Integrand size = 56, antiderivative size = 66 \[ \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \] Output:
2/3*2^(2/3)*arctanh(3^(1/2)*a^(1/6)*(a^(1/3)+2^(1/3)*b^(1/3)*x)/(-b*x^3-a) ^(1/2))*3^(1/2)/a^(1/6)/b^(1/3)
Time = 6.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {-a-b x^3}}{\sqrt {3} \left (\sqrt {a}+\sqrt [3]{2} \sqrt [6]{a} \sqrt [3]{b} x\right )}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \] Input:
Integrate[(2^(2/3)*a^(1/3) - 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*S qrt[-a - b*x^3]),x]
Output:
(2*2^(2/3)*ArcTanh[Sqrt[-a - b*x^3]/(Sqrt[3]*(Sqrt[a] + 2^(1/3)*a^(1/6)*b^ (1/3)*x))])/(Sqrt[3]*a^(1/6)*b^(1/3))
Time = 0.54 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2562, 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.
\(\displaystyle \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx\) |
\(\Big \downarrow \) 2562 |
\(\displaystyle \frac {2\ 2^{2/3} \sqrt [3]{a} \int \frac {1}{1-\frac {3 \sqrt [3]{a} \left (\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{-b x^3-a}}d\frac {\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt {-b x^3-a}}}{\sqrt [3]{b}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}}\) |
Input:
Int[(2^(2/3)*a^(1/3) - 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*Sqrt[-a - b*x^3]),x]
Output:
(2*2^(2/3)*ArcTanh[(Sqrt[3]*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/Sqrt[-a - b*x^3]])/(Sqrt[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/(1 + 3*a*x^2), x], x, (1 + 2*d*(x/c)) /Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
\[\int \frac {2^{\frac {2}{3}} a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{\left (2^{\frac {2}{3}} a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {-b \,x^{3}-a}}d x\]
Input:
int((2^(2/3)*a^(1/3)-2*b^(1/3)*x)/(2^(2/3)*a^(1/3)+b^(1/3)*x)/(-b*x^3-a)^( 1/2),x)
Output:
int((2^(2/3)*a^(1/3)-2*b^(1/3)*x)/(2^(2/3)*a^(1/3)+b^(1/3)*x)/(-b*x^3-a)^( 1/2),x)
Timed out. \[ \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\text {Timed out} \] Input:
integrate((2^(2/3)*a^(1/3)-2*b^(1/3)*x)/(2^(2/3)*a^(1/3)+b^(1/3)*x)/(-b*x^ 3-a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=- \int \left (- \frac {2^{\frac {2}{3}} \sqrt [3]{a}}{2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {- a - b x^{3}} + \sqrt [3]{b} x \sqrt {- a - b x^{3}}}\right )\, dx - \int \frac {2 \sqrt [3]{b} x}{2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {- a - b x^{3}} + \sqrt [3]{b} x \sqrt {- a - b x^{3}}}\, dx \] Input:
integrate((2**(2/3)*a**(1/3)-2*b**(1/3)*x)/(2**(2/3)*a**(1/3)+b**(1/3)*x)/ (-b*x**3-a)**(1/2),x)
Output:
-Integral(-2**(2/3)*a**(1/3)/(2**(2/3)*a**(1/3)*sqrt(-a - b*x**3) + b**(1/ 3)*x*sqrt(-a - b*x**3)), x) - Integral(2*b**(1/3)*x/(2**(2/3)*a**(1/3)*sqr t(-a - b*x**3) + b**(1/3)*x*sqrt(-a - b*x**3)), x)
\[ \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\int { -\frac {2 \, b^{\frac {1}{3}} x - 2^{\frac {2}{3}} a^{\frac {1}{3}}}{\sqrt {-b x^{3} - a} {\left (b^{\frac {1}{3}} x + 2^{\frac {2}{3}} a^{\frac {1}{3}}\right )}} \,d x } \] Input:
integrate((2^(2/3)*a^(1/3)-2*b^(1/3)*x)/(2^(2/3)*a^(1/3)+b^(1/3)*x)/(-b*x^ 3-a)^(1/2),x, algorithm="maxima")
Output:
-integrate((2*b^(1/3)*x - 2^(2/3)*a^(1/3))/(sqrt(-b*x^3 - a)*(b^(1/3)*x + 2^(2/3)*a^(1/3))), x)
Timed out. \[ \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\text {Timed out} \] Input:
integrate((2^(2/3)*a^(1/3)-2*b^(1/3)*x)/(2^(2/3)*a^(1/3)+b^(1/3)*x)/(-b*x^ 3-a)^(1/2),x, algorithm="giac")
Output:
Timed out
Time = 24.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.56 \[ \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=\frac {\sqrt {3}\,4^{1/3}\,\ln \left (\frac {{\left (\sqrt {-b\,x^3-a}+\sqrt {3}\,\sqrt {a}+2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\right )}^3\,\left (\sqrt {3}\,\sqrt {a}-\sqrt {-b\,x^3-a}+2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\right )}{{\left (2^{2/3}\,a^{1/3}+b^{1/3}\,x\right )}^6}\right )}{3\,a^{1/6}\,b^{1/3}} \] Input:
int((2^(2/3)*a^(1/3) - 2*b^(1/3)*x)/((- a - b*x^3)^(1/2)*(2^(2/3)*a^(1/3) + b^(1/3)*x)),x)
Output:
(3^(1/2)*4^(1/3)*log((((- a - b*x^3)^(1/2) + 3^(1/2)*a^(1/2) + 2^(1/3)*3^( 1/2)*a^(1/6)*b^(1/3)*x)^3*(3^(1/2)*a^(1/2) - (- a - b*x^3)^(1/2) + 2^(1/3) *3^(1/2)*a^(1/6)*b^(1/3)*x))/(2^(2/3)*a^(1/3) + b^(1/3)*x)^6))/(3*a^(1/6)* b^(1/3))
\[ \int \frac {2^{2/3} \sqrt [3]{a}-2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx=i \left (-a^{\frac {1}{3}} 2^{\frac {2}{3}} \left (\int \frac {1}{a^{\frac {1}{3}} \sqrt {b \,x^{3}+a}\, 2^{\frac {2}{3}}+b^{\frac {1}{3}} \sqrt {b \,x^{3}+a}\, x}d x \right )+2 b^{\frac {1}{3}} \left (\int \frac {x}{a^{\frac {1}{3}} \sqrt {b \,x^{3}+a}\, 2^{\frac {2}{3}}+b^{\frac {1}{3}} \sqrt {b \,x^{3}+a}\, x}d x \right )\right ) \] Input:
int((2^(2/3)*a^(1/3)-2*b^(1/3)*x)/(2^(2/3)*a^(1/3)+b^(1/3)*x)/(-b*x^3-a)^( 1/2),x)
Output:
i*( - a**(1/3)*2**(2/3)*int(1/(a**(1/3)*sqrt(a + b*x**3)*2**(2/3) + b**(1/ 3)*sqrt(a + b*x**3)*x),x) + 2*b**(1/3)*int(x/(a**(1/3)*sqrt(a + b*x**3)*2* *(2/3) + b**(1/3)*sqrt(a + b*x**3)*x),x))