Integrand size = 26, antiderivative size = 133 \[ \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=-\frac {\left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{2 a e x^2}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {c} \sqrt {e}} \] Output:
-1/2*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/a/e/x^2+1/2*(-a*d+b* c)*arctanh(c^(1/2)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/a^(1/2)/e^(1/2)) /a^(3/2)/c^(1/2)/e^(1/2)
Time = 0.68 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {-\sqrt {a} \sqrt {c} \left (a+b x^2\right ) \left (c+d x^2\right )+(b c-a d) x^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} \sqrt {c} x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \] Input:
Integrate[1/(x^3*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]
Output:
(-(Sqrt[a]*Sqrt[c]*(a + b*x^2)*(c + d*x^2)) + (b*c - a*d)*x^2*Sqrt[a + b*x ^2]*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d* x^2])])/(2*a^(3/2)*Sqrt[c]*x^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^ 2))
Time = 0.46 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2053, 2052, 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.
\(\displaystyle \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx\) |
\(\Big \downarrow \) 2053 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}dx^2\) |
\(\Big \downarrow \) 2052 |
\(\displaystyle e (b c-a d) \int \frac {1}{\left (c x^4-a e\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle e (b c-a d) \left (\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a e \left (a e-c x^4\right )}-\frac {\int \frac {1}{c x^4-a e}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 a e}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle e (b c-a d) \left (\frac {\text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {c} e^{3/2}}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 a e \left (a e-c x^4\right )}\right )\) |
Input:
Int[1/(x^3*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]
Output:
(b*c - a*d)*e*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/(2*a*e*(a*e - c*x^4)) + A rcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[a]*Sqrt[e])]/(2*a ^(3/2)*Sqrt[c]*e^(3/2)))
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[(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 = 162, normalized size of antiderivative = 1.22
method | result | size |
risch | \(-\frac {b \,x^{2}+a}{2 a \,x^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}-\frac {\left (a d -b c \right ) \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{4 a \sqrt {a c e}\, \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(162\) |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (-2 d b \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{4} \sqrt {a c}+a^{2} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d c \,x^{2}-c^{2} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) b a \,x^{2}-2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d a \,x^{2} \sqrt {a c}-2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b c \,x^{2} \sqrt {a c}+2 \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\right )}{4 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{2} c \,x^{2} \sqrt {a c}}\) | \(326\) |
Input:
int(1/x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2/a*(b*x^2+a)/x^2/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/4*(a*d-b*c)/a/(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)*(b*x^2 +a)*e)^(1/2)/(d*x^2+c)
Time = 0.22 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.50 \[ \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\left [-\frac {\sqrt {a c e} {\left (b c - a d\right )} x^{2} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} - 4 \, {\left ({\left (b c d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left (a c d x^{2} + a c^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{8 \, a^{2} c e x^{2}}, -\frac {\sqrt {-a c e} {\left (b c - a d\right )} x^{2} \arctan \left (\frac {\sqrt {-a c e} {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (a b c e x^{2} + a^{2} c e\right )}}\right ) + 2 \, {\left (a c d x^{2} + a c^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{4 \, a^{2} c e x^{2}}\right ] \] Input:
integrate(1/x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="fricas")
Output:
[-1/8*(sqrt(a*c*e)*(b*c - a*d)*x^2*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*(a*c*d*x^2 + a*c^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a^ 2*c*e*x^2), -1/4*(sqrt(-a*c*e)*(b*c - a*d)*x^2*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*(a*c*d*x^2 + a*c^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a^2*c* e*x^2)]
Timed out. \[ \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Timed out} \] Input:
integrate(1/x**3/(e*(b*x**2+a)/(d*x**2+c))**(1/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="maxima")
Output:
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. 238 vs. \(2 (113) = 226\).
Time = 0.18 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=-\frac {\frac {{\left (b c - a d\right )} \arctan \left (-\frac {\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}}{\sqrt {-a c e}}\right )}{\sqrt {-a c e} a} + \frac {{\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} b c + {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} a d + 2 \, \sqrt {b d e} a c}{{\left (a c e - {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )}^{2}\right )} a}}{2 \, \mathrm {sgn}\left (d x^{2} + c\right )} \] Input:
integrate(1/x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="giac")
Output:
-1/2*((b*c - a*d)*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) + ((sqrt(b*d*e)*x^2 - s qrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))*b*c + (sqrt(b*d*e)*x^2 - s qrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e))*a*d + 2*sqrt(b*d*e)*a*c)/( (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)*a))/sgn(d*x^2 + c)
Timed out. \[ \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int \frac {1}{x^3\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \] Input:
int(1/(x^3*((e*(a + b*x^2))/(c + d*x^2))^(1/2)),x)
Output:
int(1/(x^3*((e*(a + b*x^2))/(c + d*x^2))^(1/2)), x)
Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\sqrt {e}\, \left (-\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a c +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) a d \,x^{2}-\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {b \,x^{2}+a}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a \right ) b c \,x^{2}-\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) a d \,x^{2}+\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (x \right ) b c \,x^{2}\right )}{2 a^{2} c e \,x^{2}} \] Input:
int(1/x^3/(e*(b*x^2+a)/(d*x^2+c))^(1/2),x)
Output:
(sqrt(e)*( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c + sqrt(c)*sqrt(a)*log(s qrt(a)*sqrt(a + b*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*a*d*x**2 - sqrt(c) *sqrt(a)*log(sqrt(a)*sqrt(a + b*x**2)*c - sqrt(c)*sqrt(c + d*x**2)*a)*b*c* x**2 - sqrt(c)*sqrt(a)*log(x)*a*d*x**2 + sqrt(c)*sqrt(a)*log(x)*b*c*x**2)) /(2*a**2*c*e*x**2)