Integrand size = 38, antiderivative size = 66 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {b}} \] Output:
-1/3*arctan((a*x^3-b)^(1/2)/b^(1/2))/b^(1/2)+arctan(1/3*(a*x^3-b)^(1/2)*3^ (1/2)/b^(1/2))*3^(1/2)/b^(1/2)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )-3 \sqrt {3} \arctan \left (\frac {\sqrt {-b+a x^3}}{\sqrt {3} \sqrt {b}}\right )}{3 \sqrt {b}} \] Input:
Integrate[(-b + 4*a*x^3)/(x*Sqrt[-b + a*x^3]*(2*b + a*x^3)),x]
Output:
-1/3*(ArcTan[Sqrt[-b + a*x^3]/Sqrt[b]] - 3*Sqrt[3]*ArcTan[Sqrt[-b + a*x^3] /(Sqrt[3]*Sqrt[b])])/Sqrt[b]
Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1043, 25, 174, 73, 216, 218}
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.
\(\displaystyle \int \frac {4 a x^3-b}{x \sqrt {a x^3-b} \left (a x^3+2 b\right )} \, dx\) |
\(\Big \downarrow \) 1043 |
\(\displaystyle \frac {1}{3} \int -\frac {b-4 a x^3}{x^3 \sqrt {a x^3-b} \left (a x^3+2 b\right )}dx^3\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {b-4 a x^3}{x^3 \sqrt {a x^3-b} \left (a x^3+2 b\right )}dx^3\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{3} \left (\frac {9}{2} a \int \frac {1}{\sqrt {a x^3-b} \left (a x^3+2 b\right )}dx^3-\frac {1}{2} \int \frac {1}{x^3 \sqrt {a x^3-b}}dx^3\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (9 \int \frac {1}{x^6+3 b}d\sqrt {a x^3-b}-\frac {\int \frac {1}{\frac {x^6}{a}+\frac {b}{a}}d\sqrt {a x^3-b}}{a}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {b}}-\frac {\int \frac {1}{\frac {x^6}{a}+\frac {b}{a}}d\sqrt {a x^3-b}}{a}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{3} \left (\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{\sqrt {b}}\right )\) |
Input:
Int[(-b + 4*a*x^3)/(x*Sqrt[-b + a*x^3]*(2*b + a*x^3)),x]
Output:
(-(ArcTan[Sqrt[-b + a*x^3]/Sqrt[b]]/Sqrt[b]) + (3*Sqrt[3]*ArcTan[Sqrt[-b + a*x^3]/(Sqrt[3]*Sqrt[b])])/Sqrt[b])/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. )*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simp lify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] / ; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/ n]]
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {3 \arctan \left (\frac {\sqrt {a \,x^{3}-b}\, \sqrt {3}}{3 \sqrt {b}}\right ) \sqrt {3}-\arctan \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\) | \(50\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}+\frac {\arctan \left (\frac {\sqrt {a \,x^{3}-b}\, \sqrt {3}}{3 \sqrt {b}}\right ) \sqrt {3}}{\sqrt {b}}\) | \(54\) |
elliptic | \(\text {Expression too large to display}\) | \(1430\) |
Input:
int((4*a*x^3-b)/x/(a*x^3-b)^(1/2)/(a*x^3+2*b),x,method=_RETURNVERBOSE)
Output:
1/3*(3*arctan(1/3*(a*x^3-b)^(1/2)*3^(1/2)/b^(1/2))*3^(1/2)-arctan((a*x^3-b )^(1/2)/b^(1/2)))/b^(1/2)
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.41 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\left [\frac {3 \, \sqrt {3} b \sqrt {-\frac {1}{b}} \log \left (\frac {a x^{3} + 2 \, \sqrt {3} \sqrt {a x^{3} - b} b \sqrt {-\frac {1}{b}} - 4 \, b}{a x^{3} + 2 \, b}\right ) - \sqrt {-b} \log \left (\frac {a x^{3} + 2 \, \sqrt {a x^{3} - b} \sqrt {-b} - 2 \, b}{x^{3}}\right )}{6 \, b}, \frac {3 \, \sqrt {3} \sqrt {b} \arctan \left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \, \sqrt {b}}\right ) + \sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {a x^{3} - b}}\right )}{3 \, b}\right ] \] Input:
