\(\int \frac {\sqrt {c+d x}}{(a-b x^2)^3} \, dx\) [693]

Optimal result

Integrand size = 20, antiderivative size = 281 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}-\frac {\sqrt {c+d x} \left (a c d-\left (6 b c^2-5 a d^2\right ) x\right )}{16 a^2 \left (b c^2-a d^2\right ) \left (a-b x^2\right )}-\frac {\left (12 b c^2-18 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{5/2} b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\left (12 b c^2+18 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{5/2} b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}} \] Output:

1/4*x*(d*x+c)^(1/2)/a/(-b*x^2+a)^2-1/16*(d*x+c)^(1/2)*(a*c*d-(-5*a*d^2+6*b 
*c^2)*x)/a^2/(-a*d^2+b*c^2)/(-b*x^2+a)-1/32*(12*b*c^2-18*a^(1/2)*b^(1/2)*c 
*d+5*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^( 
5/2)/b^(3/4)/(b^(1/2)*c-a^(1/2)*d)^(3/2)+1/32*(12*b*c^2+18*a^(1/2)*b^(1/2) 
*c*d+5*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a 
^(5/2)/b^(3/4)/(b^(1/2)*c+a^(1/2)*d)^(3/2)
 

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {c+d x} \left (6 b^2 c^2 x^3+a^2 d (c+9 d x)-a b x \left (10 c^2+c d x+5 d^2 x^2\right )\right )}{\left (-b c^2+a d^2\right ) \left (a-b x^2\right )^2}-\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (12 b c^2+18 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b \left (\sqrt {b} c+\sqrt {a} d\right )^2}-\frac {\left (12 b c^2-18 \sqrt {a} \sqrt {b} c d+5 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{32 a^{5/2}} \] Input:

Integrate[Sqrt[c + d*x]/(a - b*x^2)^3,x]
 

Output:

((2*Sqrt[a]*Sqrt[c + d*x]*(6*b^2*c^2*x^3 + a^2*d*(c + 9*d*x) - a*b*x*(10*c 
^2 + c*d*x + 5*d^2*x^2)))/((-(b*c^2) + a*d^2)*(a - b*x^2)^2) - (Sqrt[-(b*c 
) - Sqrt[a]*Sqrt[b]*d]*(12*b*c^2 + 18*Sqrt[a]*Sqrt[b]*c*d + 5*a*d^2)*ArcTa 
n[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d) 
])/(b*(Sqrt[b]*c + Sqrt[a]*d)^2) - ((12*b*c^2 - 18*Sqrt[a]*Sqrt[b]*c*d + 5 
*a*d^2)*ArcTan[(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]*d)*Sqrt[-(b*c) + Sqrt[a]*Sqr 
t[b]*d]))/(32*a^(5/2))
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {494, 27, 686, 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.

\(\displaystyle \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx\)

\(\Big \downarrow \) 494

\(\displaystyle \frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}-\frac {\int -\frac {6 c+5 d x}{2 \sqrt {c+d x} \left (a-b x^2\right )^2}dx}{4 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {6 c+5 d x}{\sqrt {c+d x} \left (a-b x^2\right )^2}dx}{8 a}+\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {-\frac {\int -\frac {b \left (c \left (12 b c^2-13 a d^2\right )+d \left (6 b c^2-5 a d^2\right ) x\right )}{2 \sqrt {c+d x} \left (a-b x^2\right )}dx}{2 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a c d-x \left (6 b c^2-5 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a}+\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {c \left (12 b c^2-13 a d^2\right )+d \left (6 b c^2-5 a d^2\right ) x}{\sqrt {c+d x} \left (a-b x^2\right )}dx}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a c d-x \left (6 b c^2-5 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a}+\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}\)

