Integrand size = 52, antiderivative size = 176 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx=\frac {2}{3} \sqrt {b^2+a^2 x^3}+\frac {4}{3} b \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )+\frac {\left (-3 a b \sqrt {-a+b}+\sqrt {-a+b} c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {-a+b}}\right )}{3 a \sqrt {b}}+\frac {\left (-3 a b \sqrt {a+b}-\sqrt {a+b} c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}} \]
2/3*(a^2*x^3+b^2)^(1/2)+4/3*b*arctanh((a^2*x^3+b^2)^(1/2)/b)+1/3*(-3*a*b*( -a+b)^(1/2)+(-a+b)^(1/2)*c)*arctanh((a^2*x^3+b^2)^(1/2)/b^(1/2)/(-a+b)^(1/ 2))/a/b^(1/2)+1/3*(-3*a*b*(a+b)^(1/2)-(a+b)^(1/2)*c)*arctanh((a^2*x^3+b^2) ^(1/2)/b^(1/2)/(a+b)^(1/2))/a/b^(1/2)
Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx=\frac {1}{3} \left (2 \sqrt {b^2+a^2 x^3}-\frac {\sqrt {a-b} (3 a b-c) \arctan \left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{a \sqrt {b}}+4 b \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )-\frac {\sqrt {a+b} (3 a b+c) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{a \sqrt {b}}\right ) \]
(2*Sqrt[b^2 + a^2*x^3] - (Sqrt[a - b]*(3*a*b - c)*ArcTan[Sqrt[b^2 + a^2*x^ 3]/(Sqrt[a - b]*Sqrt[b])])/(a*Sqrt[b]) + 4*b*ArcTanh[Sqrt[b^2 + a^2*x^3]/b ] - (Sqrt[a + b]*(3*a*b + c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b])])/(a*Sqrt[b]))/3
Time = 0.89 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {a^2 x^3+b^2} \left (a^2 x^6+2 b^2+c x^3\right )}{x \left (a^2 x^6-b^2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {x^2 (3 a b+c) \sqrt {a^2 x^3+b^2}}{2 b \left (b-a x^3\right )}+\frac {x^2 (3 a b-c) \sqrt {a^2 x^3+b^2}}{2 b \left (a x^3+b\right )}-\frac {2 \sqrt {a^2 x^3+b^2}}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {a-b} (3 a b-c) \arctan \left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )}{3 a \sqrt {b}}-\frac {\sqrt {a+b} (3 a b+c) \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 a \sqrt {b}}+\frac {4}{3} b \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )+\frac {(3 a b-c) \sqrt {a^2 x^3+b^2}}{3 a b}+\frac {(3 a b+c) \sqrt {a^2 x^3+b^2}}{3 a b}-\frac {4}{3} \sqrt {a^2 x^3+b^2}\) |
(-4*Sqrt[b^2 + a^2*x^3])/3 + ((3*a*b - c)*Sqrt[b^2 + a^2*x^3])/(3*a*b) + ( (3*a*b + c)*Sqrt[b^2 + a^2*x^3])/(3*a*b) - (Sqrt[a - b]*(3*a*b - c)*ArcTan [Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt[b])])/(3*a*Sqrt[b]) + (4*b*ArcTanh[ Sqrt[b^2 + a^2*x^3]/b])/3 - (Sqrt[a + b]*(3*a*b + c)*ArcTanh[Sqrt[b^2 + a^ 2*x^3]/(Sqrt[b]*Sqrt[a + b])])/(3*a*Sqrt[b])
3.23.100.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 1.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\frac {3 \left (a b -\frac {c}{3}\right ) \left (a -b \right ) \sqrt {\left (a +b \right ) b}\, \arctan \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {\left (a -b \right ) b}}\right )}{2}+\sqrt {\left (a -b \right ) b}\, \left (\frac {3 \left (a +b \right ) \left (a b +\frac {c}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {\left (a +b \right ) b}}\right )}{2}+a \left (b \ln \left (-b +\sqrt {a^{2} x^{3}+b^{2}}\right )-b \ln \left (b +\sqrt {a^{2} x^{3}+b^{2}}\right )-\sqrt {a^{2} x^{3}+b^{2}}\right ) \sqrt {\left (a +b \right ) b}\right )\right )}{3 \sqrt {\left (a -b \right ) b}\, \sqrt {\left (a +b \right ) b}\, a}\) | \(177\) |
