Integrand size = 22, antiderivative size = 87 \[ \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2}}{2 (b c-a d) \left (a+b x^2\right )}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} (b c-a d)^{3/2}} \] Output:
-1/2*(d*x^2+c)^(1/2)/(-a*d+b*c)/(b*x^2+a)+1/2*d*arctanh(b^(1/2)*(d*x^2+c)^ (1/2)/(-a*d+b*c)^(1/2))/b^(1/2)/(-a*d+b*c)^(3/2)
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {1}{2} \left (-\frac {\sqrt {c+d x^2}}{(b c-a d) \left (a+b x^2\right )}+\frac {d \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{3/2}}\right ) \] Input:
Integrate[x/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
Output:
(-(Sqrt[c + d*x^2]/((b*c - a*d)*(a + b*x^2))) + (d*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c) + a*d)^(3/2)))/2
Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {353, 52, 73, 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.
\(\displaystyle \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (b x^2+a\right )^2 \sqrt {d x^2+c}}dx^2\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (-\frac {d \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{2 (b c-a d)}-\frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) (b c-a d)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{b c-a d}-\frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) (b c-a d)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^2}}{\left (a+b x^2\right ) (b c-a d)}\right )\) |
Input:
Int[x/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
Output:
(-(Sqrt[c + d*x^2]/((b*c - a*d)*(a + b*x^2))) + (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(Sqrt[b]*(b*c - a*d)^(3/2)))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Time = 0.80 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {x^{2} d +c}}{b \,x^{2}+a}+\frac {d \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{2 a d -2 b c}\) | \(71\) |
default | \(-\frac {\sqrt {-a b}\, \left (\frac {b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a \,b^{2}}+\frac {\sqrt {-a b}\, \left (\frac {b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a \,b^{2}}\) | \(524\) |
Input:
int(x/(b*x^2+a)^2/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/(a*d-b*c)*((d*x^2+c)^(1/2)/(b*x^2+a)+d/((a*d-b*c)*b)^(1/2)*arctan((d*x ^2+c)^(1/2)*b/((a*d-b*c)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (71) = 142\).
Time = 0.10 (sec) , antiderivative size = 404, normalized size of antiderivative = 4.64 \[ \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\left [-\frac {{\left (b d x^{2} + a d\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (b^{2} c - a b d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}, \frac {{\left (b d x^{2} + a d\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (b^{2} c - a b d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}}\right ] \] Input:
integrate(x/(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
[-1/8*((b*d*x^2 + a*d)*sqrt(b^2*c - a*b*d)*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b*d*x^2 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(b^2*c - a*b*d)*sqrt(d*x^2 + c))/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2), 1/4*((b*d*x^2 + a*d)*sqrt(-b ^2*c + a*b*d)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(-b^2*c + a*b*d)*sqr t(d*x^2 + c)/(b^2*c^2 - a*b*c*d + (b^2*c*d - a*b*d^2)*x^2)) - 2*(b^2*c - a *b*d)*sqrt(d*x^2 + c))/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2)]
\[ \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {x}{\left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)
Output:
Integral(x/((a + b*x**2)**2*sqrt(c + d*x**2)), x)
Exception generated. \[ \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x/(b*x^2+a)^2/(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(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.07 \[ \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {d \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} - \frac {\sqrt {d x^{2} + c} d}{2 \, {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} {\left (b c - a d\right )}} \] Input:
integrate(x/(b*x^2+a)^2/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
-1/2*d*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d )*(b*c - a*d)) - 1/2*sqrt(d*x^2 + c)*d/(((d*x^2 + c)*b - b*c + a*d)*(b*c - a*d))
Time = 1.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {d\,\sqrt {d\,x^2+c}}{2\,\left (a\,d-b\,c\right )\,\left (b\,\left (d\,x^2+c\right )+a\,d-b\,c\right )}+\frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}}{\sqrt {a\,d-b\,c}}\right )}{2\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}} \] Input:
int(x/((a + b*x^2)^2*(c + d*x^2)^(1/2)),x)
Output:
(d*(c + d*x^2)^(1/2))/(2*(a*d - b*c)*(b*(c + d*x^2) + a*d - b*c)) + (d*ata n((b^(1/2)*(c + d*x^2)^(1/2))/(a*d - b*c)^(1/2)))/(2*b^(1/2)*(a*d - b*c)^( 3/2))
Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.84 \[ \int \frac {x}{\left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) a d +\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) b d \,x^{2}+\sqrt {d \,x^{2}+c}\, a b d -\sqrt {d \,x^{2}+c}\, b^{2} c}{2 b \left (a^{2} b \,d^{2} x^{2}-2 a \,b^{2} c d \,x^{2}+b^{3} c^{2} x^{2}+a^{3} d^{2}-2 a^{2} b c d +a \,b^{2} c^{2}\right )} \] Input:
int(x/(b*x^2+a)^2/(d*x^2+c)^(1/2),x)
Output:
(sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(c + d*x**2)*b*x + b*c + b*d*x* *2)/(sqrt(b)*sqrt(c + d*x**2)*sqrt(a*d - b*c) + sqrt(d)*sqrt(b)*sqrt(a*d - b*c)*x))*a*d + sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(c + d*x**2)*b*x + b*c + b*d*x**2)/(sqrt(b)*sqrt(c + d*x**2)*sqrt(a*d - b*c) + sqrt(d)*sqr t(b)*sqrt(a*d - b*c)*x))*b*d*x**2 + sqrt(c + d*x**2)*a*b*d - sqrt(c + d*x* *2)*b**2*c)/(2*b*(a**3*d**2 - 2*a**2*b*c*d + a**2*b*d**2*x**2 + a*b**2*c** 2 - 2*a*b**2*c*d*x**2 + b**3*c**2*x**2))