Integrand size = 15, antiderivative size = 64 \[ \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx=\frac {(c+d x) \sqrt {a+\frac {b}{(c+d x)^2}}}{d}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{(c+d x) \sqrt {a+\frac {b}{(c+d x)^2}}}\right )}{d} \] Output:
(d*x+c)*(a+b/(d*x+c)^2)^(1/2)/d-b^(1/2)*arctanh(b^(1/2)/(d*x+c)/(a+b/(d*x+ c)^2)^(1/2))/d
Time = 10.16 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.66 \[ \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx=\frac {\sqrt {a+\frac {b}{(c+d x)^2}} \left (\sqrt {a} (c+d x) \sqrt {\frac {b+a (c+d x)^2}{a (c+d x)^2}}-\sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b}}{\sqrt {a} (c+d x)}\right )\right )}{\sqrt {a} d \sqrt {1+\frac {b}{a (c+d x)^2}}} \] Input:
Integrate[Sqrt[a + b/(c + d*x)^2],x]
Output:
(Sqrt[a + b/(c + d*x)^2]*(Sqrt[a]*(c + d*x)*Sqrt[(b + a*(c + d*x)^2)/(a*(c + d*x)^2)] - Sqrt[b]*ArcSinh[Sqrt[b]/(Sqrt[a]*(c + d*x))]))/(Sqrt[a]*d*Sq rt[1 + b/(a*(c + d*x)^2)])
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {239, 773, 247, 224, 219}
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.
\(\displaystyle \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx\) |
\(\Big \downarrow \) 239 |
\(\displaystyle \frac {\int \sqrt {a+\frac {b}{(c+d x)^2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 773 |
\(\displaystyle -\frac {\int (c+d x)^2 \sqrt {a+\frac {b}{(c+d x)^2}}d\frac {1}{c+d x}}{d}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle -\frac {b \int \frac {1}{\sqrt {a+\frac {b}{(c+d x)^2}}}d\frac {1}{c+d x}-(c+d x) \sqrt {a+\frac {b}{(c+d x)^2}}}{d}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {b \int \frac {1}{1-\frac {b}{(c+d x)^2}}d\frac {1}{(c+d x) \sqrt {a+\frac {b}{(c+d x)^2}}}-(c+d x) \sqrt {a+\frac {b}{(c+d x)^2}}}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b}}{(c+d x) \sqrt {a+\frac {b}{(c+d x)^2}}}\right )-(c+d x) \sqrt {a+\frac {b}{(c+d x)^2}}}{d}\) |
Input:
Int[Sqrt[a + b/(c + d*x)^2],x]
Output:
-((-((c + d*x)*Sqrt[a + b/(c + d*x)^2]) + Sqrt[b]*ArcTanh[Sqrt[b]/((c + d* x)*Sqrt[a + b/(c + d*x)^2])])/d)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Simp[1/Coefficient[v, x, 1 ] Subst[Int[(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, n, p}, x] && Lin earQ[v, x] && NeQ[v, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(56)=112\).
Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.08
method | result | size |
default | \(\frac {\sqrt {\frac {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}{\left (d x +c \right )^{2}}}\, \left (d x +c \right ) \left (\sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}-\sqrt {b}\, \ln \left (\frac {2 \left (\sqrt {b}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}+b \right ) d}{d x +c}\right )\right )}{\sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, d}\) | \(133\) |
risch | \(\frac {\sqrt {\frac {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}{\left (d x +c \right )^{2}}}\, \left (d x +c \right )}{d}-\frac {\sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,d^{2} \left (x +\frac {c}{d}\right )^{2}+b}}{x +\frac {c}{d}}\right ) \sqrt {\frac {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}{\left (d x +c \right )^{2}}}\, \left (d x +c \right )}{d \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}\) | \(147\) |
Input:
int((a+b/(d*x+c)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
((a*d^2*x^2+2*a*c*d*x+a*c^2+b)/(d*x+c)^2)^(1/2)*(d*x+c)/(a*d^2*x^2+2*a*c*d *x+a*c^2+b)^(1/2)/d*((a*d^2*x^2+2*a*c*d*x+a*c^2+b)^(1/2)-b^(1/2)*ln(2*(b^( 1/2)*(a*d^2*x^2+2*a*c*d*x+a*c^2+b)^(1/2)+b)*d/(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (56) = 112\).