integrate((4*a*x^3-b)/x/(a*x^3-b)^(1/2)/(a*x^3+2*b),x, algorithm="fricas")
Output:
[1/6*(3*sqrt(3)*b*sqrt(-1/b)*log((a*x^3 + 2*sqrt(3)*sqrt(a*x^3 - b)*b*sqrt (-1/b) - 4*b)/(a*x^3 + 2*b)) - sqrt(-b)*log((a*x^3 + 2*sqrt(a*x^3 - b)*sqr t(-b) - 2*b)/x^3))/b, 1/3*(3*sqrt(3)*sqrt(b)*arctan(1/3*sqrt(3)*sqrt(a*x^3 - b)/sqrt(b)) + sqrt(b)*arctan(sqrt(b)/sqrt(a*x^3 - b)))/b]
Time = 5.98 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\begin {cases} \frac {2 \left (- \frac {a \operatorname {atan}{\left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}} \right )}}{6 \sqrt {b}} + \frac {\sqrt {3} a \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \sqrt {b}} \right )}}{2 \sqrt {b}}\right )}{a} & \text {for}\: a \neq 0 \\- \frac {\log {\left (x^{3} \right )}}{6 \sqrt {- b}} & \text {otherwise} \end {cases} \] Input:
integrate((4*a*x**3-b)/x/(a*x**3-b)**(1/2)/(a*x**3+2*b),x)
Output:
Piecewise((2*(-a*atan(sqrt(a*x**3 - b)/sqrt(b))/(6*sqrt(b)) + sqrt(3)*a*at an(sqrt(3)*sqrt(a*x**3 - b)/(3*sqrt(b)))/(2*sqrt(b)))/a, Ne(a, 0)), (-log( x**3)/(6*sqrt(-b)), True))
\[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\int { \frac {4 \, a x^{3} - b}{{\left (a x^{3} + 2 \, b\right )} \sqrt {a x^{3} - b} x} \,d x } \] Input:
integrate((4*a*x^3-b)/x/(a*x^3-b)^(1/2)/(a*x^3+2*b),x, algorithm="maxima")
Output:
integrate((4*a*x^3 - b)/((a*x^3 + 2*b)*sqrt(a*x^3 - b)*x), x)
Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \, \sqrt {b}}\right )}{\sqrt {b}} - \frac {\arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}} \] Input:
integrate((4*a*x^3-b)/x/(a*x^3-b)^(1/2)/(a*x^3+2*b),x, algorithm="giac")
Output:
sqrt(3)*arctan(1/3*sqrt(3)*sqrt(a*x^3 - b)/sqrt(b))/sqrt(b) - 1/3*arctan(s qrt(a*x^3 - b)/sqrt(b))/sqrt(b)
Time = 9.81 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.45 \[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\frac {\ln \left (\frac {2\,b-a\,x^3+\sqrt {b}\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{6\,\sqrt {b}}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}\,b\,4{}\mathrm {i}+6\,\sqrt {b}\,\sqrt {a\,x^3-b}-\sqrt {3}\,a\,x^3\,1{}\mathrm {i}}{6\,a\,x^3+12\,b}\right )\,1{}\mathrm {i}}{2\,\sqrt {b}} \] Input:
int(-(b - 4*a*x^3)/(x*(a*x^3 - b)^(1/2)*(2*b + a*x^3)),x)
Output:
(log((2*b + b^(1/2)*(a*x^3 - b)^(1/2)*2i - a*x^3)/x^3)*1i)/(6*b^(1/2)) + ( 3^(1/2)*log((3^(1/2)*b*4i + 6*b^(1/2)*(a*x^3 - b)^(1/2) - 3^(1/2)*a*x^3*1i )/(12*b + 6*a*x^3))*1i)/(2*b^(1/2))
\[ \int \frac {-b+4 a x^3}{x \sqrt {-b+a x^3} \left (2 b+a x^3\right )} \, dx=\frac {-\sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {b}\, \sqrt {a \,x^{3}-b}\, a \,x^{3}-2 \sqrt {b}\, \sqrt {a \,x^{3}-b}\, b}{2 a b \,x^{3}-2 b^{2}}\right )+27 \left (\int \frac {\sqrt {a \,x^{3}-b}\, x^{2}}{a^{2} x^{6}+a b \,x^{3}-2 b^{2}}d x \right ) a b}{6 b} \] Input:
int((4*a*x^3-b)/x/(a*x^3-b)^(1/2)/(a*x^3+2*b),x)
Output:
( - sqrt(b)*atan((sqrt(b)*sqrt(a*x**3 - b)*a*x**3 - 2*sqrt(b)*sqrt(a*x**3 - b)*b)/(2*a*b*x**3 - 2*b**2)) + 27*int((sqrt(a*x**3 - b)*x**2)/(a**2*x**6 + a*b*x**3 - 2*b**2),x)*a*b)/(6*b)