| default | \(-\frac {4 \sqrt {a^{2} x^{3}+b^{2}}}{3}+\frac {4 b^{2} \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}+\frac {\left (3 a b -c \right ) \left (\sqrt {a^{2} x^{3}+b^{2}}-\frac {b \left (a -b \right ) \arctan \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {\left (a -b \right ) b}}\right )}{\sqrt {\left (a -b \right ) b}}\right )}{3 b a}+\frac {\left (3 a b +c \right ) \left (\sqrt {a^{2} x^{3}+b^{2}}-\frac {b \left (a +b \right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {\left (a +b \right ) b}}\right )}{\sqrt {\left (a +b \right ) b}}\right )}{3 b a}\) | \(181\) |
| elliptic | \(\text {Expression too large to display}\) | \(3175\) |
-2/3*(3/2*(a*b-1/3*c)*(a-b)*((a+b)*b)^(1/2)*arctan((a^2*x^3+b^2)^(1/2)/((a -b)*b)^(1/2))+((a-b)*b)^(1/2)*(3/2*(a+b)*(a*b+1/3*c)*arctanh((a^2*x^3+b^2) ^(1/2)/((a+b)*b)^(1/2))+a*(b*ln(-b+(a^2*x^3+b^2)^(1/2))-b*ln(b+(a^2*x^3+b^ 2)^(1/2))-(a^2*x^3+b^2)^(1/2))*((a+b)*b)^(1/2)))/((a-b)*b)^(1/2)/((a+b)*b) ^(1/2)/a
Time = 0.40 (sec) , antiderivative size = 735, normalized size of antiderivative = 4.18 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx=\left [\frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - {\left (3 \, a b - c\right )} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) + {\left (3 \, a b + c\right )} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, {\left (3 \, a b - c\right )} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + {\left (3 \, a b + c\right )} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {4 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 4 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, {\left (3 \, a b + c\right )} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - {\left (3 \, a b - c\right )} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) + 4 \, \sqrt {a^{2} x^{3} + b^{2}} a}{6 \, a}, \frac {2 \, a b \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 2 \, a b \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (3 \, a b + c\right )} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + {\left (3 \, a b - c\right )} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} a}{3 \, a}\right ] \]
[1/6*(4*a*b*log(b + sqrt(a^2*x^3 + b^2)) - 4*a*b*log(-b + sqrt(a^2*x^3 + b ^2)) - (3*a*b - c)*sqrt(-(a - b)/b)*log((a^2*x^3 - a*b + 2*b^2 + 2*sqrt(a^ 2*x^3 + b^2)*b*sqrt(-(a - b)/b))/(a*x^3 + b)) + (3*a*b + c)*sqrt((a + b)/b )*log((a^2*x^3 + a*b + 2*b^2 - 2*sqrt(a^2*x^3 + b^2)*b*sqrt((a + b)/b))/(a *x^3 - b)) + 4*sqrt(a^2*x^3 + b^2)*a)/a, 1/6*(4*a*b*log(b + sqrt(a^2*x^3 + b^2)) - 4*a*b*log(-b + sqrt(a^2*x^3 + b^2)) + 2*(3*a*b - c)*sqrt((a - b)/ b)*arctan(b*sqrt((a - b)/b)/sqrt(a^2*x^3 + b^2)) + (3*a*b + c)*sqrt((a + b )/b)*log((a^2*x^3 + a*b + 2*b^2 - 2*sqrt(a^2*x^3 + b^2)*b*sqrt((a + b)/b)) /(a*x^3 - b)) + 4*sqrt(a^2*x^3 + b^2)*a)/a, 1/6*(4*a*b*log(b + sqrt(a^2*x^ 3 + b^2)) - 4*a*b*log(-b + sqrt(a^2*x^3 + b^2)) + 2*(3*a*b + c)*sqrt(-(a + b)/b)*arctan(b*sqrt(-(a + b)/b)/sqrt(a^2*x^3 + b^2)) - (3*a*b - c)*sqrt(- (a - b)/b)*log((a^2*x^3 - a*b + 2*b^2 + 2*sqrt(a^2*x^3 + b^2)*b*sqrt(-(a - b)/b))/(a*x^3 + b)) + 4*sqrt(a^2*x^3 + b^2)*a)/a, 1/3*(2*a*b*log(b + sqrt (a^2*x^3 + b^2)) - 2*a*b*log(-b + sqrt(a^2*x^3 + b^2)) + (3*a*b + c)*sqrt( -(a + b)/b)*arctan(b*sqrt(-(a + b)/b)/sqrt(a^2*x^3 + b^2)) + (3*a*b - c)*s qrt((a - b)/b)*arctan(b*sqrt((a - b)/b)/sqrt(a^2*x^3 + b^2)) + 2*sqrt(a^2* x^3 + b^2)*a)/a]
