Integrand size = 50, antiderivative size = 247 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx=\frac {\sqrt {b^2+a^2 x^3} \left (-2 b^2-a^2 x^3-2 c x^3\right )}{6 b^2 x^6}+\frac {\sqrt {a-i b} \left ((-1)^{3/4} a^2 b-\sqrt [4]{-1} a c\right ) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+i b} \left (-\sqrt [4]{-1} a^2 b+(-1)^{3/4} a c\right ) \arctan \left (\frac {(-1)^{3/4} \sqrt {b^2+a^2 x^3}}{\sqrt {a+i b} \sqrt {b}}\right )}{3 b^{5/2}}+\frac {\left (a^4+4 a^2 b^2-2 a^2 c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3} \]
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+1/3*(a-I*b)^(1/2) *((-1)^(3/4)*a^2*b-(-1)^(1/4)*a*c)*arctan((-1)^(1/4)*(a^2*x^3+b^2)^(1/2)/( a-I*b)^(1/2)/b^(1/2))/b^(5/2)-1/3*(a+I*b)^(1/2)*(-(-1)^(1/4)*a^2*b+(-1)^(3 /4)*a*c)*arctan((-1)^(3/4)*(a^2*x^3+b^2)^(1/2)/(a+I*b)^(1/2)/b^(1/2))/b^(5 /2)+1/6*(a^4+4*a^2*b^2-2*a^2*c)*arctanh((a^2*x^3+b^2)^(1/2)/b)/b^3
Time = 0.84 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx=\frac {-\frac {b \sqrt {b^2+a^2 x^3} \left (2 b^2+\left (a^2+2 c\right ) x^3\right )}{x^6}+2 (-1)^{3/4} a \sqrt {a-i b} \sqrt {b} (a b+i c) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )+2 \sqrt [4]{-1} a \sqrt {a+i b} \sqrt {b} (a b-i c) \arctan \left (\frac {(-1)^{3/4} \sqrt {b^2+a^2 x^3}}{\sqrt {a+i b} \sqrt {b}}\right )+a^2 \left (a^2+4 b^2-2 c\right ) \text {arctanh}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3} \]
(-((b*Sqrt[b^2 + a^2*x^3]*(2*b^2 + (a^2 + 2*c)*x^3))/x^6) + 2*(-1)^(3/4)*a *Sqrt[a - I*b]*Sqrt[b]*(a*b + I*c)*ArcTan[((-1)^(1/4)*Sqrt[b^2 + a^2*x^3]) /(Sqrt[a - I*b]*Sqrt[b])] + 2*(-1)^(1/4)*a*Sqrt[a + I*b]*Sqrt[b]*(a*b - I* c)*ArcTan[((-1)^(3/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a + I*b]*Sqrt[b])] + a^2* (a^2 + 4*b^2 - 2*c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(6*b^3)
Leaf count is larger than twice the leaf count of optimal. \(744\) vs. \(2(247)=494\).
Time = 3.24 (sec) , antiderivative size = 744, normalized size of antiderivative = 3.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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 |
| \(\displaystyle \int \left (\frac {c \sqrt {a^2 x^3+b^2}}{b^2 x^4}+\frac {a^2 x^2 \sqrt {a^2 x^3+b^2} \left (a^2 x^3-c\right )}{b^2 \left (a^2 x^6+b^2\right )}-\frac {a^2 \sqrt {a^2 x^3+b^2}}{b^2 x}+\frac {2 \sqrt {a^2 x^3+b^2}}{x^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b}-\frac {a^2 c \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b^3}-\frac {a^2 \left (-\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\sqrt {a^2+b^2}+b}-\sqrt {2} \sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (-\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a^2 x^3+b^2}+\sqrt {b} \sqrt {\sqrt {a^2+b^2}+b}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}-\frac {c \sqrt {a^2 x^3+b^2}}{3 b^2 x^3}-\frac {a^2 \sqrt {a^2 x^3+b^2}}{6 b^2 x^3}-\frac {\sqrt {a^2 x^3+b^2}}{3 x^6}+\frac {a^2 \left (\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \log \left (-\sqrt {2} \sqrt {b} \sqrt {\sqrt {a^2+b^2}+b} \sqrt {a^2 x^3+b^2}+b \left (\sqrt {a^2+b^2}+b\right )+a^2 x^3\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}-\frac {a^2 \left (\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \log \left (\sqrt {2} \sqrt {b} \sqrt {\sqrt {a^2+b^2}+b} \sqrt {a^2 x^3+b^2}+b \left (\sqrt {a^2+b^2}+b\right )+a^2 x^3\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}+\frac {a^4 \text {arctanh}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3}\) |
