Integrand size = 19, antiderivative size = 171 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{d}+\frac {(b c-a d) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{\sqrt {3} b^{2/3} d^{4/3}}+\frac {(b c-a d) \log (a+b x)}{6 b^{2/3} d^{4/3}}+\frac {(b c-a d) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 b^{2/3} d^{4/3}} \]
(b*x+a)^(1/3)*(d*x+c)^(2/3)/d+1/6*(-a*d+b*c)*ln(b*x+a)/b^(2/3)/d^(4/3)+1/2 *(-a*d+b*c)*ln(-1+b^(1/3)*(d*x+c)^(1/3)/d^(1/3)/(b*x+a)^(1/3))/b^(2/3)/d^( 4/3)+1/3*(-a*d+b*c)*arctan(1/3*3^(1/2)+2/3*b^(1/3)*(d*x+c)^(1/3)/d^(1/3)/( b*x+a)^(1/3)*3^(1/2))/b^(2/3)/d^(4/3)*3^(1/2)
Time = 0.07 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx=\frac {6 b^{2/3} \sqrt [3]{d} \sqrt [3]{a+b x} (c+d x)^{2/3}+2 \sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )+2 (b c-a d) \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )-(b c-a d) \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{6 b^{2/3} d^{4/3}} \]
(6*b^(2/3)*d^(1/3)*(a + b*x)^(1/3)*(c + d*x)^(2/3) + 2*Sqrt[3]*(b*c - a*d) *ArcTan[(Sqrt[3]*b^(1/3)*(c + d*x)^(1/3))/(2*d^(1/3)*(a + b*x)^(1/3) + b^( 1/3)*(c + d*x)^(1/3))] + 2*(b*c - a*d)*Log[d^(1/3)*(a + b*x)^(1/3) - b^(1/ 3)*(c + d*x)^(1/3)] - (b*c - a*d)*Log[d^(2/3)*(a + b*x)^(2/3) + b^(1/3)*d^ (1/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3)])/(6*b^(2/ 3)*d^(4/3))
Time = 0.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {60, 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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{d}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}}dx}{3 d}\) |
\(\Big \downarrow \) 71 |
\(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{d}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{d}}-\frac {3 \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}}\right )}{3 d}\) |
((a + b*x)^(1/3)*(c + d*x)^(2/3))/d - ((b*c - a*d)*(-((Sqrt[3]*ArcTan[1/Sq rt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)*(a + b*x)^(1/3))])/(b ^(2/3)*d^(1/3))) - Log[a + b*x]/(2*b^(2/3)*d^(1/3)) - (3*Log[-1 + (b^(1/3) *(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(2*b^(2/3)*d^(1/3))))/(3*d)
3.31.10.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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]
\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (132) = 264\).
Time = 0.24 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.61 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx=\left [\frac {6 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b^{2} d - 3 \, \sqrt {\frac {1}{3}} {\left (b^{2} c d - a b d^{2}\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (3 \, b^{2} d x + b^{2} c + 2 \, a b d + 3 \, \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}\right ) - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) + 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right )}{6 \, b^{2} d^{2}}, \frac {6 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b^{2} d - 6 \, \sqrt {\frac {1}{3}} {\left (b^{2} c d - a b d^{2}\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}}{b^{2} d x + b^{2} c}\right ) - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) + 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b c - a d\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right )}{6 \, b^{2} d^{2}}\right ] \]
[1/6*(6*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b^2*d - 3*sqrt(1/3)*(b^2*c*d - a*b *d^2)*sqrt((-b^2*d)^(1/3)/d)*log(3*b^2*d*x + b^2*c + 2*a*b*d + 3*(-b^2*d)^ (1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b + 3*sqrt(1/3)*(2*(b*x + a)^(2/3)*( d*x + c)^(1/3)*b*d - (-b^2*d)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (-b^ 2*d)^(1/3)*(b*d*x + b*c))*sqrt((-b^2*d)^(1/3)/d)) - (-b^2*d)^(2/3)*(b*c - a*d)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d + (-b^2*d)^(2/3)*(b*x + a)^( 1/3)*(d*x + c)^(2/3) - (-b^2*d)^(1/3)*(b*d*x + b*c))/(d*x + c)) + 2*(-b^2* d)^(2/3)*(b*c - a*d)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d - (-b^2*d)^( 2/3)*(d*x + c))/(d*x + c)))/(b^2*d^2), 1/6*(6*(b*x + a)^(1/3)*(d*x + c)^(2 /3)*b^2*d - 6*sqrt(1/3)*(b^2*c*d - a*b*d^2)*sqrt(-(-b^2*d)^(1/3)/d)*arctan (sqrt(1/3)*(2*(-b^2*d)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d)^(1 /3)*(b*d*x + b*c))*sqrt(-(-b^2*d)^(1/3)/d)/(b^2*d*x + b^2*c)) - (-b^2*d)^( 2/3)*(b*c - a*d)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*b*d + (-b^2*d)^(2/3) *(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d)^(1/3)*(b*d*x + b*c))/(d*x + c) ) + 2*(-b^2*d)^(2/3)*(b*c - a*d)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*b*d - (-b^2*d)^(2/3)*(d*x + c))/(d*x + c)))/(b^2*d^2)]
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx=\int \frac {\sqrt [3]{a + b x}}{\sqrt [3]{c + d x}}\, dx \]
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (c+d\,x\right )}^{1/3}} \,d x \]