\(\Big \downarrow \) 654

\(\displaystyle \frac {\frac {\int -\frac {d \left (2 c \left (3 b c^2-4 a d^2\right )+\left (6 b c^2-5 a d^2\right ) (c+d x)\right )}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a c d-x \left (6 b c^2-5 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a}+\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\int \frac {d \left (2 c \left (3 b c^2-4 a d^2\right )+\left (6 b c^2-5 a d^2\right ) (c+d x)\right )}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a c d-x \left (6 b c^2-5 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a}+\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {d \int \frac {2 c \left (3 b c^2-4 a d^2\right )+\left (6 b c^2-5 a d^2\right ) (c+d x)}{b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2}d\sqrt {c+d x}}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a c d-x \left (6 b c^2-5 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a}+\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {-\frac {d \left (\frac {1}{2} \left (-\frac {12 b^{3/2} c^3}{\sqrt {a} d}+13 \sqrt {a} \sqrt {b} c d-5 a d^2+6 b c^2\right ) \int \frac {1}{b (c+d x)-\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}d\sqrt {c+d x}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (18 \sqrt {a} \sqrt {b} c d+5 a d^2+12 b c^2\right ) \int \frac {1}{b (c+d x)-\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}d\sqrt {c+d x}}{2 \sqrt {a} d}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a c d-x \left (6 b c^2-5 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a}+\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {d \left (-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (18 \sqrt {a} \sqrt {b} c d+5 a d^2+12 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} b^{3/4} d \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {\left (-\frac {12 b^{3/2} c^3}{\sqrt {a} d}+13 \sqrt {a} \sqrt {b} c d-5 a d^2+6 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a c d-x \left (6 b c^2-5 a d^2\right )\right )}{2 a \left (a-b x^2\right ) \left (b c^2-a d^2\right )}}{8 a}+\frac {x \sqrt {c+d x}}{4 a \left (a-b x^2\right )^2}\)

Input:

Int[Sqrt[c + d*x]/(a - b*x^2)^3,x]
 

Output:

(x*Sqrt[c + d*x])/(4*a*(a - b*x^2)^2) + (-1/2*(Sqrt[c + d*x]*(a*c*d - (6*b 
*c^2 - 5*a*d^2)*x))/(a*(b*c^2 - a*d^2)*(a - b*x^2)) - (d*(-1/2*((6*b*c^2 - 
 (12*b^(3/2)*c^3)/(Sqrt[a]*d) + 13*Sqrt[a]*Sqrt[b]*c*d - 5*a*d^2)*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]) - ((Sqrt[b]*c - Sqrt[a]*d)*(12*b*c^2 + 18*Sqrt[a]*Sqrt[b 
]*c*d + 5*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]* 
d]])/(2*Sqrt[a]*b^(3/4)*d*Sqrt[Sqrt[b]*c + Sqrt[a]*d])))/(2*a*(b*c^2 - a*d 
^2)))/(8*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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 = 0.90 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.31

method	result	size
pseudoelliptic	\(\frac {13 d^{4} \left (d \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {\left (-5 a \,d^{2}+6 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{13}+b c \left (a \,d^{2}-\frac {12 b \,c^{2}}{13}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d \left (\frac {\left (5 a \,d^{2}-6 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{13}+b c \left (a \,d^{2}-\frac {12 b \,c^{2}}{13}\right )\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {2 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \sqrt {a b \,d^{2}}\, \left (\left (-5 a b \,x^{3}+9 a^{2} x \right ) d^{2}+a c \left (-b \,x^{2}+a \right ) d -10 \left (-\frac {3 b \,x^{2}}{5}+a \right ) x b \,c^{2}\right )}{13}\right )\right ) b^{2}}{32 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (b d x +\sqrt {a b \,d^{2}}\right )^{2} a^{2} \left (b d x -\sqrt {a b \,d^{2}}\right )^{2} \left (a \,d^{2}-b \,c^{2}\right )}\)	\(369\)
derivativedivides	\(-2 d^{5} b^{3} \left (-\frac {\frac {-\frac {\sqrt {a b \,d^{2}}\, \left (6 b c +5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{2} \left (b c +\sqrt {a b \,d^{2}}\right )}+\frac {\sqrt {a b \,d^{2}}\, \left (6 b c +7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{3}}}{{\left (-d x +\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}+\frac {\left (5 a \,d^{2}+12 b \,c^{2}+18 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 \left (b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 \sqrt {a b \,d^{2}}\, d^{4} a^{2} b^{2}}+\frac {\frac {\frac {\sqrt {a b \,d^{2}}\, \left (6 b c -5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{2} \left (b c -\sqrt {a b \,d^{2}}\right )}-\frac {\sqrt {a b \,d^{2}}\, \left (6 b c -7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{3}}}{{\left (-d x -\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}-\frac {\left (-5 a \,d^{2}-12 b \,c^{2}+18 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 \left (-b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 \sqrt {a b \,d^{2}}\, d^{4} a^{2} b^{2}}\right )\)	\(430\)
default	\(-2 d^{5} b^{3} \left (-\frac {\frac {-\frac {\sqrt {a b \,d^{2}}\, \left (6 b c +5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{2} \left (b c +\sqrt {a b \,d^{2}}\right )}+\frac {\sqrt {a b \,d^{2}}\, \left (6 b c +7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{3}}}{{\left (-d x +\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}+\frac {\left (5 a \,d^{2}+12 b \,c^{2}+18 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 \left (b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 \sqrt {a b \,d^{2}}\, d^{4} a^{2} b^{2}}+\frac {\frac {\frac {\sqrt {a b \,d^{2}}\, \left (6 b c -5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{2} \left (b c -\sqrt {a b \,d^{2}}\right )}-\frac {\sqrt {a b \,d^{2}}\, \left (6 b c -7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{3}}}{{\left (-d x -\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}-\frac {\left (-5 a \,d^{2}-12 b \,c^{2}+18 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 \left (-b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 \sqrt {a b \,d^{2}}\, d^{4} a^{2} b^{2}}\right )\)	\(430\)

