Integrand size = 42, antiderivative size = 36 \[ \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {d} x \left (a+b x^2\right )^{3/2}}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}} \] Output:
e*arctanh(d^(1/2)*x*(b*x^2+a)^(3/2)/c^(1/2))/c^(1/2)/d^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 17.21 (sec) , antiderivative size = 23155, normalized size of antiderivative = 643.19 \[ \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(Sqrt[a + b*x^2]*(a*e + 4*b*e*x^2))/(c - d*x^2*(a + b*x^2)^3),x]
Output:
Result too large to show
Time = 0.84 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {7260, 221}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 7260 |
| \(\displaystyle e \int \frac {1}{c-d x^2 \left (b x^2+a\right )^3}d\left (x \left (b x^2+a\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {e \text {arctanh}\left (\frac {\sqrt {d} x \left (a+b x^2\right )^{3/2}}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d}}\) |
Input:
Int[(Sqrt[a + b*x^2]*(a*e + 4*b*e*x^2))/(c - d*x^2*(a + b*x^2)^3),x]
Output:
(e*ArcTanh[(Sqrt[d]*x*(a + b*x^2)^(3/2))/Sqrt[c]])/(Sqrt[c]*Sqrt[d])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(u_)*(v_)^(r_.)*((a_) + (b_.)*(v_)^(p_.)*(w_)^(q_.))^(m_.), x_Symbol] : > With[{c = Simplify[u/(p*w*D[v, x] + q*v*D[w, x])]}, Simp[c*(p/(r + 1)) Subst[Int[(a + b*x^(p/(r + 1)))^m, x], x, v^(r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q, r}, x] && EqQ[p, q*(r + 1)] && NeQ[r, -1] && Integ erQ[p/(r + 1)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.64
| method | result | size |
| default | \(-\frac {e \,a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\left (-a^{4} d -4 b c \right ) \textit {\_Z}^{6}+6 c \,b^{2} \textit {\_Z}^{4}-4 c \,b^{3} \textit {\_Z}^{2}+b^{4} c \right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{4}+2 \textit {\_R}^{2} b -3 b^{2}\right ) \ln \left (\frac {-\textit {\_R} x +\sqrt {b \,x^{2}+a}}{x}\right )}{-3 \textit {\_R}^{4} a^{4} d +4 \textit {\_R}^{6} c -12 \textit {\_R}^{4} b c +12 \textit {\_R}^{2} b^{2} c -4 b^{3} c}\right )}{2}\) | \(131\) |
| pseudoelliptic | \(-\frac {e \,a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\left (-a^{4} d -4 b c \right ) \textit {\_Z}^{6}+6 c \,b^{2} \textit {\_Z}^{4}-4 c \,b^{3} \textit {\_Z}^{2}+b^{4} c \right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{4}+2 \textit {\_R}^{2} b -3 b^{2}\right ) \ln \left (\frac {-\textit {\_R} x +\sqrt {b \,x^{2}+a}}{x}\right )}{-3 \textit {\_R}^{4} a^{4} d +4 \textit {\_R}^{6} c -12 \textit {\_R}^{4} b c +12 \textit {\_R}^{2} b^{2} c -4 b^{3} c}\right )}{2}\) | \(131\) |
Input:
int((b*x^2+a)^(1/2)*(4*b*e*x^2+a*e)/(c-d*x^2*(b*x^2+a)^3),x,method=_RETURN VERBOSE)
Output:
-1/2*e*a^2*sum(_R*(_R^4+2*_R^2*b-3*b^2)*ln((-_R*x+(b*x^2+a)^(1/2))/x)/(-3* _R^4*a^4*d+4*_R^6*c-12*_R^4*b*c+12*_R^2*b^2*c-4*b^3*c),_R=RootOf(c*_Z^8+(- a^4*d-4*b*c)*_Z^6+6*c*b^2*_Z^4-4*c*b^3*_Z^2+b^4*c))
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (26) = 52\).
