Integrand size = 44, antiderivative size = 52 \[ \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \] Output:
-2/3*arctanh(1/3*(a^(1/3)-b^(1/3)*x)^2/a^(1/6)/(-b*x^3+a)^(1/2))/a^(1/6)/b ^(1/3)
Time = 3.39 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {3 \sqrt [6]{a} \sqrt {a-b x^3}}{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \] Input:
Integrate[(a^(1/3) - b^(1/3)*x)/((2*a^(1/3) + b^(1/3)*x)*Sqrt[a - b*x^3]), x]
Output:
(-2*ArcTanh[(3*a^(1/6)*Sqrt[a - b*x^3])/(a^(1/3) - b^(1/3)*x)^2])/(3*a^(1/ 6)*b^(1/3))
Time = 0.46 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2563, 219}
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.
\(\displaystyle \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx\) |
\(\Big \downarrow \) 2563 |
\(\displaystyle -\frac {2 \sqrt [3]{a} \int \frac {1}{9-\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^4}{\sqrt [3]{a} \left (a-b x^3\right )}}d\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{a^{2/3} \sqrt {a-b x^3}}}{\sqrt [3]{b}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}}\) |
Input:
Int[(a^(1/3) - b^(1/3)*x)/((2*a^(1/3) + b^(1/3)*x)*Sqrt[a - b*x^3]),x]
Output:
(-2*ArcTanh[(a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*Sqrt[a - b*x^3])])/(3*a^(1/ 6)*b^(1/3))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[-2*(e/d) Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & & EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
\[\int \frac {a^{\frac {1}{3}}-b^{\frac {1}{3}} x}{\left (2 a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {-b \,x^{3}+a}}d x\]
Input:
int((a^(1/3)-b^(1/3)*x)/(2*a^(1/3)+b^(1/3)*x)/(-b*x^3+a)^(1/2),x)
Output:
int((a^(1/3)-b^(1/3)*x)/(2*a^(1/3)+b^(1/3)*x)/(-b*x^3+a)^(1/2),x)
Timed out. \[ \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Timed out} \] Input:
integrate((a^(1/3)-b^(1/3)*x)/(2*a^(1/3)+b^(1/3)*x)/(-b*x^3+a)^(1/2),x, al gorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=- \int \left (- \frac {\sqrt [3]{a}}{2 \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\right )\, dx - \int \frac {\sqrt [3]{b} x}{2 \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx \] Input:
integrate((a**(1/3)-b**(1/3)*x)/(2*a**(1/3)+b**(1/3)*x)/(-b*x**3+a)**(1/2) ,x)
Output:
-Integral(-a**(1/3)/(2*a**(1/3)*sqrt(a - b*x**3) + b**(1/3)*x*sqrt(a - b*x **3)), x) - Integral(b**(1/3)*x/(2*a**(1/3)*sqrt(a - b*x**3) + b**(1/3)*x* sqrt(a - b*x**3)), x)
\[ \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\int { -\frac {b^{\frac {1}{3}} x - a^{\frac {1}{3}}}{\sqrt {-b x^{3} + a} {\left (b^{\frac {1}{3}} x + 2 \, a^{\frac {1}{3}}\right )}} \,d x } \] Input:
integrate((a^(1/3)-b^(1/3)*x)/(2*a^(1/3)+b^(1/3)*x)/(-b*x^3+a)^(1/2),x, al gorithm="maxima")
Output:
-integrate((b^(1/3)*x - a^(1/3))/(sqrt(-b*x^3 + a)*(b^(1/3)*x + 2*a^(1/3)) ), x)
Timed out. \[ \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Timed out} \] Input:
integrate((a^(1/3)-b^(1/3)*x)/(2*a^(1/3)+b^(1/3)*x)/(-b*x^3+a)^(1/2),x, al gorithm="giac")
Output:
Timed out
Time = 22.67 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {a-b\,x^3}-\sqrt {a}\right )\,{\left (\sqrt {a-b\,x^3}+\sqrt {a}+2\,a^{1/6}\,b^{1/3}\,x\right )}^3}{x^3\,{\left (b^{1/3}\,x+2\,a^{1/3}\right )}^3}\right )}{3\,a^{1/6}\,b^{1/3}} \] Input:
int(-(b^(1/3)*x - a^(1/3))/((b^(1/3)*x + 2*a^(1/3))*(a - b*x^3)^(1/2)),x)
Output:
log((((a - b*x^3)^(1/2) - a^(1/2))*((a - b*x^3)^(1/2) + a^(1/2) + 2*a^(1/6 )*b^(1/3)*x)^3)/(x^3*(b^(1/3)*x + 2*a^(1/3))^3))/(3*a^(1/6)*b^(1/3))
\[ \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=a^{\frac {1}{3}} \left (\int \frac {\sqrt {-b \,x^{3}+a}}{2 a^{\frac {4}{3}}-2 a^{\frac {1}{3}} b \,x^{3}+b^{\frac {1}{3}} a x -b^{\frac {4}{3}} x^{4}}d x \right )-b^{\frac {1}{3}} \left (\int \frac {\sqrt {-b \,x^{3}+a}\, x}{2 a^{\frac {4}{3}}-2 a^{\frac {1}{3}} b \,x^{3}+b^{\frac {1}{3}} a x -b^{\frac {4}{3}} x^{4}}d x \right ) \] Input:
int((a^(1/3)-b^(1/3)*x)/(2*a^(1/3)+b^(1/3)*x)/(-b*x^3+a)^(1/2),x)
Output:
a**(1/3)*int(sqrt(a - b*x**3)/(2*a**(1/3)*a - 2*a**(1/3)*b*x**3 + b**(1/3) *a*x - b**(1/3)*b*x**4),x) - b**(1/3)*int((sqrt(a - b*x**3)*x)/(2*a**(1/3) *a - 2*a**(1/3)*b*x**3 + b**(1/3)*a*x - b**(1/3)*b*x**4),x)