Input:

int((d*x+c)^(1/2)/(-b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

13/32*d^4/(a*b*d^2)^(1/2)*(d*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(1/13*(-5*a*d 
^2+6*b*c^2)*(a*b*d^2)^(1/2)+b*c*(a*d^2-12/13*b*c^2))*(-b*x^2+a)^2*arctan(b 
*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+((-b*c+(a*b*d^2)^(1/2))*b 
)^(1/2)*(d*(1/13*(5*a*d^2-6*b*c^2)*(a*b*d^2)^(1/2)+b*c*(a*d^2-12/13*b*c^2) 
)*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+2/ 
13*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(d*x+c)^(1/2)*(a*b*d^2)^(1/2)*((-5*a*b* 
x^3+9*a^2*x)*d^2+a*c*(-b*x^2+a)*d-10*(-3/5*b*x^2+a)*x*b*c^2)))*b^2/((b*c+( 
a*b*d^2)^(1/2))*b)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(b*d*x+(a*b*d^2) 
^(1/2))^2/a^2/(b*d*x-(a*b*d^2)^(1/2))^2/(a*d^2-b*c^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3787 vs. \(2 (223) = 446\).

Time = 0.44 (sec) , antiderivative size = 3787, normalized size of antiderivative = 13.48 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)/(-b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\int { -\frac {\sqrt {d x + c}}{{\left (b x^{2} - a\right )}^{3}} \,d x } \] Input:

integrate((d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-integrate(sqrt(d*x + c)/(b*x^2 - a)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1064 vs. \(2 (223) = 446\).

Time = 0.28 (sec) , antiderivative size = 1064, normalized size of antiderivative = 3.79 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^(1/2)/(-b*x^2+a)^3,x, algorithm="giac")
 

Output:

-1/32*((a^2*b*c^2*d - a^3*d^3)^2*(6*b*c^2*d - 5*a*d^3)*abs(b) + 2*(3*sqrt( 
a*b)*a*b^2*c^5*d - 7*sqrt(a*b)*a^2*b*c^3*d^3 + 4*sqrt(a*b)*a^3*c*d^5)*abs( 
a^2*b*c^2*d - a^3*d^3)*abs(b) - (12*a^3*b^4*c^8*d - 37*a^4*b^3*c^6*d^3 + 3 
8*a^5*b^2*c^4*d^5 - 13*a^6*b*c^2*d^7)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-( 
a^2*b^2*c^3 - a^3*b*c*d^2 + sqrt((a^2*b^2*c^3 - a^3*b*c*d^2)^2 - (a^2*b^2* 
c^4 - 2*a^3*b*c^2*d^2 + a^4*d^4)*(a^2*b^2*c^2 - a^3*b*d^2)))/(a^2*b^2*c^2 
- a^3*b*d^2)))/((a^4*b^3*c^4*d - 2*a^5*b^2*c^2*d^3 + a^6*b*d^5 - sqrt(a*b) 
*a^3*b^3*c^5 + 2*sqrt(a*b)*a^4*b^2*c^3*d^2 - sqrt(a*b)*a^5*b*c*d^4)*sqrt(- 
b^2*c - sqrt(a*b)*b*d)*abs(a^2*b*c^2*d - a^3*d^3)) - 1/32*((a^2*b*c^2*d - 
a^3*d^3)^2*(6*b*c^2*d - 5*a*d^3)*abs(b) - 2*(3*sqrt(a*b)*a*b^2*c^5*d - 7*s 
qrt(a*b)*a^2*b*c^3*d^3 + 4*sqrt(a*b)*a^3*c*d^5)*abs(a^2*b*c^2*d - a^3*d^3) 
*abs(b) - (12*a^3*b^4*c^8*d - 37*a^4*b^3*c^6*d^3 + 38*a^5*b^2*c^4*d^5 - 13 
*a^6*b*c^2*d^7)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a^2*b^2*c^3 - a^3*b*c* 
d^2 - sqrt((a^2*b^2*c^3 - a^3*b*c*d^2)^2 - (a^2*b^2*c^4 - 2*a^3*b*c^2*d^2 
+ a^4*d^4)*(a^2*b^2*c^2 - a^3*b*d^2)))/(a^2*b^2*c^2 - a^3*b*d^2)))/((a^4*b 
^3*c^4*d - 2*a^5*b^2*c^2*d^3 + a^6*b*d^5 + sqrt(a*b)*a^3*b^3*c^5 - 2*sqrt( 
a*b)*a^4*b^2*c^3*d^2 + sqrt(a*b)*a^5*b*c*d^4)*sqrt(-b^2*c + sqrt(a*b)*b*d) 
*abs(a^2*b*c^2*d - a^3*d^3)) - 1/16*(6*(d*x + c)^(7/2)*b^2*c^2*d - 18*(d*x 
 + c)^(5/2)*b^2*c^3*d + 18*(d*x + c)^(3/2)*b^2*c^4*d - 6*sqrt(d*x + c)*b^2 
*c^5*d - 5*(d*x + c)^(7/2)*a*b*d^3 + 14*(d*x + c)^(5/2)*a*b*c*d^3 - 23*...
 

