*b*c*f + a*d*f)*Log[d^(1/3) - (b^(1/ 3)*(c + d*x)^(1/3))/(a + b*x)^(1/3)] + (3*b*d*e - 2*b*c*f - a*d*f)*Log[d^( 2/3) + (b^(1/3)*d^(1/3)*(c + d*x)^(1/3))/(a + b*x)^(1/3) + (b^(2/3)*(c + d *x)^(2/3))/(a + b*x)^(2/3)])/(6*b^(4/3)*d^(5/3))
Time = 0.22 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {90, 71}
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.
\(\displaystyle \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {(-a d f-2 b c f+3 b d e) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}}dx}{3 b d}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d}\) |
\(\Big \downarrow \) 71 |
\(\displaystyle \frac {(-a d f-2 b c f+3 b d e) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 \sqrt [3]{b} d^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}}\right )}{3 b d}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d}\) |
(f*(a + b*x)^(2/3)*(c + d*x)^(1/3))/(b*d) + ((3*b*d*e - 2*b*c*f - a*d*f)*( -((Sqrt[3]*ArcTan[1/Sqrt[3] + (2*d^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*b^(1/3) *(c + d*x)^(1/3))])/(b^(1/3)*d^(2/3))) - Log[c + d*x]/(2*b^(1/3)*d^(2/3)) - (3*Log[-1 + (d^(1/3)*(a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3))])/(2*b^( 1/3)*d^(2/3))))/(3*b*d)
3.31.17.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
\[\int \frac {f x +e}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}d x\]
Time = 0.26 (sec) , antiderivative size = 660, normalized size of antiderivative = 3.30 \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\left [\frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} f - 3 \, \sqrt {\frac {1}{3}} {\left (3 \, b^{2} d^{2} e - {\left (2 \, b^{2} c d + a b d^{2}\right )} f\right )} \sqrt {-\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}} \log \left (-3 \, b d^{2} x - 2 \, b c d - a d^{2} + 3 \, \left (b d^{2}\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} d + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {-\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}}\right ) - 2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + \left (b d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right )}{6 \, b^{2} d^{3}}, \frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} f + 6 \, \sqrt {\frac {1}{3}} {\left (3 \, b^{2} d^{2} e - {\left (2 \, b^{2} c d + a b d^{2}\right )} f\right )} \sqrt {\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}}}{b d^{2} x + a d^{2}}\right ) - 2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + \left (b d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right )}{6 \, b^{2} d^{3}}\right ] \]
[1/6*(6*(b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d^2*f - 3*sqrt(1/3)*(3*b^2*d^2*e - (2*b^2*c*d + a*b*d^2)*f)*sqrt(-(b*d^2)^(1/3)/b)*log(-3*b*d^2*x - 2*b*c* d - a*d^2 + 3*(b*d^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3)*d + 3*sqrt(1/3 )*(2*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d - (b*d^2)^(2/3)*(b*x + a)^(2/3)*( d*x + c)^(1/3) - (b*d^2)^(1/3)*(b*d*x + a*d))*sqrt(-(b*d^2)^(1/3)/b)) - 2* (b*d^2)^(2/3)*(3*b*d*e - (2*b*c + a*d)*f)*log(((b*x + a)^(2/3)*(d*x + c)^( 1/3)*b*d - (b*d^2)^(2/3)*(b*x + a))/(b*x + a)) + (b*d^2)^(2/3)*(3*b*d*e - (2*b*c + a*d)*f)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d + (b*d^2)^(2/3)* (b*x + a)^(2/3)*(d*x + c)^(1/3) + (b*d^2)^(1/3)*(b*d*x + a*d))/(b*x + a))) /(b^2*d^3), 1/6*(6*(b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d^2*f + 6*sqrt(1/3)*( 3*b^2*d^2*e - (2*b^2*c*d + a*b*d^2)*f)*sqrt((b*d^2)^(1/3)/b)*arctan(sqrt(1 /3)*(2*(b*d^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b*d^2)^(1/3)*(b*d* x + a*d))*sqrt((b*d^2)^(1/3)/b)/(b*d^2*x + a*d^2)) - 2*(b*d^2)^(2/3)*(3*b* d*e - (2*b*c + a*d)*f)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d - (b*d^2)^ (2/3)*(b*x + a))/(b*x + a)) + (b*d^2)^(2/3)*(3*b*d*e - (2*b*c + a*d)*f)*lo g(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d + (b*d^2)^(2/3)*(b*x + a)^(2/3)*(d* x + c)^(1/3) + (b*d^2)^(1/3)*(b*d*x + a*d))/(b*x + a)))/(b^2*d^3)]
\[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {e + f x}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}}}\, dx \]
\[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int { \frac {f x + e}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int { \frac {f x + e}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {e+f\,x}{{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \]