Integrand size = 28, antiderivative size = 68 \[ \int \frac {a+b x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {b x \sqrt {-c+d x} \sqrt {c+d x}}{2 d^2}+\frac {\left (b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^3} \]
(2*a*d^2+b*c^2)*arctanh((d*x-c)^(1/2)/(d*x+c)^(1/2))/d^3+1/2*b*x*(d*x-c)^( 1/2)*(d*x+c)^(1/2)/d^2
Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {b d x \sqrt {-c+d x} \sqrt {c+d x}+2 \left (b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{2 d^3} \]
(b*d*x*Sqrt[-c + d*x]*Sqrt[c + d*x] + 2*(b*c^2 + 2*a*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/(2*d^3)
Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {646, 45, 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.
\(\displaystyle \int \frac {a+b x^2}{\sqrt {d x-c} \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 646 |
\(\displaystyle \frac {\left (2 a d^2+b c^2\right ) \int \frac {1}{\sqrt {d x-c} \sqrt {c+d x}}dx}{2 d^2}+\frac {b x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\) |
\(\Big \downarrow \) 45 |
\(\displaystyle \frac {\left (2 a d^2+b c^2\right ) \int \frac {1}{d-\frac {d (d x-c)}{c+d x}}d\frac {\sqrt {d x-c}}{\sqrt {c+d x}}}{d^2}+\frac {b x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\left (2 a d^2+b c^2\right ) \text {arctanh}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^3}+\frac {b x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\) |
(b*x*Sqrt[-c + d*x]*Sqrt[c + d*x])/(2*d^2) + ((b*c^2 + 2*a*d^2)*ArcTanh[Sq rt[-c + d*x]/Sqrt[c + d*x]])/d^3
3.4.62.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
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_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) ^2), x_Symbol] :> Simp[b*x*(c + d*x)^(m + 1)*((e + f*x)^(n + 1)/(d*f*(2*m + 3))), x] - Simp[(b*c*e - a*d*f*(2*m + 3))/(d*f*(2*m + 3)) Int[(c + d*x)^ m*(e + f*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && !LtQ[m, -1]
Time = 4.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.63
method | result | size |
risch | \(-\frac {\left (-d x +c \right ) \sqrt {d x +c}\, b x}{2 d^{2} \sqrt {d x -c}}+\frac {\left (2 a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {x \,d^{2}}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{2 d^{2} \sqrt {d^{2}}\, \sqrt {d x -c}\, \sqrt {d x +c}}\) | \(111\) |
default | \(\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (\operatorname {csgn}\left (d \right ) d \sqrt {d^{2} x^{2}-c^{2}}\, b x +\ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{2}+2 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) a \,d^{2}\right ) \operatorname {csgn}\left (d \right )}{2 d^{3} \sqrt {d^{2} x^{2}-c^{2}}}\) | \(124\) |
-1/2*(-d*x+c)*(d*x+c)^(1/2)*b*x/d^2/(d*x-c)^(1/2)+1/2*(2*a*d^2+b*c^2)/d^2* ln(x*d^2/(d^2)^(1/2)+(d^2*x^2-c^2)^(1/2))/(d^2)^(1/2)*((d*x-c)*(d*x+c))^(1 /2)/(d*x-c)^(1/2)/(d*x+c)^(1/2)
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\sqrt {d x + c} \sqrt {d x - c} b d x - {\left (b c^{2} + 2 \, a d^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{2 \, d^{3}} \]
1/2*(sqrt(d*x + c)*sqrt(d*x - c)*b*d*x - (b*c^2 + 2*a*d^2)*log(-d*x + sqrt (d*x + c)*sqrt(d*x - c)))/d^3
Timed out. \[ \int \frac {a+b x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.31 \[ \int \frac {a+b x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {b c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{2 \, d^{3}} + \frac {a \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{d} + \frac {\sqrt {d^{2} x^{2} - c^{2}} b x}{2 \, d^{2}} \]
1/2*b*c^2*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*d)/d^3 + a*log(2*d^2*x + 2*s qrt(d^2*x^2 - c^2)*d)/d + 1/2*sqrt(d^2*x^2 - c^2)*b*x/d^2
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.16 \[ \int \frac {a+b x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\sqrt {d x + c} \sqrt {d x - c} {\left (\frac {{\left (d x + c\right )} b}{d^{2}} - \frac {b c}{d^{2}}\right )} - \frac {2 \, {\left (b c^{2} + 2 \, a d^{2}\right )} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{2}}}{2 \, d} \]
1/2*(sqrt(d*x + c)*sqrt(d*x - c)*((d*x + c)*b/d^2 - b*c/d^2) - 2*(b*c^2 + 2*a*d^2)*log(abs(-sqrt(d*x + c) + sqrt(d*x - c)))/d^2)/d
Time = 15.57 (sec) , antiderivative size = 417, normalized size of antiderivative = 6.13 \[ \int \frac {a+b x^2}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\frac {2\,b\,c^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{\sqrt {-c}-\sqrt {d\,x-c}}+\frac {14\,b\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^3}+\frac {14\,b\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^5}+\frac {2\,b\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^7}}{d^3-\frac {4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {6\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}}+\frac {4\,a\,\mathrm {atan}\left (\frac {d\,\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}{\sqrt {-d^2}\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}\right )}{\sqrt {-d^2}}-\frac {2\,b\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{d^3} \]
((2*b*c^2*((c + d*x)^(1/2) - c^(1/2)))/((-c)^(1/2) - (d*x - c)^(1/2)) + (1 4*b*c^2*((c + d*x)^(1/2) - c^(1/2))^3)/((-c)^(1/2) - (d*x - c)^(1/2))^3 + (14*b*c^2*((c + d*x)^(1/2) - c^(1/2))^5)/((-c)^(1/2) - (d*x - c)^(1/2))^5 + (2*b*c^2*((c + d*x)^(1/2) - c^(1/2))^7)/((-c)^(1/2) - (d*x - c)^(1/2))^7 )/(d^3 - (4*d^3*((c + d*x)^(1/2) - c^(1/2))^2)/((-c)^(1/2) - (d*x - c)^(1/ 2))^2 + (6*d^3*((c + d*x)^(1/2) - c^(1/2))^4)/((-c)^(1/2) - (d*x - c)^(1/2 ))^4 - (4*d^3*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c)^(1/2) )^6 + (d^3*((c + d*x)^(1/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 ) + (4*a*atan((d*((-c)^(1/2) - (d*x - c)^(1/2)))/((-d^2)^(1/2)*((c + d*x)^ (1/2) - c^(1/2)))))/(-d^2)^(1/2) - (2*b*c^2*atanh(((c + d*x)^(1/2) - c^(1/ 2))/((-c)^(1/2) - (d*x - c)^(1/2))))/d^3