Integrand size = 24, antiderivative size = 127 \[ \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\frac {\left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{2 d}-\frac {(b c-a d) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{\sqrt {b} \sqrt {e}}\right )}{2 \sqrt {b} d^{3/2}} \] Output:
1/2*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/d-1/2*(-a*d+b*c)*e^(1 /2)*arctanh(d^(1/2)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/b^(1/2)/e^(1/2) )/b^(1/2)/d^(3/2)
Time = 0.38 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.04 \[ \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {b} \sqrt {d} \sqrt {a+b x^2} \left (c+d x^2\right )-(b c-a d) \sqrt {c+d x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )\right )}{2 \sqrt {b} d^{3/2} \sqrt {a+b x^2}} \] Input:
Integrate[x*Sqrt[(e*(a + b*x^2))/(c + d*x^2)],x]
Output:
(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^2]*(c + d *x^2) - (b*c - a*d)*Sqrt[c + d*x^2]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqr t[b]*Sqrt[c + d*x^2])]))/(2*Sqrt[b]*d^(3/2)*Sqrt[a + b*x^2])
Time = 0.43 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2053, 2051, 252, 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.
\(\displaystyle \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx\) |
\(\Big \downarrow \) 2053 |
\(\displaystyle \frac {1}{2} \int \sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}dx^2\) |
\(\Big \downarrow \) 2051 |
\(\displaystyle e (b c-a d) \int \frac {x^4}{\left (b e-d x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle e (b c-a d) \left (\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 d \left (b e-d x^4\right )}-\frac {\int \frac {1}{b e-d x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 d}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle e (b c-a d) \left (\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 d \left (b e-d x^4\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 \sqrt {b} d^{3/2} \sqrt {e}}\right )\) |
Input:
Int[x*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*d*(b*e - d*x^4)) - Arc Tanh[(Sqrt[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[b]*Sqrt[e])]/(2*Sqr t[b]*d^(3/2)*Sqrt[e]))
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[((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[(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_ Symbol] :> With[{q = Denominator[p]}, Simp[q*e*((b*c - a*d)/n) Subst[Int[ x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n - 1)/(b*e - d*x^q)^(1/n + 1)), x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n]
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 = 0.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.21
method | result | size |
risch | \(\frac {\left (d \,x^{2}+c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{2 d}+\frac {\left (a d -b c \right ) \ln \left (\frac {\frac {1}{2} a d e +\frac {1}{2} b c e +b d \,x^{2} e}{\sqrt {b d e}}+\sqrt {b d e \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{4 d \sqrt {b d e}\, \left (b \,x^{2}+a \right )}\) | \(154\) |
default | \(\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (a \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) d -b \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) c +2 \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\right )}{4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, d \sqrt {b d}}\) | \(200\) |
Input:
int(x*(e*(b*x^2+a)/(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/d*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/4*(a*d-b*c)/d*ln((1/2*a*d* e+1/2*b*c*e+b*d*x^2*e)/(b*d*e)^(1/2)+(b*d*e*x^4+(a*d*e+b*c*e)*x^2+a*c*e)^( 1/2))/(b*d*e)^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*(b*x^2+a)*e)^ (1/2)/(b*x^2+a)
Time = 0.09 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.46 \[ \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\left [-\frac {{\left (b c - a d\right )} \sqrt {\frac {e}{b d}} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} e x^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e + 4 \, {\left (2 \, b^{2} d^{3} x^{4} + b^{2} c^{2} d + a b c d^{2} + {\left (3 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {e}{b d}}\right ) - 4 \, {\left (d x^{2} + c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{8 \, d}, \frac {{\left (b c - a d\right )} \sqrt {-\frac {e}{b d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {e}{b d}}}{2 \, {\left (b e x^{2} + a e\right )}}\right ) + 2 \, {\left (d x^{2} + c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{4 \, d}\right ] \] Input:
integrate(x*(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="fricas")
Output:
[-1/8*((b*c - a*d)*sqrt(e/(b*d))*log(8*b^2*d^2*e*x^4 + 8*(b^2*c*d + a*b*d^ 2)*e*x^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e + 4*(2*b^2*d^3*x^4 + b^2*c^2* d + a*b*c*d^2 + (3*b^2*c*d^2 + a*b*d^3)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))*sqrt(e/(b*d))) - 4*(d*x^2 + c)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/d, 1/4*((b*c - a*d)*sqrt(-e/(b*d))*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt((b *e*x^2 + a*e)/(d*x^2 + c))*sqrt(-e/(b*d))/(b*e*x^2 + a*e)) + 2*(d*x^2 + c) *sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/d]
Timed out. \[ \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\text {Timed out} \] Input:
integrate(x*(e*(b*x**2+a)/(d*x**2+c))**(1/2),x)
Output:
Timed out
Exception generated. \[ \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*(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
Time = 0.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.09 \[ \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\frac {1}{4} \, {\left (\frac {2 \, \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}}{d} + \frac {{\left (b c e - a d e\right )} \sqrt {b d e} \log \left ({\left | -2 \, {\left (\sqrt {b d e} x^{2} - \sqrt {b d e x^{4} + b c e x^{2} + a d e x^{2} + a c e}\right )} b d - \sqrt {b d e} b c - \sqrt {b d e} a d \right |}\right )}{b d^{2} e}\right )} \mathrm {sgn}\left (d x^{2} + c\right ) \] Input:
integrate(x*(e*(b*x^2+a)/(d*x^2+c))^(1/2),x, algorithm="giac")
Output:
1/4*(2*sqrt(b*d*e*x^4 + b*c*e*x^2 + a*d*e*x^2 + a*c*e)/d + (b*c*e - a*d*e) *sqrt(b*d*e)*log(abs(-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))*b*d - sqrt(b*d*e)*b*c - sqrt(b*d*e)*a*d))/(b*d^2*e))*sgn (d*x^2 + c)
Timed out. \[ \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\int x\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}} \,d x \] Input:
int(x*((e*(a + b*x^2))/(c + d*x^2))^(1/2),x)
Output:
int(x*((e*(a + b*x^2))/(c + d*x^2))^(1/2), x)
Time = 0.31 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \, dx=\frac {\sqrt {e}\, \left (\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d +\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d -\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) a d -\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {b \,x^{2}+a}\, d -\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b \right ) b c \right )}{2 b \,d^{2}} \] Input:
int(x*(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)*b*d + sqrt(d)*sqrt(b)*log( - s qrt(b)*sqrt(a + b*x**2)*d - sqrt(d)*sqrt(c + d*x**2)*b)*a*d - sqrt(d)*sqrt (b)*log( - sqrt(b)*sqrt(a + b*x**2)*d - sqrt(d)*sqrt(c + d*x**2)*b)*b*c))/ (2*b*d**2)