Integrand size = 19, antiderivative size = 127 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}}-\frac {3 \log \left (1-\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 \sqrt [3]{b} d^{2/3}} \] Output:
-3^(1/2)*arctan(1/3*3^(1/2)+2/3*d^(1/3)*(b*x+a)^(1/3)*3^(1/2)/b^(1/3)/(d*x +c)^(1/3))/b^(1/3)/d^(2/3)-1/2*ln(d*x+c)/b^(1/3)/d^(2/3)-3/2*ln(1-d^(1/3)* (b*x+a)^(1/3)/b^(1/3)/(d*x+c)^(1/3))/b^(1/3)/d^(2/3)
Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )-2 \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b} d^{2/3}} \] Input:
Integrate[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)),x]
Output:
(2*Sqrt[3]*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*Log[d^(1/3)*(a + b*x)^(1/3) - b^(1/3) *(c + d*x)^(1/3)] + 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)])/(2*b^(1/3)*d^(2/3))
Time = 0.17 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.99, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {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.
\(\displaystyle \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx\) |
\(\Big \downarrow \) 71 |
\(\displaystyle -\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 \sqrt [3]{b} d^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}}\) |
Input:
Int[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)),x]
Output:
-((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))
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]
\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (x d +c \right )^{\frac {2}{3}}}d x\]
Input:
int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3),x)
Output:
int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3),x)
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (91) = 182\).
Time = 0.10 (sec) , antiderivative size = 521, normalized size of antiderivative = 4.10 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx =\text {Too large to display} \] Input:
integrate(1/(b*x+a)^(1/3)/(d*x+c)^(2/3),x, algorithm="fricas")
Output:
[1/2*(sqrt(3)*b*d*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 - sqrt(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)*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)*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*d^2), 1/2*(2*sqrt(3)*b*d*sqrt(-(-b*d^2)^(1/3)/b)*arctan(1/ 3*sqrt(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)*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)*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*d^2)]
\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {1}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}}}\, dx \] Input:
integrate(1/(b*x+a)**(1/3)/(d*x+c)**(2/3),x)
Output:
Integral(1/((a + b*x)**(1/3)*(c + d*x)**(2/3)), x)
\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/3)/(d*x+c)^(2/3),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)), x)
\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate(1/(b*x+a)^(1/3)/(d*x+c)^(2/3),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)), x)
Timed out. \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \] Input:
int(1/((a + b*x)^(1/3)*(c + d*x)^(2/3)),x)
Output:
int(1/((a + b*x)^(1/3)*(c + d*x)^(2/3)), x)
\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {1}{\left (d x +c \right )^{\frac {2}{3}} \left (b x +a \right )^{\frac {1}{3}}}d x \] Input:
int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3),x)
Output:
int(1/((c + d*x)**(2/3)*(a + b*x)**(1/3)),x)