Integrand size = 26, antiderivative size = 218 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=-\frac {(b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac {(b c-a d) (3 b c+a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^2 c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(b c-a d) (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{8 a^{5/2} c^{3/2} \sqrt {e}} \]
-1/8*(-a*d+b*c)*(a*d+3*b*c)*arctanh(c^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/ a^(1/2)/e^(1/2))/a^(5/2)/c^(3/2)/e^(1/2)-1/4*(-a*d+b*c)^2*e*(e*(b*x^2+a)/( d*x^2+c))^(1/2)/a/c/(a*e-c*e*(b*x^2+a)/(d*x^2+c))^2-1/8*(-a*d+b*c)*(a*d+3* b*c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a^2/c/(a*e-c*e*(b*x^2+a)/(d*x^2+c))
Time = 1.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {a} \sqrt {c} \left (a+b x^2\right ) \sqrt {c+d x^2} \left (3 b c x^2-a \left (2 c+d x^2\right )\right )-\left (3 b^2 c^2-2 a b c d-a^2 d^2\right ) x^4 \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} c^{3/2} x^4 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \]
(Sqrt[a]*Sqrt[c]*(a + b*x^2)*Sqrt[c + d*x^2]*(3*b*c*x^2 - a*(2*c + d*x^2)) - (3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*x^4*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[c]* Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(5/2)*c^(3/2)*x^4*Sqrt[( e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])
Time = 0.33 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2053, 2052, 25, 298, 215, 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.
| \(\displaystyle \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle e (b c-a d) \int -\frac {b e-d x^4}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\left (e (b c-a d) \int \frac {b e-d x^4}{\left (a e-c x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\right )\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle e (b c-a d) \left (-\frac {(a d+3 b c) \int \frac {1}{\left (a e-c x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 a c}-\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle e (b c-a d) \left (-\frac {(a d+3 b c) \left (\frac {\int \frac {1}{a e-c x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 a e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a e \left (a e-c x^4\right )}\right )}{4 a c}-\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle e (b c-a d) \left (-\frac {(a d+3 b c) \left (\frac {\text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {c} e^{3/2}}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a e \left (a e-c x^4\right )}\right )}{4 a c}-\frac {(b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-c x^4\right )^2}\right )\) |
(b*c - a*d)*e*(-1/4*((b*c - a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(a*c*( a*e - c*x^4)^2) - ((3*b*c + a*d)*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/(2*a*e *(a*e - c*x^4)) + ArcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqr t[a]*Sqrt[e])]/(2*a^(3/2)*Sqrt[c]*e^(3/2))))/(4*a*c))
3.4.1.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.16 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(-\frac {\left (b \,x^{2}+a \right ) \left (a d \,x^{2}-3 b c \,x^{2}+2 a c \right )}{8 a^{2} x^{4} c \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) \ln \left (\frac {2 a c e +\left (e d a +e b c \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (e d a +e b c \right ) x^{2}+a c e}}{x^{2}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{16 c \,a^{2} \sqrt {a c e}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(200\) |
| default | \(-\frac {\left (b \,x^{2}+a \right ) \left (2 b \,d^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{6} a \sqrt {a c}+10 b^{2} d \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{6} c \sqrt {a c}-a^{3} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d^{2} c \,x^{4}-2 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d b \,a^{2} c^{2} x^{4}+3 c^{3} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) b^{2} a \,x^{4}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d^{2} a^{2} x^{4} \sqrt {a c}+8 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b c d \,x^{4}+10 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{2} c^{2} x^{4} \sqrt {a c}-2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d a \,x^{2} \sqrt {a c}-10 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b c \,x^{2} \sqrt {a c}+4 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} a c \sqrt {a c}\right )}{16 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{3} c^{2} x^{4} \sqrt {a c}}\) | \(559\) |
-1/8*(b*x^2+a)*(a*d*x^2-3*b*c*x^2+2*a*c)/a^2/x^4/c/(e*(b*x^2+a)/(d*x^2+c)) ^(1/2)+1/16*(a^2*d^2+2*a*b*c*d-3*b^2*c^2)/c/a^2/(a*c*e)^(1/2)*ln((2*a*c*e+ (a*d*e+b*c*e)*x^2+2*(a*c*e)^(1/2)*(b*d*e*x^4+(a*d*e+b*c*e)*x^2+a*c*e)^(1/2 ))/x^2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*e*(b*x^2+a))^(1/2)/(d*x^2 +c)
Time = 0.88 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.03 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\left [-\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {a c e} x^{4} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left ({\left (b c d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left (2 \, a^{2} c^{3} - {\left (3 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, a^{3} c^{2} e x^{4}}, \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-a c e} x^{4} \arctan \left (\frac {\sqrt {-a c e} {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (a b c e x^{2} + a^{2} c e\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{3} - {\left (3 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, a^{3} c^{2} e x^{4}}\right ] \]
[-1/32*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(a*c*e)*x^4*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e*x^4 + 8*a^2*c^2*e + 8*(a*b*c^2 + a^2*c*d)*e*x^2 + 4 *((b*c*d + a*d^2)*x^4 + 2*a*c^2 + (b*c^2 + 3*a*c*d)*x^2)*sqrt(a*c*e)*sqrt( (b*e*x^2 + a*e)/(d*x^2 + c)))/x^4) + 4*(2*a^2*c^3 - (3*a*b*c^2*d - a^2*c*d ^2)*x^4 - 3*(a*b*c^3 - a^2*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/ (a^3*c^2*e*x^4), 1/16*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(-a*c*e)*x^4* arctan(1/2*sqrt(-a*c*e)*((b*c + a*d)*x^2 + 2*a*c)*sqrt((b*e*x^2 + a*e)/(d* x^2 + c))/(a*b*c*e*x^2 + a^2*c*e)) - 2*(2*a^2*c^3 - (3*a*b*c^2*d - a^2*c*d ^2)*x^4 - 3*(a*b*c^3 - a^2*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/ (a^3*c^2*e*x^4)]
Timed out. \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (194) = 388\).
Time = 0.39 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.46 \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}}{\sqrt {-a c e}}\right )}{\sqrt {-a c e} a^{2} c} + \frac {5 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} a b^{2} c^{3} e + 10 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} a^{2} b c^{2} d e + {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} a^{3} c d^{2} e + 8 \, \sqrt {b d e} a^{2} b c^{3} e - 3 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{3} b^{2} c^{2} + 2 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{3} a b c d + {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{3} a^{2} d^{2} + 8 \, \sqrt {b d e} {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{2} a^{2} c d}{{\left (a c e - {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{2}\right )}^{2} a^{2} c}}{8 \, \mathrm {sgn}\left (d x^{2} + c\right )} \]
1/8*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*arctan(-(sqrt(b*d*e)*x^2 - sqrt(b*d *e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))/sqrt(-a*c*e))/(sqrt(-a*c*e)*a^2*c ) + (5*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e)) *a*b^2*c^3*e + 10*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^ 2 + a*c*e))*a^2*b*c^2*d*e + (sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))*a^3*c*d^2*e + 8*sqrt(b*d*e)*a^2*b*c^3*e - 3*(sqrt(b* d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^3*b^2*c^2 + 2* (sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^3*a*b* c*d + (sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^ 3*a^2*d^2 + 8*sqrt(b*d*e)*(sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^2*a^2*c*d)/((a*c*e - (sqrt(b*d*e)*x^2 - sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))^2)^2*a^2*c))/sgn(d*x^2 + c)
Timed out. \[ \int \frac {1}{x^5 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int \frac {1}{x^5\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \]