Integrand size = 21, antiderivative size = 82 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}+\frac {c \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} \sqrt {b c-a d}} \] Output:
1/2*x*(d*x^2+c)^(1/2)/a/(b*x^2+a)+1/2*c*arctan((-a*d+b*c)^(1/2)*x/a^(1/2)/ (d*x^2+c)^(1/2))/a^(3/2)/(-a*d+b*c)^(1/2)
Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {x \sqrt {c+d x^2}}{2 a^2+2 a b x^2}-\frac {c \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{3/2} \sqrt {b c-a d}} \] Input:
Integrate[Sqrt[c + d*x^2]/(a + b*x^2)^2,x]
Output:
(x*Sqrt[c + d*x^2])/(2*a^2 + 2*a*b*x^2) - (c*ArcTan[(a*Sqrt[d] + b*x*(Sqrt [d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(2*a^(3/2)*Sqrt[b*c - a*d])
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {292, 291, 218}
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.
\(\displaystyle \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 292 |
\(\displaystyle \frac {c \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 a}+\frac {x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {c \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{2 a}+\frac {x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {c \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} \sqrt {b c-a d}}+\frac {x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}\) |
Input:
Int[Sqrt[c + d*x^2]/(a + b*x^2)^2,x]
Output:
(x*Sqrt[c + d*x^2])/(2*a*(a + b*x^2)) + (c*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqr t[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*Sqrt[b*c - a*d])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
Time = 0.71 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(-\frac {c \left (-\frac {\sqrt {x^{2} d +c}\, x}{c \left (b \,x^{2}+a \right )}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 a}\) | \(73\) |
default | \(\text {Expression too large to display}\) | \(1965\) |
Input:
int((d*x^2+c)^(1/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*c/a*(-(d*x^2+c)^(1/2)*x/c/(b*x^2+a)-1/((a*d-b*c)*a)^(1/2)*arctanh((d* x^2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (66) = 132\).
Time = 0.14 (sec) , antiderivative size = 369, normalized size of antiderivative = 4.50 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c} x - {\left (b c x^{2} + a c\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{8 \, {\left (a^{3} b c - a^{4} d + {\left (a^{2} b^{2} c - a^{3} b d\right )} x^{2}\right )}}, \frac {2 \, {\left (a b c - a^{2} d\right )} \sqrt {d x^{2} + c} x + {\left (b c x^{2} + a c\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right )}{4 \, {\left (a^{3} b c - a^{4} d + {\left (a^{2} b^{2} c - a^{3} b d\right )} x^{2}\right )}}\right ] \] Input:
integrate((d*x^2+c)^(1/2)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/8*(4*(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x - (b*c*x^2 + a*c)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqr t(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/(a^3*b*c - a^4*d + (a^2*b^2*c - a^3*b*d)*x^2), 1/4*(2*(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x + (b*c*x^2 + a*c )*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)))/( a^3*b*c - a^4*d + (a^2*b^2*c - a^3*b*d)*x^2)]
\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \] Input:
integrate((d*x**2+c)**(1/2)/(b*x**2+a)**2,x)
Output:
Integral(sqrt(c + d*x**2)/(a + b*x**2)**2, x)
\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
integrate(sqrt(d*x^2 + c)/(b*x^2 + a)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (66) = 132\).
Time = 0.33 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.66 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {c \sqrt {d} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} a} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} a b} \] Input:
integrate((d*x^2+c)^(1/2)/(b*x^2+a)^2,x, algorithm="giac")
Output:
-1/2*c*sqrt(d)*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d )/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a) - ((sqrt(d)*x - sqr t(d*x^2 + c))^2*b*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d^(3/2) - b*c^2*sqrt(d))/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt (d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*a*b)
Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:
int((c + d*x^2)^(1/2)/(a + b*x^2)^2,x)
Output:
int((c + d*x^2)^(1/2)/(a + b*x^2)^2, x)
Time = 0.23 (sec) , antiderivative size = 419, normalized size of antiderivative = 5.11 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a c +\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b c \,x^{2}+\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a c +\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b c \,x^{2}-\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) a c -\sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) b c \,x^{2}+2 \sqrt {d \,x^{2}+c}\, a^{2} d x -2 \sqrt {d \,x^{2}+c}\, a b c x}{4 a^{2} \left (a b d \,x^{2}-b^{2} c \,x^{2}+a^{2} d -a b c \right )} \] Input:
int((d*x^2+c)^(1/2)/(b*x^2+a)^2,x)
Output:
(sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2 *a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*c + sqrt(a)* sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b* c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*x**2 + sqrt(a)*sqrt (a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sq rt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*c + sqrt(a)*sqrt(a*d - b*c)* log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*x**2 - sqrt(a)*sqrt(a*d - b*c)*log(2*s qrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*a*c - sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*b*c*x**2 + 2* sqrt(c + d*x**2)*a**2*d*x - 2*sqrt(c + d*x**2)*a*b*c*x)/(4*a**2*(a**2*d - a*b*c + a*b*d*x**2 - b**2*c*x**2))