\(\int \frac {\sqrt {c+d x}}{(a-b x^2)^2} \, dx\) [644]

Optimal result

Integrand size = 20, antiderivative size = 194 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx=\frac {x \sqrt {c+d x}}{2 a \left (a-b x^2\right )}-\frac {\left (\frac {2 \sqrt {b} c}{\sqrt {a}}-d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 a b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {\left (\frac {2 \sqrt {b} c}{\sqrt {a}}+d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 a b^{3/4} \sqrt {\sqrt {b} c+\sqrt {a} d}} \] Output:

1/2*x*(d*x+c)^(1/2)/a/(-b*x^2+a)-1/4*(2*b^(1/2)*c/a^(1/2)-d)*arctanh(b^(1/ 
4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a/b^(3/4)/(b^(1/2)*c-a^(1/2) 
*d)^(1/2)+1/4*(2*b^(1/2)*c/a^(1/2)+d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/ 
2)*c+a^(1/2)*d)^(1/2))/a/b^(3/4)/(b^(1/2)*c+a^(1/2)*d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} x \sqrt {c+d x}}{a-b x^2}+\frac {\left (2 \sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {\left (2 \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{4 a^{3/2}} \] Input:

Integrate[Sqrt[c + d*x]/(a - b*x^2)^2,x]
 

Output:

((2*Sqrt[a]*x*Sqrt[c + d*x])/(a - b*x^2) + ((2*Sqrt[b]*c + Sqrt[a]*d)*ArcT 
an[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d 
)])/(Sqrt[b]*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]) - ((2*Sqrt[b]*c - Sqrt[a]*d 
)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqr 
t[a]*d)])/(Sqrt[b]*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/(4*a^(3/2))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {494, 27, 654, 25, 27, 1480, 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.

\(\displaystyle \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx\)

\(\Big \downarrow \) 494

\(\displaystyle \frac {x \sqrt {c+d x}}{2 a \left (a-b x^2\right )}-\frac {\int -\frac {2 c+d x}{2 \sqrt {c+d x} \left (a-b x^2\right )}dx}{2 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {2 c+d x}{\sqrt {c+d x} \left (a-b x^2\right )}dx}{4 a}+\frac {x \sqrt {c+d x}}{2 a \left (a-b x^2\right )}\)

\(\Big \downarrow \) 654

\(\displaystyle \frac {\int -\frac {d (2 c+d x)}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a}+\frac {x \sqrt {c+d x}}{2 a \left (a-b x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x \sqrt {c+d x}}{2 a \left (a-b x^2\right )}-\frac {\int \frac {d (2 c+d x)}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {c+d x}}{2 a \left (a-b x^2\right )}-\frac {d \int \frac {2 c+d x}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {x \sqrt {c+d x}}{2 a \left (a-b x^2\right )}-\frac {d \left (\frac {1}{2} \left (1-\frac {2 \sqrt {b} c}{\sqrt {a} d}\right ) \int \frac {1}{b (c+d x)-\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}d\sqrt {c+d x}+\frac {1}{2} \left (\frac {2 \sqrt {b} c}{\sqrt {a} d}+1\right ) \int \frac {1}{b (c+d x)-\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}d\sqrt {c+d x}\right )}{2 a}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {x \sqrt {c+d x}}{2 a \left (a-b x^2\right )}-\frac {d \left (-\frac {\left (1-\frac {2 \sqrt {b} c}{\sqrt {a} d}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\left (\frac {2 \sqrt {b} c}{\sqrt {a} d}+1\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 a}\)

Input:

Int[Sqrt[c + d*x]/(a - b*x^2)^2,x]
 

Output:

(x*Sqrt[c + d*x])/(2*a*(a - b*x^2)) - (d*(-1/2*((1 - (2*Sqrt[b]*c)/(Sqrt[a 
]*d))*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(b^(3/ 
4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - ((1 + (2*Sqrt[b]*c)/(Sqrt[a]*d))*ArcTanh 
[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*b^(3/4)*Sqrt[Sqr 
t[b]*c + Sqrt[a]*d])))/(2*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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]
 

rule 494

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-x)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[1/(2*a*(p + 
 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^(p + 1)*(c*(2*p + 3) + d*(n + 2*p 
+ 3)*x), x], x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 0] && (Lt 
Q[n, 1] || (ILtQ[n + 2*p + 3, 0] && NeQ[n, 2])) && IntQuadraticQ[a, 0, b, c 
, d, n, p, x]
 

rule 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(\frac {d \,b^{2} \left (-\frac {\sqrt {d x +c}}{2 b^{2} \left (b d x +\sqrt {a b \,d^{2}}\right )}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right ) c}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, b \sqrt {a b \,d^{2}}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b^{2} \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\frac {\sqrt {a b \,d^{2}}\, \sqrt {d x +c}}{-b d x +\sqrt {a b \,d^{2}}}+\frac {\left (\sqrt {a b \,d^{2}}+2 b c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{2 b^{2} \sqrt {a b \,d^{2}}}\right )}{2 a}\)	\(246\)
derivativedivides	\(2 d^{3} b^{2} \left (\frac {\frac {\sqrt {a b \,d^{2}}\, \sqrt {d x +c}}{2 b^{2} \left (-d x +\frac {\sqrt {a b \,d^{2}}}{b}\right )}+\frac {\left (\sqrt {a b \,d^{2}}+2 b c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{4 b a \,d^{2} \sqrt {a b \,d^{2}}}+\frac {\frac {\sqrt {a b \,d^{2}}\, \sqrt {d x +c}}{2 b^{2} \left (-d x -\frac {\sqrt {a b \,d^{2}}}{b}\right )}-\frac {\left (\sqrt {a b \,d^{2}}-2 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{4 b a \,d^{2} \sqrt {a b \,d^{2}}}\right )\)	\(251\)
default	\(2 d^{3} b^{2} \left (\frac {\frac {\sqrt {a b \,d^{2}}\, \sqrt {d x +c}}{2 b^{2} \left (-d x +\frac {\sqrt {a b \,d^{2}}}{b}\right )}+\frac {\left (\sqrt {a b \,d^{2}}+2 b c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{4 b a \,d^{2} \sqrt {a b \,d^{2}}}+\frac {\frac {\sqrt {a b \,d^{2}}\, \sqrt {d x +c}}{2 b^{2} \left (-d x -\frac {\sqrt {a b \,d^{2}}}{b}\right )}-\frac {\left (\sqrt {a b \,d^{2}}-2 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{4 b a \,d^{2} \sqrt {a b \,d^{2}}}\right )\)	\(251\)

Input:

int((d*x+c)^(1/2)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*d*b^2/a*(-1/2/b^2*(d*x+c)^(1/2)/(b*d*x+(a*b*d^2)^(1/2))+1/((-b*c+(a*b* 
d^2)^(1/2))*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/ 
2))*c/b/(a*b*d^2)^(1/2)-1/2/b^2/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)*arctan(b* 
(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+1/2/b^2/(a*b*d^2)^(1/2)*(( 
a*b*d^2)^(1/2)*(d*x+c)^(1/2)/(-b*d*x+(a*b*d^2)^(1/2))+((a*b*d^2)^(1/2)+2*b 
*c)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2 
)^(1/2))*b)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1385 vs. \(2 (143) = 286\).

Time = 0.12 (sec) , antiderivative size = 1385, normalized size of antiderivative = 7.14 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="fricas")
 

Output:

-1/8*((a*b*x^2 - a^2)*sqrt((4*b*c^3 - 3*a*c*d^2 + (a^3*b^2*c^2 - a^4*b*d^2 
)*sqrt(d^6/(a^3*b^5*c^4 - 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4)))/(a^3*b^2*c^2 
- a^4*b*d^2))*log(-(4*b*c^2*d^3 - a*d^5)*sqrt(d*x + c) + (a^2*b*c*d^4 - (2 
*a^3*b^4*c^4 - 3*a^4*b^3*c^2*d^2 + a^5*b^2*d^4)*sqrt(d^6/(a^3*b^5*c^4 - 2* 
a^4*b^4*c^2*d^2 + a^5*b^3*d^4)))*sqrt((4*b*c^3 - 3*a*c*d^2 + (a^3*b^2*c^2 
- a^4*b*d^2)*sqrt(d^6/(a^3*b^5*c^4 - 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4)))/(a 
^3*b^2*c^2 - a^4*b*d^2))) - (a*b*x^2 - a^2)*sqrt((4*b*c^3 - 3*a*c*d^2 + (a 
^3*b^2*c^2 - a^4*b*d^2)*sqrt(d^6/(a^3*b^5*c^4 - 2*a^4*b^4*c^2*d^2 + a^5*b^ 
3*d^4)))/(a^3*b^2*c^2 - a^4*b*d^2))*log(-(4*b*c^2*d^3 - a*d^5)*sqrt(d*x + 
c) - (a^2*b*c*d^4 - (2*a^3*b^4*c^4 - 3*a^4*b^3*c^2*d^2 + a^5*b^2*d^4)*sqrt 
(d^6/(a^3*b^5*c^4 - 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4)))*sqrt((4*b*c^3 - 3*a 
*c*d^2 + (a^3*b^2*c^2 - a^4*b*d^2)*sqrt(d^6/(a^3*b^5*c^4 - 2*a^4*b^4*c^2*d 
^2 + a^5*b^3*d^4)))/(a^3*b^2*c^2 - a^4*b*d^2))) + (a*b*x^2 - a^2)*sqrt((4* 
b*c^3 - 3*a*c*d^2 - (a^3*b^2*c^2 - a^4*b*d^2)*sqrt(d^6/(a^3*b^5*c^4 - 2*a^ 
4*b^4*c^2*d^2 + a^5*b^3*d^4)))/(a^3*b^2*c^2 - a^4*b*d^2))*log(-(4*b*c^2*d^ 
3 - a*d^5)*sqrt(d*x + c) + (a^2*b*c*d^4 + (2*a^3*b^4*c^4 - 3*a^4*b^3*c^2*d 
^2 + a^5*b^2*d^4)*sqrt(d^6/(a^3*b^5*c^4 - 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4) 
))*sqrt((4*b*c^3 - 3*a*c*d^2 - (a^3*b^2*c^2 - a^4*b*d^2)*sqrt(d^6/(a^3*b^5 
*c^4 - 2*a^4*b^4*c^2*d^2 + a^5*b^3*d^4)))/(a^3*b^2*c^2 - a^4*b*d^2))) - (a 
*b*x^2 - a^2)*sqrt((4*b*c^3 - 3*a*c*d^2 - (a^3*b^2*c^2 - a^4*b*d^2)*sqr...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)/(-b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \] Input:

integrate((d*x+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x + c)/(b*x^2 - a)^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (143) = 286\).

Time = 0.20 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.79 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx=-\frac {{\left (d x + c\right )}^{\frac {3}{2}} d - \sqrt {d x + c} c d}{2 \, {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )} a} + \frac {{\left (2 \, a b c^{2} d {\left | b \right |} - a^{2} d^{3} {\left | b \right |} - \sqrt {a b} c d {\left | a \right |} {\left | b \right |} {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c + \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{4 \, {\left (a^{2} b d - \sqrt {a b} a b c\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} + \frac {{\left (a b c d {\left | a \right |} {\left | b \right |} {\left | d \right |} + 2 \, \sqrt {a b} a b c^{2} d {\left | b \right |} - \sqrt {a b} a^{2} d^{3} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c - \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{4 \, {\left (a^{2} b^{2} c + \sqrt {a b} a^{2} b d\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | a \right |} {\left | d \right |}} \] Input:

integrate((d*x+c)^(1/2)/(-b*x^2+a)^2,x, algorithm="giac")
 

Output:

-1/2*((d*x + c)^(3/2)*d - sqrt(d*x + c)*c*d)/(((d*x + c)^2*b - 2*(d*x + c) 
*b*c + b*c^2 - a*d^2)*a) + 1/4*(2*a*b*c^2*d*abs(b) - a^2*d^3*abs(b) - sqrt 
(a*b)*c*d*abs(a)*abs(b)*abs(d))*arctan(sqrt(d*x + c)/sqrt(-(a*b*c + sqrt(a 
^2*b^2*c^2 - (a*b*c^2 - a^2*d^2)*a*b))/(a*b)))/((a^2*b*d - sqrt(a*b)*a*b*c 
)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(a)*abs(d)) + 1/4*(a*b*c*d*abs(a)*abs(b) 
*abs(d) + 2*sqrt(a*b)*a*b*c^2*d*abs(b) - sqrt(a*b)*a^2*d^3*abs(b))*arctan( 
sqrt(d*x + c)/sqrt(-(a*b*c - sqrt(a^2*b^2*c^2 - (a*b*c^2 - a^2*d^2)*a*b))/ 
(a*b)))/((a^2*b^2*c + sqrt(a*b)*a^2*b*d)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs( 
a)*abs(d))
 

Mupad [B] (verification not implemented)

Time = 10.14 (sec) , antiderivative size = 2332, normalized size of antiderivative = 12.02 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int((c + d*x)^(1/2)/(a - b*x^2)^2,x)
 

Output:

