Integrand size = 52, antiderivative size = 207 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+a^2 x^3+2 c x^3\right )}{6 b^2 x^6}+\frac {\left (-a^4+12 a^2 b^2+2 a^2 c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}-\frac {\sqrt {-a+b} \left (3 a^2 b-a c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {-a+b}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+b} \left (3 a^2 b+a c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}} \]
1/6*(a^2*x^3+b^2)^(1/2)*(a^2*x^3+2*c*x^3+2*b^2)/b^2/x^6+1/6*(-a^4+12*a^2*b ^2+2*a^2*c)*arctanh((a^2*x^3+b^2)^(1/2)/b)/b^3-1/3*(-a+b)^(1/2)*(3*a^2*b-a *c)*arctanh((a^2*x^3+b^2)^(1/2)/b^(1/2)/(-a+b)^(1/2))/b^(5/2)-1/3*(a+b)^(1 /2)*(3*a^2*b+a*c)*arctanh((a^2*x^3+b^2)^(1/2)/b^(1/2)/(a+b)^(1/2))/b^(5/2)
Time = 0.69 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=-\frac {-\frac {b \sqrt {b^2+a^2 x^3} \left (2 b^2+\left (a^2+2 c\right ) x^3\right )}{x^6}+2 a \sqrt {a-b} \sqrt {b} (3 a b-c) \arctan \left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )+a^2 \left (a^2-2 \left (6 b^2+c\right )\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )+2 a \sqrt {b} \sqrt {a+b} (3 a b+c) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{6 b^3} \]
-1/6*(-((b*Sqrt[b^2 + a^2*x^3]*(2*b^2 + (a^2 + 2*c)*x^3))/x^6) + 2*a*Sqrt[ a - b]*Sqrt[b]*(3*a*b - c)*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt[b] )] + a^2*(a^2 - 2*(6*b^2 + c))*ArcTanh[Sqrt[b^2 + a^2*x^3]/b] + 2*a*Sqrt[b ]*Sqrt[a + b]*(3*a*b + c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b] )])/b^3
Time = 1.09 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.73, 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.
| \(\displaystyle \int \frac {\sqrt {a^2 x^3+b^2} \left (a^2 x^6+2 b^2+c x^3\right )}{x^7 \left (a^2 x^6-b^2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {c \sqrt {a^2 x^3+b^2}}{b^2 x^4}-\frac {3 a^2 \sqrt {a^2 x^3+b^2}}{b^2 x}-\frac {2 \sqrt {a^2 x^3+b^2}}{x^7}-\frac {a^2 x^2 (3 a b+c) \sqrt {a^2 x^3+b^2}}{2 b^3 \left (b-a x^3\right )}+\frac {a^2 x^2 (3 a b-c) \sqrt {a^2 x^3+b^2}}{2 b^3 \left (a x^3+b\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \sqrt {a-b} (3 a b-c) \arctan \left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )}{3 b^{5/2}}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{b}-\frac {a \sqrt {a+b} (3 a b+c) \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}}+\frac {a^2 c \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b^3}+\frac {c \sqrt {a^2 x^3+b^2}}{3 b^2 x^3}-\frac {2 a^2 \sqrt {a^2 x^3+b^2}}{b^2}+\frac {a^2 \sqrt {a^2 x^3+b^2}}{6 b^2 x^3}+\frac {\sqrt {a^2 x^3+b^2}}{3 x^6}+\frac {a (3 a b-c) \sqrt {a^2 x^3+b^2}}{3 b^3}+\frac {a (3 a b+c) \sqrt {a^2 x^3+b^2}}{3 b^3}-\frac {a^4 \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3}\) |
(-2*a^2*Sqrt[b^2 + a^2*x^3])/b^2 + (a*(3*a*b - c)*Sqrt[b^2 + a^2*x^3])/(3* b^3) + (a*(3*a*b + c)*Sqrt[b^2 + a^2*x^3])/(3*b^3) + Sqrt[b^2 + a^2*x^3]/( 3*x^6) + (a^2*Sqrt[b^2 + a^2*x^3])/(6*b^2*x^3) + (c*Sqrt[b^2 + a^2*x^3])/( 3*b^2*x^3) - (a*Sqrt[a - b]*(3*a*b - c)*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt[b])])/(3*b^(5/2)) - (a^4*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(6*b^3 ) + (2*a^2*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/b + (a^2*c*ArcTanh[Sqrt[b^2 + a ^2*x^3]/b])/(3*b^3) - (a*Sqrt[a + b]*(3*a*b + c)*ArcTanh[Sqrt[b^2 + a^2*x^ 3]/(Sqrt[b]*Sqrt[a + b])])/(3*b^(5/2))
3.25.96.3.1 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 = 1.24 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.99
| method | result | size |
| risch | \(\frac {\sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 c \,x^{3}+2 b^{2}\right )}{6 b^{2} x^{6}}+\frac {a^{2} \left (-\frac {2 \left (a^{2}-12 b^{2}-2 c \right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}+\frac {2 \left (-6 a^{2} b +6 a \,b^{2}+2 a c -2 b c \right ) \arctan \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a -b \right )}}\right )}{3 \sqrt {b \left (a -b \right )}\, a}-\frac {2 \left (6 a^{2} b +6 a \,b^{2}+2 a c +2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a +b \right )}}\right )}{3 \sqrt {b \left (a +b \right )}\, a}\right )}{4 b^{2}}\) | \(204\) |
