Integrand size = 22, antiderivative size = 72 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {1}{(b c-a d) \sqrt {c+d x^2}}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}} \] Output:
1/(-a*d+b*c)/(d*x^2+c)^(1/2)-b^(1/2)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d +b*c)^(1/2))/(-a*d+b*c)^(3/2)
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {1}{(b c-a d) \sqrt {c+d x^2}}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}} \] Input:
Integrate[x/((a + b*x^2)*(c + d*x^2)^(3/2)),x]
Output:
1/((b*c - a*d)*Sqrt[c + d*x^2]) - (Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2] )/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2)
Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {353, 61, 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.
\(\displaystyle \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{2} \left (\frac {b \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{b c-a d}+\frac {2}{\sqrt {c+d x^2} (b c-a d)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {2 b \int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{d (b c-a d)}+\frac {2}{\sqrt {c+d x^2} (b c-a d)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{\sqrt {c+d x^2} (b c-a d)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}\right )\) |
Input:
Int[x/((a + b*x^2)*(c + d*x^2)^(3/2)),x]
Output:
(2/((b*c - a*d)*Sqrt[c + d*x^2]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d* x^2])/Sqrt[b*c - a*d]])/(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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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 = 0.70 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {-\frac {1}{\sqrt {x^{2} d +c}}-\frac {b \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}}{a d -b c}\) | \(63\) |
default | \(\frac {-\frac {b}{\left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {2 d \sqrt {-a b}\, \left (2 d \left (x -\frac {\sqrt {-a b}}{b}\right )+\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{2 b}+\frac {-\frac {b}{\left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}-\frac {2 d \sqrt {-a b}\, \left (2 d \left (x +\frac {\sqrt {-a b}}{b}\right )-\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{2 b}\) | \(727\) |
Input:
int(x/(b*x^2+a)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/(a*d-b*c)*(-1/(d*x^2+c)^(1/2)-b/((a*d-b*c)*b)^(1/2)*arctan((d*x^2+c)^(1/ 2)*b/((a*d-b*c)*b)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 323, normalized size of antiderivative = 4.49 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (d x^{2} + c\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt {d x^{2} + c}}{4 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}, \frac {{\left (d x^{2} + c\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + 2 \, \sqrt {d x^{2} + c}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ] \] Input:
integrate(x/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
[-1/4*((d*x^2 + c)*sqrt(b/(b*c - a*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*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d))) /(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*sqrt(d*x^2 + c))/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2), 1/2*((d*x^2 + c)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) + 2*s qrt(d*x^2 + c))/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)]
Time = 6.73 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.58 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {d}{2 \sqrt {c + d x^{2}} \left (a d - b c\right )} - \frac {d \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{2 \sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )}\right )}{d} & \text {for}\: d \neq 0 \\\begin {cases} \frac {x^{2}}{2 a c^{\frac {3}{2}}} & \text {for}\: b = 0 \\\tilde {\infty } x^{2} & \text {for}\: c^{\frac {3}{2}} = 0 \\\frac {\log {\left (2 a c^{\frac {3}{2}} + 2 b c^{\frac {3}{2}} x^{2} \right )}}{2 b c^{\frac {3}{2}}} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases} \] Input:
integrate(x/(b*x**2+a)/(d*x**2+c)**(3/2),x)
Output:
Piecewise((2*(-d/(2*sqrt(c + d*x**2)*(a*d - b*c)) - d*atan(sqrt(c + d*x**2 )/sqrt((a*d - b*c)/b))/(2*sqrt((a*d - b*c)/b)*(a*d - b*c)))/d, Ne(d, 0)), (Piecewise((x**2/(2*a*c**(3/2)), Eq(b, 0)), (zoo*x**2, Eq(c**(3/2), 0)), ( log(2*a*c**(3/2) + 2*b*c**(3/2)*x**2)/(2*b*c**(3/2)), True)), True))
Exception generated. \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x/(b*x^2+a)/(d*x^2+c)^(3/2),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 = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {b \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} + \frac {1}{\sqrt {d x^{2} + c} {\left (b c - a d\right )}} \] Input:
integrate(x/(b*x^2+a)/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
b*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/(sqrt(d*x^2 + c)*(b*c - a*d))
Time = 1.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {1}{\sqrt {d\,x^2+c}\,\left (a\,d-b\,c\right )}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}}{\sqrt {a\,d-b\,c}}\right )}{{\left (a\,d-b\,c\right )}^{3/2}} \] Input:
int(x/((a + b*x^2)*(c + d*x^2)^(3/2)),x)
Output:
- 1/((c + d*x^2)^(1/2)*(a*d - b*c)) - (b^(1/2)*atan((b^(1/2)*(c + d*x^2)^( 1/2))/(a*d - b*c)^(1/2)))/(a*d - b*c)^(3/2)
Time = 0.24 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.33 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) c -\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) d \,x^{2}-\sqrt {d \,x^{2}+c}\, a d +\sqrt {d \,x^{2}+c}\, b c}{a^{2} d^{3} x^{2}-2 a b c \,d^{2} x^{2}+b^{2} c^{2} d \,x^{2}+a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}} \] Input:
int(x/(b*x^2+a)/(d*x^2+c)^(3/2),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))*c - 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)*sq rt(b)*sqrt(a*d - b*c)*x))*d*x**2 - sqrt(c + d*x**2)*a*d + sqrt(c + d*x**2) *b*c)/(a**2*c*d**2 + a**2*d**3*x**2 - 2*a*b*c**2*d - 2*a*b*c*d**2*x**2 + b **2*c**3 + b**2*c**2*d*x**2)