Integrand size = 25, antiderivative size = 63 \[ \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sqrt {a c^2-d^2} x}{a c+d \sqrt {a+b x^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}} \] Output:
arctan(b^(1/2)*(a*c^2-d^2)^(1/2)*x/(a*c+d*(b*x^2+a)^(1/2)))/b^(1/2)/(a*c^2 -d^2)^(1/2)
Time = 0.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=-\frac {2 \arctan \left (\frac {d+c \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}} \] Input:
Integrate[(a*c + b*c*x^2 + d*Sqrt[a + b*x^2])^(-1),x]
Output:
(-2*ArcTan[(d + c*(-(Sqrt[b]*x) + Sqrt[a + b*x^2]))/Sqrt[a*c^2 - d^2]])/(S qrt[b]*Sqrt[a*c^2 - d^2])
Time = 0.42 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.63, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2587, 27, 218, 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.
\(\displaystyle \int \frac {1}{d \sqrt {a+b x^2}+a c+b c x^2} \, dx\) |
\(\Big \downarrow \) 2587 |
\(\displaystyle a c \int \frac {1}{a b c^2 x^2+a \left (a c^2-d^2\right )}dx-a d \int \frac {1}{a \sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a c \int \frac {1}{a b c^2 x^2+a \left (a c^2-d^2\right )}dx-d \int \frac {1}{\sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}-d \int \frac {1}{\sqrt {b x^2+a} \left (b x^2 c^2+a c^2-d^2\right )}dx\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}-d \int \frac {1}{a c^2-d^2-\frac {\left (b \left (a c^2-d^2\right )-a b c^2\right ) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {b} c x}{\sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}-\frac {\arctan \left (\frac {\sqrt {b} d x}{\sqrt {a+b x^2} \sqrt {a c^2-d^2}}\right )}{\sqrt {b} \sqrt {a c^2-d^2}}\) |
Input:
Int[(a*c + b*c*x^2 + d*Sqrt[a + b*x^2])^(-1),x]
Output:
ArcTan[(Sqrt[b]*c*x)/Sqrt[a*c^2 - d^2]]/(Sqrt[b]*Sqrt[a*c^2 - d^2]) - ArcT an[(Sqrt[b]*d*x)/(Sqrt[a*c^2 - d^2]*Sqrt[a + b*x^2])]/(Sqrt[b]*Sqrt[a*c^2 - d^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(u_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_ Symbol] :> Simp[c Int[u/(c^2 - a*e^2 + c*d*x^n), x], x] - Simp[a*e Int[ u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x], x] /; FreeQ[{a, b, c, d, e , n}, x] && EqQ[b*c - a*d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1721\) vs. \(2(53)=106\).
Time = 0.02 (sec) , antiderivative size = 1722, normalized size of antiderivative = 27.33
Input:
int(1/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x,method=_RETURNVERBOSE)
Output:
d*(-1/2*b*c^2/(-a*b)^(1/2)/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/( (-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*(((x-(-a*b)^(1/2)/b)^2*b+2*(- a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(1/2)+(-a*b)^(1/2)*ln(((x-(-a*b)^(1/2)/b)*b +(-a*b)^(1/2))/b^(1/2)+((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1 /2)/b))^(1/2))/b^(1/2))+1/2*b*c^2/(-a*b)^(1/2)/((-a*b)^(1/2)*c^2+(-(a*c^2- d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*(((x+(-a* b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2)-(-a*b)^(1/2)*ln(( (x+(-a*b)^(1/2)/b)*b-(-a*b)^(1/2))/b^(1/2)+((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b )^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2))/b^(1/2))+1/2*b*c^4/((-a*b)^(1/2)*c^2+(- (a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))/(- (a*c^2-d^2)*b*c^2)^(1/2)*(((x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2*b+2/c^2* (-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+d^2/c^2)^( 1/2)+1/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*ln((1/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2) +(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)*b)/b^(1/2)+((x-(-(a*c^2-d^2)*b*c^2)^ (1/2)/b/c^2)^2*b+2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^ (1/2)/b/c^2)+d^2/c^2)^(1/2))/b^(1/2)-d^2/c^2/(d^2/c^2)^(1/2)*ln((2*d^2/c^2 +2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+2*( d^2/c^2)^(1/2)*((x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2*b+2/c^2*(-(a*c^2-d^ 2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+d^2/c^2)^(1/2))/(x-(- (a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)))-1/2*b*c^4/((-a*b)^(1/2)*c^2+(-(a*c^2-...
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (53) = 106\).