-1/3*Sqrt[b^2 + a^2*x^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^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])/(3*b) - (a^2*c*ArcTanh[Sqrt [b^2 + a^2*x^3]/b])/(3*b^3) - (a^2*(a^2*b + b^3 - Sqrt[a^2 + b^2]*(b^2 - c ))*ArcTanh[(Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[b^2 + a^2*x^3 ])/(Sqrt[b]*Sqrt[b - Sqrt[a^2 + b^2]])])/(3*Sqrt[2]*b^(5/2)*Sqrt[a^2 + b^2 ]*Sqrt[b - Sqrt[a^2 + b^2]]) + (a^2*(a^2*b + b^3 - Sqrt[a^2 + b^2]*(b^2 - c))*ArcTanh[(Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]] + Sqrt[2]*Sqrt[b^2 + a^2*x^ 3])/(Sqrt[b]*Sqrt[b - Sqrt[a^2 + b^2]])])/(3*Sqrt[2]*b^(5/2)*Sqrt[a^2 + b^ 2]*Sqrt[b - Sqrt[a^2 + b^2]]) + (a^2*(a^2*b + b^3 + Sqrt[a^2 + b^2]*(b^2 - c))*Log[b*(b + Sqrt[a^2 + b^2]) + a^2*x^3 - Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt [a^2 + b^2]]*Sqrt[b^2 + a^2*x^3]])/(6*Sqrt[2]*b^(5/2)*Sqrt[a^2 + b^2]*Sqrt [b + Sqrt[a^2 + b^2]]) - (a^2*(a^2*b + b^3 + Sqrt[a^2 + b^2]*(b^2 - c))*Lo g[b*(b + Sqrt[a^2 + b^2]) + a^2*x^3 + Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[b^2 + a^2*x^3]])/(6*Sqrt[2]*b^(5/2)*Sqrt[a^2 + b^2]*Sqrt[b + Sq rt[a^2 + b^2]])
3.28.9.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(199)=398\).
Time = 1.49 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.98
| method | result | size |
| risch | \(-\frac {\sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 c \,x^{3}+2 b^{2}\right )}{6 b^{2} x^{6}}-\frac {a^{2} \left (-\frac {2 \left (a^{2}+4 b^{2}-2 c \right ) \operatorname {arctanh}\left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}-\frac {4 \left (-\frac {\left (-c \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+b^{2} \left (a^{2}+c \right )\right ) \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}\, \left (\ln \left (b^{2}+a^{2} x^{3}+\sqrt {a^{2} x^{3}+b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}+\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right )-\ln \left (-a^{2} x^{3}+\sqrt {a^{2} x^{3}+b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}-b^{2}-\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right )\right ) \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}}{4}+\left (\arctan \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}-\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}}{\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}}\right )+\arctan \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}+\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}}{\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}}\right )\right ) a^{2} \left (b^{2}-c -\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right ) b^{2}\right )}{3 \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}\, a^{2} b^{2}}\right )}{4 b^{2}}\) | \(488\) |
| pseudoelliptic | \(-\frac {a^{4} \left (-\left (-c \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+b^{2} \left (a^{2}+c \right )\right ) \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}\, x^{6} \ln \left (-a^{2} x^{3}+\sqrt {a^{2} x^{3}+b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}-b^{2}-\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right )+\left (-c \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+b^{2} \left (a^{2}+c \right )\right ) \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}\, x^{6} \ln \left (b^{2}+a^{2} x^{3}+\sqrt {a^{2} x^{3}+b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}+\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right )+b \left (4 a^{2} b \,x^{6} \left (b^{2}-c -\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right ) \arctan \left (\frac {-2 \sqrt {a^{2} x^{3}+b^{2}}+\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}}{\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}}\right )-4 a^{2} b \,x^{6} \left (b^{2}-c -\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right ) \arctan \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}+\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}}{\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}}\right )+\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}\, \left (a^{2} x^{6} \left (a^{2}+4 b^{2}-2 c \right ) \ln \left (-b +\sqrt {a^{2} x^{3}+b^{2}}\right )-a^{2} x^{6} \left (a^{2}+4 b^{2}-2 c \right ) \ln \left (b +\sqrt {a^{2} x^{3}+b^{2}}\right )+2 \left (2 b^{2}+x^{3} \left (a^{2}+2 c \right )\right ) b \sqrt {a^{2} x^{3}+b^{2}}\right )\right )\right )}{12 \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}\, b^{4} \left (b -\sqrt {a^{2} x^{3}+b^{2}}\right )^{2} \left (b +\sqrt {a^{2} x^{3}+b^{2}}\right )^{2}}\) | \(675\) |
| 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 {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 {4 a^{2} b^{2} \left (b^{2}-c -\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right ) \arctan \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}-\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}}{\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}}\right )+\left (-c \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+b^{2} \left (a^{2}+c \right )\right ) \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}\, \ln \left (b^{2}+a^{2} x^{3}-\sqrt {a^{2} x^{3}+b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}+\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right )-\left (-c \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+b^{2} \left (a^{2}+c \right )\right ) \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}\, \ln \left (b^{2}+a^{2} x^{3}+\sqrt {a^{2} x^{3}+b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}+\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right )+4 a^{2} b^{2} \left (\left (b^{2}-c -\sqrt {b^{2} \left (a^{2}+b^{2}\right )}\right ) \arctan \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}+\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}+2 b^{2}}}{\sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}}\right )+2 \sqrt {a^{2} x^{3}+b^{2}}\, \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}\right )}{12 b^{4} \sqrt {2 \sqrt {b^{2} \left (a^{2}+b^{2}\right )}-2 b^{2}}}\) | \(716\) |
| elliptic | \(\text {Expression too large to display}\) | \(12546\) |
-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+4*b^2-2*c)*arctanh((a^2*x^3+b^2)^(1/2)/(b^2)^(1/2))/(b^2)^(1/2)-4/3* (-1/4*(-c*(b^2*(a^2+b^2))^(1/2)+b^2*(a^2+c))*(2*(b^2*(a^2+b^2))^(1/2)+2*b^ 2)^(1/2)*(ln(b^2+a^2*x^3+(a^2*x^3+b^2)^(1/2)*(2*(b^2*(a^2+b^2))^(1/2)+2*b^ 2)^(1/2)+(b^2*(a^2+b^2))^(1/2))-ln(-a^2*x^3+(a^2*x^3+b^2)^(1/2)*(2*(b^2*(a ^2+b^2))^(1/2)+2*b^2)^(1/2)-b^2-(b^2*(a^2+b^2))^(1/2)))*(2*(b^2*(a^2+b^2)) ^(1/2)-2*b^2)^(1/2)+(arctan((2*(a^2*x^3+b^2)^(1/2)-(2*(b^2*(a^2+b^2))^(1/2 )+2*b^2)^(1/2))/(2*(b^2*(a^2+b^2))^(1/2)-2*b^2)^(1/2))+arctan((2*(a^2*x^3+ b^2)^(1/2)+(2*(b^2*(a^2+b^2))^(1/2)+2*b^2)^(1/2))/(2*(b^2*(a^2+b^2))^(1/2) -2*b^2)^(1/2)))*a^2*(b^2-c-(b^2*(a^2+b^2))^(1/2))*b^2)/(2*(b^2*(a^2+b^2))^ (1/2)-2*b^2)^(1/2)/a^2/b^2)
Leaf count of result is larger than twice the leaf count of optimal. 1523 vs. \(2 (194) = 388\).