| pseudoelliptic | \(\frac {a^{4} \left (-12 \left (a -b \right ) x^{6} \left (a b -\frac {c}{3}\right ) a \sqrt {b \left (a +b \right )}\, b \arctan \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a -b \right )}}\right )+\left (-12 x^{6} a \left (a +b \right ) \left (a b +\frac {c}{3}\right ) b \,\operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a +b \right )}}\right )+\left (a^{2} x^{6} \left (a^{2}-12 b^{2}-2 c \right ) \ln \left (-b +\sqrt {a^{2} x^{3}+b^{2}}\right )-a^{2} x^{6} \left (a^{2}-12 b^{2}-2 c \right ) \ln \left (\sqrt {a^{2} x^{3}+b^{2}}+b \right )+2 b \sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 c \,x^{3}+2 b^{2}\right )\right ) \sqrt {b \left (a +b \right )}\right ) \sqrt {b \left (a -b \right )}\right )}{12 \sqrt {b \left (a +b \right )}\, \sqrt {b \left (a -b \right )}\, b^{3} \left (\sqrt {a^{2} x^{3}+b^{2}}+b \right )^{2} \left (b -\sqrt {a^{2} x^{3}+b^{2}}\right )^{2}}\) | \(279\) |
| default | \(\frac {\sqrt {a^{2} x^{3}+b^{2}}}{3 x^{6}}+\frac {a^{2} \sqrt {a^{2} x^{3}+b^{2}}}{6 b^{2} x^{3}}-\frac {a^{4} \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{6 b^{2} \sqrt {b^{2}}}-\frac {c \left (-\frac {\sqrt {a^{2} x^{3}+b^{2}}}{3 x^{3}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{b^{2}}-\frac {3 a^{2} \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3}-\frac {2 b^{2} \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{b^{2}}+\frac {a \left (3 a b +c \right ) \left (\sqrt {a^{2} x^{3}+b^{2}}-\frac {b \left (a +b \right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a +b \right )}}\right )}{\sqrt {b \left (a +b \right )}}\right )}{3 b^{3}}+\frac {a \left (3 a b -c \right ) \left (\sqrt {a^{2} x^{3}+b^{2}}-\frac {b \left (a -b \right ) \arctan \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b \left (a -b \right )}}\right )}{\sqrt {b \left (a -b \right )}}\right )}{3 b^{3}}\) | \(316\) |
| elliptic | \(\text {Expression too large to display}\) | \(3271\) |
1/6*(a^2*x^3+b^2)^(1/2)*(a^2*x^3+2*c*x^3+2*b^2)/b^2/x^6+1/4/b^2*a^2*(-2/3* (a^2-12*b^2-2*c)*arctanh((a^2*x^3+b^2)^(1/2)/(b^2)^(1/2))/(b^2)^(1/2)+2/3* (-6*a^2*b+6*a*b^2+2*a*c-2*b*c)/(b*(a-b))^(1/2)*arctan((a^2*x^3+b^2)^(1/2)/ (b*(a-b))^(1/2))/a-2/3*(6*a^2*b+6*a*b^2+2*a*c+2*b*c)/(b*(a+b))^(1/2)*arcta nh((a^2*x^3+b^2)^(1/2)/(b*(a+b))^(1/2))/a)
Time = 0.77 (sec) , antiderivative size = 1051, normalized size of antiderivative = 5.08 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\left [-\frac {2 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \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 ) - 2 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \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 (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 2 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \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 (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - 2 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \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 (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 4 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}\right ] \]
[-1/12*(2*(3*a^2*b^2 - a*b*c)*x^6*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)) - 2*(3*a^2*b^ 2 + a*b*c)*x^6*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)) + (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6 *log(b + sqrt(a^2*x^3 + b^2)) - (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(-b + sqrt(a^2*x^3 + b^2)) - 2*sqrt(a^2*x^3 + b^2)*((a^2*b + 2*b*c)*x^3 + 2*b^3) )/(b^3*x^6), 1/12*(4*(3*a^2*b^2 - a*b*c)*x^6*sqrt((a - b)/b)*arctan(b*sqrt ((a - b)/b)/sqrt(a^2*x^3 + b^2)) + 2*(3*a^2*b^2 + a*b*c)*x^6*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)) - (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(b + sqrt(a^2*x^3 + b^2) ) + (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(-b + sqrt(a^2*x^3 + b^2)) + 2*sqr t(a^2*x^3 + b^2)*((a^2*b + 2*b*c)*x^3 + 2*b^3))/(b^3*x^6), 1/12*(4*(3*a^2* b^2 + a*b*c)*x^6*sqrt(-(a + b)/b)*arctan(b*sqrt(-(a + b)/b)/sqrt(a^2*x^3 + b^2)) - 2*(3*a^2*b^2 - a*b*c)*x^6*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)) - (a^4 - 12* a^2*b^2 - 2*a^2*c)*x^6*log(b + sqrt(a^2*x^3 + b^2)) + (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(-b + sqrt(a^2*x^3 + b^2)) + 2*sqrt(a^2*x^3 + b^2)*((a^2*b + 2*b*c)*x^3 + 2*b^3))/(b^3*x^6), 1/12*(4*(3*a^2*b^2 + a*b*c)*x^6*sqrt(-( a + b)/b)*arctan(b*sqrt(-(a + b)/b)/sqrt(a^2*x^3 + b^2)) + 4*(3*a^2*b^2 - a*b*c)*x^6*sqrt((a - b)/b)*arctan(b*sqrt((a - b)/b)/sqrt(a^2*x^3 + b^2)...