- atan((((8*b^3*c*d^3 - 64*a*b^4*c*d^2*(c + d*x)^(1/2)*(-(d^3*(a^9*b^3)^(1 
/2) - 4*a^3*b^3*c^3 + 3*a^4*b^2*c*d^2)/(64*(a^6*b^4*c^2 - a^7*b^3*d^2)))^( 
1/2))*(-(d^3*(a^9*b^3)^(1/2) - 4*a^3*b^3*c^3 + 3*a^4*b^2*c*d^2)/(64*(a^6*b 
^4*c^2 - a^7*b^3*d^2)))^(1/2) + ((a*b^2*d^4 + 4*b^3*c^2*d^2)*(c + d*x)^(1/ 
2))/a^2)*(-(d^3*(a^9*b^3)^(1/2) - 4*a^3*b^3*c^3 + 3*a^4*b^2*c*d^2)/(64*(a^ 
6*b^4*c^2 - a^7*b^3*d^2)))^(1/2)*1i - ((8*b^3*c*d^3 + 64*a*b^4*c*d^2*(c + 
d*x)^(1/2)*(-(d^3*(a^9*b^3)^(1/2) - 4*a^3*b^3*c^3 + 3*a^4*b^2*c*d^2)/(64*( 
a^6*b^4*c^2 - a^7*b^3*d^2)))^(1/2))*(-(d^3*(a^9*b^3)^(1/2) - 4*a^3*b^3*c^3 
 + 3*a^4*b^2*c*d^2)/(64*(a^6*b^4*c^2 - a^7*b^3*d^2)))^(1/2) - ((a*b^2*d^4 
+ 4*b^3*c^2*d^2)*(c + d*x)^(1/2))/a^2)*(-(d^3*(a^9*b^3)^(1/2) - 4*a^3*b^3* 
c^3 + 3*a^4*b^2*c*d^2)/(64*(a^6*b^4*c^2 - a^7*b^3*d^2)))^(1/2)*1i)/(((8*b^ 
3*c*d^3 - 64*a*b^4*c*d^2*(c + d*x)^(1/2)*(-(d^3*(a^9*b^3)^(1/2) - 4*a^3*b^ 
3*c^3 + 3*a^4*b^2*c*d^2)/(64*(a^6*b^4*c^2 - a^7*b^3*d^2)))^(1/2))*(-(d^3*( 
a^9*b^3)^(1/2) - 4*a^3*b^3*c^3 + 3*a^4*b^2*c*d^2)/(64*(a^6*b^4*c^2 - a^7*b 
^3*d^2)))^(1/2) + ((a*b^2*d^4 + 4*b^3*c^2*d^2)*(c + d*x)^(1/2))/a^2)*(-(d^ 
3*(a^9*b^3)^(1/2) - 4*a^3*b^3*c^3 + 3*a^4*b^2*c*d^2)/(64*(a^6*b^4*c^2 - a^ 
7*b^3*d^2)))^(1/2) - (4*b^2*c^2*d^3 - a*b*d^5)/(4*a^3) + ((8*b^3*c*d^3 + 6 
4*a*b^4*c*d^2*(c + d*x)^(1/2)*(-(d^3*(a^9*b^3)^(1/2) - 4*a^3*b^3*c^3 + 3*a 
^4*b^2*c*d^2)/(64*(a^6*b^4*c^2 - a^7*b^3*d^2)))^(1/2))*(-(d^3*(a^9*b^3)^(1 
/2) - 4*a^3*b^3*c^3 + 3*a^4*b^2*c*d^2)/(64*(a^6*b^4*c^2 - a^7*b^3*d^2))...
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 908, normalized size of antiderivative = 4.68 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(1/2)/(-b*x^2+a)^2,x)
 

Output:

( - 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b 
)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*d**2 + 4*sqrt(a)*sqrt(sqrt(b)*sqrt( 
a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)) 
)*a*b*c**2 + 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b 
)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b*d**2*x**2 - 4*sqrt(a)*sqrt( 
sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt 
(a)*d - b*c)))*b**2*c**2*x**2 + 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*at 
an((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*c*d - 2 
*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqr 
t(sqrt(b)*sqrt(a)*d - b*c)))*a*b*c*d*x**2 - sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d 
 + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a**2 
*d**2 + 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a 
)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*b*c**2 + sqrt(a)*sqrt(sqrt(b)*sqrt(a 
)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a 
*b*d**2*x**2 - 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b) 
*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*b**2*c**2*x**2 + sqrt(a)*sqrt(s 
qrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c 
 + d*x))*a**2*d**2 - 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt 
(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*b*c**2 - sqrt(a)*sqrt(sqrt 
(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c...