Integrand size = 20, antiderivative size = 111 \[ \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx=\frac {3 \sqrt [3]{a+b x}}{b \sqrt [3]{a-b x}}-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a-b x}}{\sqrt {3} \sqrt [3]{a+b x}}\right )}{b}-\frac {\log (a+b x)}{2 b}-\frac {3 \log \left (1+\frac {\sqrt [3]{a-b x}}{\sqrt [3]{a+b x}}\right )}{2 b} \] Output:
3*(b*x+a)^(1/3)/b/(-b*x+a)^(1/3)+3^(1/2)*arctan(-1/3*3^(1/2)+2/3*(-b*x+a)^ (1/3)*3^(1/2)/(b*x+a)^(1/3))/b-1/2*ln(b*x+a)/b-3/2*ln(1+(-b*x+a)^(1/3)/(b* x+a)^(1/3))/b
Time = 0.16 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx=\frac {\frac {6 \sqrt [3]{a+b x}}{\sqrt [3]{a-b x}}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{-2 \sqrt [3]{a-b x}+\sqrt [3]{a+b x}}\right )-2 \log \left (b \left (\sqrt [3]{a-b x}+\sqrt [3]{a+b x}\right )\right )+\log \left ((a-b x)^{2/3}-\sqrt [3]{a-b x} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{2 b} \] Input:
Integrate[(a + b*x)^(1/3)/(a - b*x)^(4/3),x]
Output:
((6*(a + b*x)^(1/3))/(a - b*x)^(1/3) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*(a + b*x) ^(1/3))/(-2*(a - b*x)^(1/3) + (a + b*x)^(1/3))] - 2*Log[b*((a - b*x)^(1/3) + (a + b*x)^(1/3))] + Log[(a - b*x)^(2/3) - (a - b*x)^(1/3)*(a + b*x)^(1/ 3) + (a + b*x)^(2/3)])/(2*b)
Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {57, 72}
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.
\(\displaystyle \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {3 \sqrt [3]{a+b x}}{b \sqrt [3]{a-b x}}-\int \frac {1}{\sqrt [3]{a-b x} (a+b x)^{2/3}}dx\) |
\(\Big \downarrow \) 72 |
\(\displaystyle -\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a-b x}}{\sqrt {3} \sqrt [3]{a+b x}}\right )}{b}+\frac {3 \sqrt [3]{a+b x}}{b \sqrt [3]{a-b x}}-\frac {\log (a+b x)}{2 b}-\frac {3 \log \left (\frac {\sqrt [3]{a-b x}}{\sqrt [3]{a+b x}}+1\right )}{2 b}\) |
Input:
Int[(a + b*x)^(1/3)/(a - b*x)^(4/3),x]
Output:
(3*(a + b*x)^(1/3))/(b*(a - b*x)^(1/3)) - (Sqrt[3]*ArcTan[1/Sqrt[3] - (2*( a - b*x)^(1/3))/(Sqrt[3]*(a + b*x)^(1/3))])/b - Log[a + b*x]/(2*b) - (3*Lo g[1 + (a - b*x)^(1/3)/(a + b*x)^(1/3)])/(2*b)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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[1/Sqrt[3] - 2*q*((a + b* x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))], 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])] /; F reeQ[{a, b, c, d}, x] && NegQ[d/b]
\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (-b x +a \right )^{\frac {4}{3}}}d x\]
Input:
int((b*x+a)^(1/3)/(-b*x+a)^(4/3),x)
Output:
int((b*x+a)^(1/3)/(-b*x+a)^(4/3),x)
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (90) = 180\).
Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx=\frac {2 \, \sqrt {3} {\left (b x - a\right )} \arctan \left (\frac {\sqrt {3} {\left (b x - a\right )} + 2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (b x - a\right )}}\right ) + {\left (b x - a\right )} \log \left (\frac {b x - {\left (b x + a\right )}^{\frac {2}{3}} {\left (-b x + a\right )}^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} - a}{b x - a}\right ) - 2 \, {\left (b x - a\right )} \log \left (-\frac {b x - {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} - a}{b x - a}\right ) - 6 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}}{2 \, {\left (b^{2} x - a b\right )}} \] Input:
integrate((b*x+a)^(1/3)/(-b*x+a)^(4/3),x, algorithm="fricas")
Output:
1/2*(2*sqrt(3)*(b*x - a)*arctan(1/3*(sqrt(3)*(b*x - a) + 2*sqrt(3)*(b*x + a)^(1/3)*(-b*x + a)^(2/3))/(b*x - a)) + (b*x - a)*log((b*x - (b*x + a)^(2/ 3)*(-b*x + a)^(1/3) + (b*x + a)^(1/3)*(-b*x + a)^(2/3) - a)/(b*x - a)) - 2 *(b*x - a)*log(-(b*x - (b*x + a)^(1/3)*(-b*x + a)^(2/3) - a)/(b*x - a)) - 6*(b*x + a)^(1/3)*(-b*x + a)^(2/3))/(b^2*x - a*b)
\[ \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx=\int \frac {\sqrt [3]{a + b x}}{\left (a - b x\right )^{\frac {4}{3}}}\, dx \] Input:
integrate((b*x+a)**(1/3)/(-b*x+a)**(4/3),x)
Output:
Integral((a + b*x)**(1/3)/(a - b*x)**(4/3), x)
\[ \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (-b x + a\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((b*x+a)^(1/3)/(-b*x+a)^(4/3),x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/3)/(-b*x + a)^(4/3), x)
\[ \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (-b x + a\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((b*x+a)^(1/3)/(-b*x+a)^(4/3),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/3)/(-b*x + a)^(4/3), x)
Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (a-b\,x\right )}^{4/3}} \,d x \] Input:
int((a + b*x)^(1/3)/(a - b*x)^(4/3),x)
Output:
int((a + b*x)^(1/3)/(a - b*x)^(4/3), x)
\[ \int \frac {\sqrt [3]{a+b x}}{(a-b x)^{4/3}} \, dx=\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (-b x +a \right )^{\frac {1}{3}} a -\left (-b x +a \right )^{\frac {1}{3}} b x}d x \] Input:
int((b*x+a)^(1/3)/(-b*x+a)^(4/3),x)
Output:
int((a + b*x)**(1/3)/((a - b*x)**(1/3)*a - (a - b*x)**(1/3)*b*x),x)