Timed out. \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\int { \frac {{\left (a^{2} x^{6} + c x^{3} + 2 \, b^{2}\right )} \sqrt {a^{2} x^{3} + b^{2}}}{{\left (a^{2} x^{6} - b^{2}\right )} x^{7}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=-\frac {{\left (3 \, a^{3} b - 3 \, a^{2} b^{2} - a^{2} c + a b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}}\right )}{3 \, \sqrt {a b - b^{2}} b^{2}} + \frac {{\left (3 \, a^{3} b + 3 \, a^{2} b^{2} + a^{2} c + a b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {-a b - b^{2}}}\right )}{3 \, \sqrt {-a b - b^{2}} b^{2}} - \frac {{\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} \log \left ({\left | b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right )}{12 \, b^{3}} + \frac {{\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} \log \left ({\left | -b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right )}{12 \, b^{3}} + \frac {\sqrt {a^{2} x^{3} + b^{2}} a^{4} b^{2} + {\left (a^{2} x^{3} + b^{2}\right )}^{\frac {3}{2}} a^{4} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} a^{2} b^{2} c + 2 \, {\left (a^{2} x^{3} + b^{2}\right )}^{\frac {3}{2}} a^{2} c}{6 \, a^{4} b^{2} x^{6}} \]
-1/3*(3*a^3*b - 3*a^2*b^2 - a^2*c + a*b*c)*arctan(sqrt(a^2*x^3 + b^2)/sqrt (a*b - b^2))/(sqrt(a*b - b^2)*b^2) + 1/3*(3*a^3*b + 3*a^2*b^2 + a^2*c + a* b*c)*arctan(sqrt(a^2*x^3 + b^2)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*b^2) - 1/12*(a^4 - 12*a^2*b^2 - 2*a^2*c)*log(abs(b + sqrt(a^2*x^3 + b^2)))/b^3 + 1/12*(a^4 - 12*a^2*b^2 - 2*a^2*c)*log(abs(-b + sqrt(a^2*x^3 + b^2)))/b^3 + 1/6*(sqrt(a^2*x^3 + b^2)*a^4*b^2 + (a^2*x^3 + b^2)^(3/2)*a^4 - 2*sqrt(a^ 2*x^3 + b^2)*a^2*b^2*c + 2*(a^2*x^3 + b^2)^(3/2)*a^2*c)/(a^4*b^2*x^6)
Time = 15.00 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (-b^2+a^2 x^6\right )} \, dx=\frac {\sqrt {a^2\,x^3+b^2}}{3\,x^6}+\frac {a^2\,\ln \left (\frac {{\left (b+\sqrt {a^2\,x^3+b^2}\right )}^3\,\left (b-\sqrt {a^2\,x^3+b^2}\right )}{x^6}\right )\,\left (-a^2+12\,b^2+2\,c\right )}{12\,b^3}+\frac {\sqrt {a^2\,x^3+b^2}\,\left (a^2+2\,c\right )}{6\,b^2\,x^3}+\frac {a\,\ln \left (\frac {a\,b+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a+b}}{b-a\,x^3}\right )\,\sqrt {a+b}\,\left (c+3\,a\,b\right )}{6\,b^{5/2}}+\frac {a\,\ln \left (\frac {2\,b^2-a\,b+a^2\,x^3+2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {b-a}}{a\,x^3+b}\right )\,\sqrt {b-a}\,\left (c-3\,a\,b\right )}{6\,b^{5/2}} \]
(b^2 + a^2*x^3)^(1/2)/(3*x^6) + (a^2*log(((b + (b^2 + a^2*x^3)^(1/2))^3*(b - (b^2 + a^2*x^3)^(1/2)))/x^6)*(2*c - a^2 + 12*b^2))/(12*b^3) + ((b^2 + a ^2*x^3)^(1/2)*(2*c + a^2))/(6*b^2*x^3) + (a*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*b^(5/2)) + (a*log((2*b^2 - a*b + a^2*x^3 + 2*b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(b - a)^(1/2))/(b + a*x^3))*(b - a)^(1/2)*(c - 3*a*b))/(6* b^(5/2))