Time = 0.10 (sec) , antiderivative size = 510, normalized size of antiderivative = 8.10 \[ \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\left [-\frac {\sqrt {-a b c^{2} + b d^{2}} \log \left (\frac {a^{4} c^{4} - 2 \, a^{3} c^{2} d^{2} + a^{2} d^{4} + {\left (a^{2} b^{2} c^{4} - 8 \, a b^{2} c^{2} d^{2} + 8 \, b^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{3} b c^{4} - 5 \, a^{2} b c^{2} d^{2} + 4 \, a b d^{4}\right )} x^{2} - 4 \, \sqrt {-a b c^{2} + b d^{2}} {\left ({\left (a b c^{2} d - 2 \, b d^{3}\right )} x^{3} + {\left (a^{2} c^{2} d - a d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{b^{2} c^{4} x^{4} + a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4} + 2 \, {\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2}}\right ) + 2 \, \sqrt {-a b c^{2} + b d^{2}} \log \left (\frac {b c^{2} x^{2} - a c^{2} - 2 \, \sqrt {-a b c^{2} + b d^{2}} c x + d^{2}}{b c^{2} x^{2} + a c^{2} - d^{2}}\right )}{4 \, {\left (a b c^{2} - b d^{2}\right )}}, -\frac {2 \, \sqrt {a b c^{2} - b d^{2}} \arctan \left (-\frac {\sqrt {a b c^{2} - b d^{2}} c x}{a c^{2} - d^{2}}\right ) - \sqrt {a b c^{2} - b d^{2}} \arctan \left (\frac {{\left (a^{2} c^{2} - a d^{2} + {\left (a b c^{2} - 2 \, b d^{2}\right )} x^{2}\right )} \sqrt {a b c^{2} - b d^{2}} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (a b^{2} c^{2} d - b^{2} d^{3}\right )} x^{3} + {\left (a^{2} b c^{2} d - a b d^{3}\right )} x\right )}}\right )}{2 \, {\left (a b c^{2} - b d^{2}\right )}}\right ] \] Input:
integrate(1/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="fricas")
Output:
[-1/4*(sqrt(-a*b*c^2 + b*d^2)*log((a^4*c^4 - 2*a^3*c^2*d^2 + a^2*d^4 + (a^ 2*b^2*c^4 - 8*a*b^2*c^2*d^2 + 8*b^2*d^4)*x^4 + 2*(a^3*b*c^4 - 5*a^2*b*c^2* d^2 + 4*a*b*d^4)*x^2 - 4*sqrt(-a*b*c^2 + b*d^2)*((a*b*c^2*d - 2*b*d^3)*x^3 + (a^2*c^2*d - a*d^3)*x)*sqrt(b*x^2 + a))/(b^2*c^4*x^4 + a^2*c^4 - 2*a*c^ 2*d^2 + d^4 + 2*(a*b*c^4 - b*c^2*d^2)*x^2)) + 2*sqrt(-a*b*c^2 + b*d^2)*log ((b*c^2*x^2 - a*c^2 - 2*sqrt(-a*b*c^2 + b*d^2)*c*x + d^2)/(b*c^2*x^2 + a*c ^2 - d^2)))/(a*b*c^2 - b*d^2), -1/2*(2*sqrt(a*b*c^2 - b*d^2)*arctan(-sqrt( a*b*c^2 - b*d^2)*c*x/(a*c^2 - d^2)) - sqrt(a*b*c^2 - b*d^2)*arctan(1/2*(a^ 2*c^2 - a*d^2 + (a*b*c^2 - 2*b*d^2)*x^2)*sqrt(a*b*c^2 - b*d^2)*sqrt(b*x^2 + a)/((a*b^2*c^2*d - b^2*d^3)*x^3 + (a^2*b*c^2*d - a*b*d^3)*x)))/(a*b*c^2 - b*d^2)]
\[ \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\int \frac {1}{a c + b c x^{2} + d \sqrt {a + b x^{2}}}\, dx \] Input:
integrate(1/(a*c+b*c*x**2+d*(b*x**2+a)**(1/2)),x)
Output:
Integral(1/(a*c + b*c*x**2 + d*sqrt(a + b*x**2)), x)
\[ \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\int { \frac {1}{b c x^{2} + a c + \sqrt {b x^{2} + a} d} \,d x } \] Input:
integrate(1/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/(b*c*x^2 + a*c + sqrt(b*x^2 + a)*d), x)
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (53) = 106\).