Time = 0.12 (sec) , antiderivative size = 272, normalized size of antiderivative = 4.25 \[ \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx=\left [\frac {2 \, {\left (d x + c\right )} \sqrt {\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \sqrt {b} \log \left (-\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} - 2 \, {\left (d x + c\right )} \sqrt {b} \sqrt {\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + 2 \, b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, d}, \frac {{\left (d x + c\right )} \sqrt {\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} + \sqrt {-b} \arctan \left (\frac {{\left (d x + c\right )} \sqrt {-b} \sqrt {\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{b}\right )}{d}\right ] \] Input:
integrate((a+b/(d*x+c)^2)^(1/2),x, algorithm="fricas")
Output:
[1/2*(2*(d*x + c)*sqrt((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c* d*x + c^2)) + sqrt(b)*log(-(a*d^2*x^2 + 2*a*c*d*x + a*c^2 - 2*(d*x + c)*sq rt(b)*sqrt((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2)) + 2*b)/(d^2*x^2 + 2*c*d*x + c^2)))/d, ((d*x + c)*sqrt((a*d^2*x^2 + 2*a*c*d *x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2)) + sqrt(-b)*arctan((d*x + c)*sqr t(-b)*sqrt((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2))/ b))/d]
\[ \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx=\int \sqrt {a + \frac {b}{\left (c + d x\right )^{2}}}\, dx \] Input:
integrate((a+b/(d*x+c)**2)**(1/2),x)
Output:
Integral(sqrt(a + b/(c + d*x)**2), x)
\[ \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx=\int { \sqrt {a + \frac {b}{{\left (d x + c\right )}^{2}}} \,d x } \] Input:
integrate((a+b/(d*x+c)^2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a + b/(d*x + c)^2), x)
Time = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.67 \[ \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx=\frac {2 \, b \arctan \left (-\frac {\sqrt {a} c {\left | d \right |} + {\left (\sqrt {a d^{2}} x - \sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}\right )} d}{\sqrt {-b} d}\right ) \mathrm {sgn}\left (d x + c\right )}{\sqrt {-b} d} + \frac {\sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b} \mathrm {sgn}\left (d x + c\right )}{d} \] Input:
integrate((a+b/(d*x+c)^2)^(1/2),x, algorithm="giac")
Output:
2*b*arctan(-(sqrt(a)*c*abs(d) + (sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d* x + a*c^2 + b))*d)/(sqrt(-b)*d))*sgn(d*x + c)/(sqrt(-b)*d) + sqrt(a*d^2*x^ 2 + 2*a*c*d*x + a*c^2 + b)*sgn(d*x + c)/d
Time = 8.83 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx=\frac {\sqrt {a+\frac {b}{{\left (c+d\,x\right )}^2}}\,\left (c+d\,x+\frac {\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,1{}\mathrm {i}}{\sqrt {a}\,\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{\sqrt {a}\,\sqrt {\frac {b}{a\,{\left (c+d\,x\right )}^2}+1}}\right )}{d} \] Input:
int((a + b/(c + d*x)^2)^(1/2),x)
Output:
((a + b/(c + d*x)^2)^(1/2)*(c + d*x + (b^(1/2)*asin((b^(1/2)*1i)/(a^(1/2)* (c + d*x)))*1i)/(a^(1/2)*(b/(a*(c + d*x)^2) + 1)^(1/2))))/d
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.81 \[ \int \sqrt {a+\frac {b}{(c+d x)^2}} \, dx=\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x -\sqrt {b}}{\sqrt {b}}\right )-\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x +\sqrt {b}}{\sqrt {b}}\right )}{d} \] Input:
int((a+b/(d*x+c)^2)^(1/2),x)
Output:
(sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b) + sqrt(b)*log((sqrt(a*c**2 + 2 *a*c*d*x + a*d**2*x**2 + b) + sqrt(a)*c + sqrt(a)*d*x - sqrt(b))/sqrt(b)) - sqrt(b)*log((sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b) + sqrt(a)*c + sq rt(a)*d*x + sqrt(b))/sqrt(b)))/d