Time = 50.98 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx=\begin {cases} \frac {2 \left (- \frac {a^{2} b \log {\left (b - \sqrt {a^{2} x^{3} + b^{2}} \right )}}{3} + \frac {a^{2} b \log {\left (b + \sqrt {a^{2} x^{3} + b^{2}} \right )}}{3} + \frac {a^{2} \sqrt {a^{2} x^{3} + b^{2}}}{3} - \frac {a \left (a - b\right ) \left (3 a b - c\right ) \operatorname {atan}{\left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}} \right )}}{6 \sqrt {a b - b^{2}}} + \frac {a \left (a + b\right ) \left (3 a b + c\right ) \operatorname {atan}{\left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {- a b - b^{2}}} \right )}}{6 \sqrt {- a b - b^{2}}}\right )}{a^{2}} & \text {for}\: a^{2} \neq 0 \\- \frac {2 b^{2} \sqrt {b^{2}} \log {\left (c x^{3} \sqrt {b^{2}} \right )} + c x^{3} \sqrt {b^{2}}}{3 b^{2}} & \text {otherwise} \end {cases} \]
Piecewise((2*(-a**2*b*log(b - sqrt(a**2*x**3 + b**2))/3 + a**2*b*log(b + s qrt(a**2*x**3 + b**2))/3 + a**2*sqrt(a**2*x**3 + b**2)/3 - a*(a - b)*(3*a* b - c)*atan(sqrt(a**2*x**3 + b**2)/sqrt(a*b - b**2))/(6*sqrt(a*b - b**2)) + a*(a + b)*(3*a*b + c)*atan(sqrt(a**2*x**3 + b**2)/sqrt(-a*b - b**2))/(6* sqrt(-a*b - b**2)))/a**2, Ne(a**2, 0)), (-(2*b**2*sqrt(b**2)*log(c*x**3*sq rt(b**2)) + c*x**3*sqrt(b**2))/(3*b**2), True))
\[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx=\int { \frac {{\left (a^{2} x^{6} + c x^{3} + 2 \, b^{2}\right )} \sqrt {a^{2} x^{3} + b^{2}}}{{\left (a^{2} x^{6} - b^{2}\right )} x} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx=\frac {2}{3} \, b \log \left ({\left | b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right ) - \frac {2}{3} \, b \log \left ({\left | -b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right ) + \frac {2}{3} \, \sqrt {a^{2} x^{3} + b^{2}} - \frac {{\left (3 \, a^{2} b - 3 \, a b^{2} - a c + b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}}\right )}{3 \, \sqrt {a b - b^{2}} a} + \frac {{\left (3 \, a^{2} b + 3 \, a b^{2} + a c + b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {-a b - b^{2}}}\right )}{3 \, \sqrt {-a b - b^{2}} a} \]
2/3*b*log(abs(b + sqrt(a^2*x^3 + b^2))) - 2/3*b*log(abs(-b + sqrt(a^2*x^3 + b^2))) + 2/3*sqrt(a^2*x^3 + b^2) - 1/3*(3*a^2*b - 3*a*b^2 - a*c + b*c)*a rctan(sqrt(a^2*x^3 + b^2)/sqrt(a*b - b^2))/(sqrt(a*b - b^2)*a) + 1/3*(3*a^ 2*b + 3*a*b^2 + a*c + b*c)*arctan(sqrt(a^2*x^3 + b^2)/sqrt(-a*b - b^2))/(s qrt(-a*b - b^2)*a)
Time = 15.68 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x \left (-b^2+a^2 x^6\right )} \, dx=\frac {2\,\sqrt {a^2\,x^3+b^2}}{3}+\frac {2\,b\,\ln \left (\frac {{\left (b+\sqrt {a^2\,x^3+b^2}\right )}^3\,\left (b-\sqrt {a^2\,x^3+b^2}\right )}{x^6}\right )}{3}+\frac {\ln \left (\frac {a\,b+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a+b}}{b-a\,x^3}\right )\,\sqrt {a+b}\,\left (c+3\,a\,b\right )}{6\,a\,\sqrt {b}}+\frac {\ln \left (\frac {2\,b^2-a\,b+a^2\,x^3+\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a-b}\,2{}\mathrm {i}}{a\,x^3+b}\right )\,\sqrt {a-b}\,\left (c-3\,a\,b\right )\,1{}\mathrm {i}}{6\,a\,\sqrt {b}} \]
(2*(b^2 + a^2*x^3)^(1/2))/3 + (2*b*log(((b + (b^2 + a^2*x^3)^(1/2))^3*(b - (b^2 + a^2*x^3)^(1/2)))/x^6))/3 + (log((2*b^2 - a*b + a^2*x^3 + b^(1/2)*( b^2 + a^2*x^3)^(1/2)*(a - b)^(1/2)*2i)/(b + a*x^3))*(a - b)^(1/2)*(c - 3*a *b)*1i)/(6*a*b^(1/2)) + (log((a*b + 2*b^2 + a^2*x^3 - 2*b^(1/2)*(b^2 + a^2 *x^3)^(1/2)*(a + b)^(1/2))/(b - a*x^3))*(a + b)^(1/2)*(c + 3*a*b))/(6*a*b^ (1/2))