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.70 \[ \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\frac {\arctan \left (\frac {b c x}{\sqrt {a b c^{2} - b d^{2}}}\right )}{\sqrt {a b c^{2} - b d^{2}}} + \frac {\arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} c^{2} + a c^{2} - 2 \, d^{2}}{2 \, \sqrt {a c^{2} - d^{2}} d}\right )}{\sqrt {a c^{2} - d^{2}} \sqrt {b}} \] Input:
integrate(1/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="giac")
Output:
arctan(b*c*x/sqrt(a*b*c^2 - b*d^2))/sqrt(a*b*c^2 - b*d^2) + arctan(1/2*((s qrt(b)*x - sqrt(b*x^2 + a))^2*c^2 + a*c^2 - 2*d^2)/(sqrt(a*c^2 - d^2)*d))/ (sqrt(a*c^2 - d^2)*sqrt(b))
Timed out. \[ \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\left \{\begin {array}{cl} \frac {\mathrm {atan}\left (\frac {b\,c\,x}{\sqrt {a\,b\,c^2-b\,d^2}}\right )}{\sqrt {a\,b\,c^2-b\,d^2}}-\frac {d\,x}{\sqrt {a}\,\left (a\,c^2-d^2\right )} & \text {\ if\ \ }b=0\vee d=0\\ \frac {\mathrm {atan}\left (\frac {b\,c\,x}{\sqrt {a\,b\,c^2-b\,d^2}}\right )}{\sqrt {a\,b\,c^2-b\,d^2}}-\frac {d\,\mathrm {atan}\left (\frac {x\,\sqrt {a\,b\,c^2-b\,\left (a\,c^2-d^2\right )}}{\sqrt {a\,c^2-d^2}\,\sqrt {b\,x^2+a}}\right )}{\sqrt {-\left (a\,c^2-d^2\right )\,\left (b\,\left (a\,c^2-d^2\right )-a\,b\,c^2\right )}} & \text {\ if\ \ }0<b\,d^2\\ \frac {\mathrm {atan}\left (\frac {b\,c\,x}{\sqrt {a\,b\,c^2-b\,d^2}}\right )}{\sqrt {a\,b\,c^2-b\,d^2}}-\frac {d\,\ln \left (\frac {\sqrt {\left (a\,c^2-d^2\right )\,\left (b\,x^2+a\right )}+x\,\sqrt {b\,\left (a\,c^2-d^2\right )-a\,b\,c^2}}{\sqrt {\left (a\,c^2-d^2\right )\,\left (b\,x^2+a\right )}-x\,\sqrt {b\,\left (a\,c^2-d^2\right )-a\,b\,c^2}}\right )}{2\,\sqrt {\left (a\,c^2-d^2\right )\,\left (b\,\left (a\,c^2-d^2\right )-a\,b\,c^2\right )}} & \text {\ if\ \ }b\,d^2<0\\ \int \frac {1}{a\,c+d\,\sqrt {b\,x^2+a}+b\,c\,x^2} \,d x & \text {\ if\ \ }b\,d^2\notin \mathbb {R} \end {array}\right . \] Input:
int(1/(a*c + d*(a + b*x^2)^(1/2) + b*c*x^2),x)
Output:
piecewise(b == 0 | d == 0, atan((b*c*x)/(- b*d^2 + a*b*c^2)^(1/2))/(- b*d^ 2 + a*b*c^2)^(1/2) - (d*x)/(a^(1/2)*(a*c^2 - d^2)), 0 < b*d^2, atan((b*c*x )/(- b*d^2 + a*b*c^2)^(1/2))/(- b*d^2 + a*b*c^2)^(1/2) - (d*atan((x*(- b*( a*c^2 - d^2) + a*b*c^2)^(1/2))/((a*c^2 - d^2)^(1/2)*(a + b*x^2)^(1/2))))/( -(a*c^2 - d^2)*(b*(a*c^2 - d^2) - a*b*c^2))^(1/2), b*d^2 < 0, atan((b*c*x) /(- b*d^2 + a*b*c^2)^(1/2))/(- b*d^2 + a*b*c^2)^(1/2) - (d*log((((a*c^2 - d^2)*(a + b*x^2))^(1/2) + x*(b*(a*c^2 - d^2) - a*b*c^2)^(1/2))/(((a*c^2 - d^2)*(a + b*x^2))^(1/2) - x*(b*(a*c^2 - d^2) - a*b*c^2)^(1/2))))/(2*((a*c^ 2 - d^2)*(b*(a*c^2 - d^2) - a*b*c^2))^(1/2)), ~in(b*d^2, 'real'), int(1/(a *c + d*(a + b*x^2)^(1/2) + b*c*x^2), x))
Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {1}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx=\frac {2 \sqrt {b}\, \sqrt {a \,c^{2}-d^{2}}\, \mathit {atan} \left (\frac {\sqrt {b \,x^{2}+a}\, c +\sqrt {b}\, c x +d}{\sqrt {a \,c^{2}-d^{2}}}\right )}{b \left (a \,c^{2}-d^{2}\right )} \] Input:
int(1/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x)
Output:
(2*sqrt(b)*sqrt(a*c**2 - d**2)*atan((sqrt(a + b*x**2)*c + sqrt(b)*c*x + d) /sqrt(a*c**2 - d**2)))/(b*(a*c**2 - d**2))