Time = 0.76 (sec) , antiderivative size = 451, normalized size of antiderivative = 12.53 \[ \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx=\left [\frac {\sqrt {c d} e \log \left (\frac {b^{6} d^{2} x^{16} + 6 \, a b^{5} d^{2} x^{14} + 15 \, a^{2} b^{4} d^{2} x^{12} + 20 \, a^{3} b^{3} d^{2} x^{10} + 3 \, {\left (5 \, a^{4} b^{2} d^{2} + 2 \, b^{3} c d\right )} x^{8} + 6 \, a^{3} c d x^{2} + 6 \, {\left (a^{5} b d^{2} + 3 \, a b^{2} c d\right )} x^{6} + {\left (a^{6} d^{2} + 18 \, a^{2} b c d\right )} x^{4} + 4 \, {\left (b^{4} d x^{11} + 4 \, a b^{3} d x^{9} + 6 \, a^{2} b^{2} d x^{7} + 4 \, a^{3} b d x^{5} + {\left (a^{4} d + b c\right )} x^{3} + a c x\right )} \sqrt {b x^{2} + a} \sqrt {c d} + c^{2}}{b^{6} d^{2} x^{16} + 6 \, a b^{5} d^{2} x^{14} + 15 \, a^{2} b^{4} d^{2} x^{12} + 20 \, a^{3} b^{3} d^{2} x^{10} + {\left (15 \, a^{4} b^{2} d^{2} - 2 \, b^{3} c d\right )} x^{8} - 2 \, a^{3} c d x^{2} + 6 \, {\left (a^{5} b d^{2} - a b^{2} c d\right )} x^{6} + {\left (a^{6} d^{2} - 6 \, a^{2} b c d\right )} x^{4} + c^{2}}\right )}{4 \, c d}, -\frac {\sqrt {-c d} e \arctan \left (\frac {{\left (b^{3} d x^{8} + 3 \, a b^{2} d x^{6} + 3 \, a^{2} b d x^{4} + a^{3} d x^{2} + c\right )} \sqrt {b x^{2} + a} \sqrt {-c d}}{2 \, {\left (b^{2} c d x^{5} + 2 \, a b c d x^{3} + a^{2} c d x\right )}}\right )}{2 \, c d}\right ] \] Input:
integrate((b*x^2+a)^(1/2)*(4*b*e*x^2+a*e)/(c-d*x^2*(b*x^2+a)^3),x, algorit hm="fricas")
Output:
[1/4*sqrt(c*d)*e*log((b^6*d^2*x^16 + 6*a*b^5*d^2*x^14 + 15*a^2*b^4*d^2*x^1 2 + 20*a^3*b^3*d^2*x^10 + 3*(5*a^4*b^2*d^2 + 2*b^3*c*d)*x^8 + 6*a^3*c*d*x^ 2 + 6*(a^5*b*d^2 + 3*a*b^2*c*d)*x^6 + (a^6*d^2 + 18*a^2*b*c*d)*x^4 + 4*(b^ 4*d*x^11 + 4*a*b^3*d*x^9 + 6*a^2*b^2*d*x^7 + 4*a^3*b*d*x^5 + (a^4*d + b*c) *x^3 + a*c*x)*sqrt(b*x^2 + a)*sqrt(c*d) + c^2)/(b^6*d^2*x^16 + 6*a*b^5*d^2 *x^14 + 15*a^2*b^4*d^2*x^12 + 20*a^3*b^3*d^2*x^10 + (15*a^4*b^2*d^2 - 2*b^ 3*c*d)*x^8 - 2*a^3*c*d*x^2 + 6*(a^5*b*d^2 - a*b^2*c*d)*x^6 + (a^6*d^2 - 6* a^2*b*c*d)*x^4 + c^2))/(c*d), -1/2*sqrt(-c*d)*e*arctan(1/2*(b^3*d*x^8 + 3* a*b^2*d*x^6 + 3*a^2*b*d*x^4 + a^3*d*x^2 + c)*sqrt(b*x^2 + a)*sqrt(-c*d)/(b ^2*c*d*x^5 + 2*a*b*c*d*x^3 + a^2*c*d*x))/(c*d)]
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(1/2)*(4*b*e*x**2+a*e)/(c-d*x**2*(b*x**2+a)**3),x)
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx=\int { -\frac {{\left (4 \, b e x^{2} + a e\right )} \sqrt {b x^{2} + a}}{{\left (b x^{2} + a\right )}^{3} d x^{2} - c} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(4*b*e*x^2+a*e)/(c-d*x^2*(b*x^2+a)^3),x, algorit hm="maxima")
Output:
-integrate((4*b*e*x^2 + a*e)*sqrt(b*x^2 + a)/((b*x^2 + a)^3*d*x^2 - c), x)
\[ \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx=\int { -\frac {{\left (4 \, b e x^{2} + a e\right )} \sqrt {b x^{2} + a}}{{\left (b x^{2} + a\right )}^{3} d x^{2} - c} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(4*b*e*x^2+a*e)/(c-d*x^2*(b*x^2+a)^3),x, algorit hm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (4\,b\,e\,x^2+a\,e\right )}{c-d\,x^2\,{\left (b\,x^2+a\right )}^3} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(a*e + 4*b*e*x^2))/(c - d*x^2*(a + b*x^2)^3),x)
Output:
int(((a + b*x^2)^(1/2)*(a*e + 4*b*e*x^2))/(c - d*x^2*(a + b*x^2)^3), x)
\[ \int \frac {\sqrt {a+b x^2} \left (a e+4 b e x^2\right )}{c-d x^2 \left (a+b x^2\right )^3} \, dx=e \left (-\left (\int \frac {\sqrt {b \,x^{2}+a}}{b^{3} d \,x^{8}+3 a \,b^{2} d \,x^{6}+3 a^{2} b d \,x^{4}+a^{3} d \,x^{2}-c}d x \right ) a -4 \left (\int \frac {\sqrt {b \,x^{2}+a}\, x^{2}}{b^{3} d \,x^{8}+3 a \,b^{2} d \,x^{6}+3 a^{2} b d \,x^{4}+a^{3} d \,x^{2}-c}d x \right ) b \right ) \] Input:
int((b*x^2+a)^(1/2)*(4*b*e*x^2+a*e)/(c-d*x^2*(b*x^2+a)^3),x)
Output:
e*( - int(sqrt(a + b*x**2)/(a**3*d*x**2 + 3*a**2*b*d*x**4 + 3*a*b**2*d*x** 6 + b**3*d*x**8 - c),x)*a - 4*int((sqrt(a + b*x**2)*x**2)/(a**3*d*x**2 + 3 *a**2*b*d*x**4 + 3*a*b**2*d*x**6 + b**3*d*x**8 - c),x)*b)