Time = 0.75 (sec) , antiderivative size = 1523, normalized size of antiderivative = 6.17 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx=\text {Too large to display} \]
1/12*(2*b^3*x^6*sqrt((a^4*b^2 - 2*a^4*c + b^4*sqrt(-(a^10*b^4 + 4*a^8*b^4* c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) - a^2*c^2 )/b^4)*log(-(a^8*b^4 + 2*a^6*b^4*c + 2*a^4*b^2*c^3 - a^4*c^4)*sqrt(a^2*x^3 + b^2) + (a^6*b^6 + 2*a^4*b^6*c - a^4*b^4*c^2 - b^8*c*sqrt(-(a^10*b^4 + 4 *a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10)) *sqrt((a^4*b^2 - 2*a^4*c + b^4*sqrt(-(a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c ^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) - a^2*c^2)/b^4)) - 2*b^3 *x^6*sqrt((a^4*b^2 - 2*a^4*c + b^4*sqrt(-(a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b ^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) - a^2*c^2)/b^4)*log( -(a^8*b^4 + 2*a^6*b^4*c + 2*a^4*b^2*c^3 - a^4*c^4)*sqrt(a^2*x^3 + b^2) - ( a^6*b^6 + 2*a^4*b^6*c - a^4*b^4*c^2 - b^8*c*sqrt(-(a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10))*sqrt((a^4* b^2 - 2*a^4*c + b^4*sqrt(-(a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^ 4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) - a^2*c^2)/b^4)) + 2*b^3*x^6*sqrt(( a^4*b^2 - 2*a^4*c - b^4*sqrt(-(a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^ 6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) - a^2*c^2)/b^4)*log(-(a^8*b^4 + 2*a^6*b^4*c + 2*a^4*b^2*c^3 - a^4*c^4)*sqrt(a^2*x^3 + b^2) + (a^6*b^6 + 2 *a^4*b^6*c - a^4*b^4*c^2 + b^8*c*sqrt(-(a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2 *c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10))*sqrt((a^4*b^2 - 2*a^4 *c - b^4*sqrt(-(a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a...
Timed out. \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx=\int { \frac {{\left (a^{2} x^{6} + c x^{3} + 2 \, b^{2}\right )} \sqrt {a^{2} x^{3} + b^{2}}}{{\left (a^{2} x^{6} + b^{2}\right )} x^{7}} \,d x } \]
Exception generated. \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con st gen &
Time = 18.87 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx=\frac {a^2\,\ln \left (\frac {{\left (b+\sqrt {a^2\,x^3+b^2}\right )}^3\,\left (b-\sqrt {a^2\,x^3+b^2}\right )}{x^6}\right )\,\left (a^2+4\,b^2-2\,c\right )}{12\,b^3}-\frac {\sqrt {a^2\,x^3+b^2}\,\left (a^2+2\,c\right )}{6\,b^2\,x^3}-\frac {\sqrt {a^2\,x^3+b^2}}{3\,x^6}+\frac {a\,\ln \left (\frac {2\,b^2-a\,b\,1{}\mathrm {i}+a^2\,x^3+\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {-b+a\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a\,x^3+b\,1{}\mathrm {i}}\right )\,\left (-a\,b+c\,1{}\mathrm {i}\right )\,\sqrt {-b+a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{6\,b^{5/2}}+\frac {a\,\ln \left (\frac {a\,b\,1{}\mathrm {i}+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {b+a\,1{}\mathrm {i}}}{-a\,x^3+b\,1{}\mathrm {i}}\right )\,\left (a\,b+c\,1{}\mathrm {i}\right )\,\sqrt {b+a\,1{}\mathrm {i}}}{6\,b^{5/2}} \]
(a^2*log(((b + (b^2 + a^2*x^3)^(1/2))^3*(b - (b^2 + a^2*x^3)^(1/2)))/x^6)* (a^2 - 2*c + 4*b^2))/(12*b^3) - ((b^2 + a^2*x^3)^(1/2)*(2*c + a^2))/(6*b^2 *x^3) - (b^2 + a^2*x^3)^(1/2)/(3*x^6) + (a*log((2*b^2 - a*b*1i + a^2*x^3 + b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(a*1i - b)^(1/2)*2i)/(b*1i + a*x^3))*(c*1i - a*b)*(a*1i - b)^(1/2)*1i)/(6*b^(5/2)) + (a*log((a*b*1i + 2*b^2 + a^2*x^3 - 2*b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(a*1i + b)^(1/2))/(b*1i - a*x^3))*(c*1i + a*b)*(a*1i + b)^(1/2))/(6*b^(5/2))