Integrand size = 29, antiderivative size = 350 \[ \int \frac {x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {d^2 x}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {2 \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b-\sqrt {b^2-4 a c}} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2}}+\frac {2 \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b+\sqrt {b^2-4 a c}} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c e^{3/2}} \]
arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/c/e^(3/2)-d^2*x/e/(a*e^2-b*d*e+c*d^2)/( e*x^2+d)^(1/2)+2*arctan(x*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d) ^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b^2-a*c-b*(-3*a*c+b^2)/(-4*a*c+b^2)^ (1/2))/c/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(3/2)/(b-(-4*a*c+b^2)^(1/2))^(1/ 2)+2*arctan(x*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b+(- 4*a*c+b^2)^(1/2))^(1/2))*(b^2-a*c+b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/c/(2* c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(3/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(10546\) vs. \(2(350)=700\).
Time = 21.27 (sec) , antiderivative size = 10546, normalized size of antiderivative = 30.13 \[ \int \frac {x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]
Time = 2.33 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.30, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1620, 252, 224, 219, 2246, 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.
| \(\displaystyle \int \frac {x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1620 |
| \(\displaystyle \frac {d^2 \int \frac {x^2}{\left (e x^2+d\right )^{3/2}}dx}{a e^2-b d e+c d^2}-\frac {\int \frac {x^2 \left ((b d-a e) x^2+a d\right )}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {d^2 \left (\frac {\int \frac {1}{\sqrt {e x^2+d}}dx}{e}-\frac {x}{e \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}-\frac {\int \frac {x^2 \left ((b d-a e) x^2+a d\right )}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^2 \left (\frac {\int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{e}-\frac {x}{e \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}-\frac {\int \frac {x^2 \left ((b d-a e) x^2+a d\right )}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}-\frac {x}{e \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}-\frac {\int \frac {x^2 \left ((b d-a e) x^2+a d\right )}{\sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 2246 |
| \(\displaystyle \frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}-\frac {x}{e \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}-\frac {\int \left (\frac {b d-a e}{c \sqrt {e x^2+d}}-\frac {\left (d b^2-a e b-a c d\right ) x^2+a (b d-a e)}{c \sqrt {e x^2+d} \left (c x^4+b x^2+a\right )}\right )dx}{a e^2-b d e+c d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}-\frac {x}{e \sqrt {d+e x^2}}\right )}{a e^2-b d e+c d^2}-\frac {-\frac {\left (-\frac {2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a c d+b^2 d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (\frac {2 a^2 c e-a b^2 e-3 a b c d+b^3 d}{\sqrt {b^2-4 a c}}-a b e-a c d+b^2 d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{c \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c \sqrt {e}}}{a e^2-b d e+c d^2}\) |
(d^2*(-(x/(e*Sqrt[d + e*x^2])) + ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/e^(3 /2)))/(c*d^2 - b*d*e + a*e^2) - (-(((b^2*d - a*c*d - a*b*e - (b^3*d - 3*a* b*c*d - a*b^2*e + 2*a^2*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/( c*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e])) - ((b^2*d - a*c*d - a*b*e + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)/Sqrt[b ^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[ 2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) + ((b*d - a*e)*ArcTanh[(Sqrt[e]*x)/Sqr t[d + e*x^2]])/(c*Sqrt[e]))/(c*d^2 - b*d*e + a*e^2)
3.4.94.3.1 Defintions of rubi rules used
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Simp[d^2*(f^4/(c*d^2 - b*d*e + a*e^2)) Int[(f *x)^(m - 4)*(d + e*x^2)^q, x], x] - Simp[f^4/(c*d^2 - b*d*e + a*e^2) Int[ (f*x)^(m - 4)*(d + e*x^2)^(q + 1)*(Simp[a*d + (b*d - a*e)*x^2, x]/(a + b*x^ 2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && !IntegerQ[q] && LtQ[q, -1] && GtQ[m, 3]
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) ^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x ] && PolyQ[Px, x] && IntegerQ[p]
Time = 0.61 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.43
| method | result | size |
| pseudoelliptic | \(-\frac {\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, a \sqrt {2}\, \sqrt {e \,x^{2}+d}\, \left (\frac {\left (-e^{\frac {3}{2}} b d +a \,e^{\frac {5}{2}}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (d \left (a c -\frac {b^{2}}{2}\right ) e^{\frac {3}{2}}+\frac {a b \,e^{\frac {5}{2}}}{2}\right )\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (a \sqrt {2}\, \left (\frac {\left (e^{\frac {3}{2}} b d -a \,e^{\frac {5}{2}}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+d \left (d \left (a c -\frac {b^{2}}{2}\right ) e^{\frac {3}{2}}+\frac {a b \,e^{\frac {5}{2}}}{2}\right )\right ) \sqrt {e \,x^{2}+d}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\left (\sqrt {e \,x^{2}+d}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )-\sqrt {e}\, c \,d^{2} x \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e^{\frac {3}{2}} \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) c}\) | \(502\) |
| default | \(\frac {-\frac {x}{e \sqrt {e \,x^{2}+d}}+\frac {\ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{e^{\frac {3}{2}}}}{c}-\frac {b x}{c^{2} d \sqrt {e \,x^{2}+d}}-\frac {\left (\left (a e -b d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (\left (2 a c -b^{2}\right ) d +a b e \right ) d \right ) a \sqrt {2}\, c d \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (\left (a e -b d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-2 a c +b^{2}\right ) d^{2}-a b d e \right ) a \sqrt {2}\, c d \sqrt {e \,x^{2}+d}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+2 e \left (d \left (a c -b^{2}\right )+a b e \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, x \right )}{2 c^{2} \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(508\) |
-1/(-4*d^2*(a*c-1/4*b^2))^(1/2)/e^(3/2)/((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2) )^(1/2))*a)^(1/2)/((2*a*e-b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*(((-2 *a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2)*a*2^(1/2)*(e*x^2+d)^(1/2)* (1/2*(-e^(3/2)*b*d+a*e^(5/2))*(-4*d^2*(a*c-1/4*b^2))^(1/2)+d*(d*(a*c-1/2*b ^2)*e^(3/2)+1/2*a*b*e^(5/2)))*arctanh(a/x*(e*x^2+d)^(1/2)*2^(1/2)/((2*a*e- b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2))*a)^(1/2))+((2*a*e-b*d+(-4*d^2*(a*c-1/4*b ^2))^(1/2))*a)^(1/2)*(a*2^(1/2)*(1/2*(e^(3/2)*b*d-a*e^(5/2))*(-4*d^2*(a*c- 1/4*b^2))^(1/2)+d*(d*(a*c-1/2*b^2)*e^(3/2)+1/2*a*b*e^(5/2)))*(e*x^2+d)^(1/ 2)*arctan(a/x*(e*x^2+d)^(1/2)*2^(1/2)/((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^ (1/2))*a)^(1/2))-((e*x^2+d)^(1/2)*(a*e^2-b*d*e+c*d^2)*arctanh((e*x^2+d)^(1 /2)/x/e^(1/2))-e^(1/2)*c*d^2*x)*((-2*a*e+b*d+(-4*d^2*(a*c-1/4*b^2))^(1/2)) *a)^(1/2)*(-4*d^2*(a*c-1/4*b^2))^(1/2)))/(e*x^2+d)^(1/2)/(a*e^2-b*d*e+c*d^ 2)/c
Timed out. \[ \int \frac {x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {x^{6}}{\left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {x^{6}}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^6}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {x^6}{{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]