Mupad [B] (verification not implemented)

Time = 10.82 (sec) , antiderivative size = 6163, normalized size of antiderivative = 21.93 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((c + d*x)^(1/2)/(a - b*x^2)^3,x)
 

Output:

- atan(((((32768*a^7*b^3*c*d^7 + 24576*a^5*b^5*c^5*d^3 - 57344*a^6*b^4*c^3 
*d^5)/(4096*(a^8*d^4 + a^6*b^2*c^4 - 2*a^7*b*c^2*d^2)) - ((c + d*x)^(1/2)* 
(4096*a^7*b^4*c*d^6 + 4096*a^5*b^6*c^5*d^2 - 8192*a^6*b^5*c^3*d^4)*((144*a 
^5*b^5*c^7 - 25*a*d^7*(a^15*b^3)^(1/2) - 105*a^8*b^2*c*d^6 - 420*a^6*b^4*c 
^5*d^2 + 385*a^7*b^3*c^3*d^4 + 21*b*c^2*d^5*(a^15*b^3)^(1/2))/(4096*(a^10* 
b^6*c^6 - a^13*b^3*d^6 - 3*a^11*b^5*c^4*d^2 + 3*a^12*b^4*c^2*d^4)))^(1/2)) 
/(64*(a^6*d^4 + a^4*b^2*c^4 - 2*a^5*b*c^2*d^2)))*((144*a^5*b^5*c^7 - 25*a* 
d^7*(a^15*b^3)^(1/2) - 105*a^8*b^2*c*d^6 - 420*a^6*b^4*c^5*d^2 + 385*a^7*b 
^3*c^3*d^4 + 21*b*c^2*d^5*(a^15*b^3)^(1/2))/(4096*(a^10*b^6*c^6 - a^13*b^3 
*d^6 - 3*a^11*b^5*c^4*d^2 + 3*a^12*b^4*c^2*d^4)))^(1/2) + ((c + d*x)^(1/2) 
*(25*a^3*b^2*d^8 + 144*b^5*c^6*d^2 - 276*a*b^4*c^4*d^4 + 109*a^2*b^3*c^2*d 
^6))/(64*(a^6*d^4 + a^4*b^2*c^4 - 2*a^5*b*c^2*d^2)))*((144*a^5*b^5*c^7 - 2 
5*a*d^7*(a^15*b^3)^(1/2) - 105*a^8*b^2*c*d^6 - 420*a^6*b^4*c^5*d^2 + 385*a 
^7*b^3*c^3*d^4 + 21*b*c^2*d^5*(a^15*b^3)^(1/2))/(4096*(a^10*b^6*c^6 - a^13 
*b^3*d^6 - 3*a^11*b^5*c^4*d^2 + 3*a^12*b^4*c^2*d^4)))^(1/2)*1i - (((32768* 
a^7*b^3*c*d^7 + 24576*a^5*b^5*c^5*d^3 - 57344*a^6*b^4*c^3*d^5)/(4096*(a^8* 
d^4 + a^6*b^2*c^4 - 2*a^7*b*c^2*d^2)) + ((c + d*x)^(1/2)*(4096*a^7*b^4*c*d 
^6 + 4096*a^5*b^6*c^5*d^2 - 8192*a^6*b^5*c^3*d^4)*((144*a^5*b^5*c^7 - 25*a 
*d^7*(a^15*b^3)^(1/2) - 105*a^8*b^2*c*d^6 - 420*a^6*b^4*c^5*d^2 + 385*a^7* 
b^3*c^3*d^4 + 21*b*c^2*d^5*(a^15*b^3)^(1/2))/(4096*(a^10*b^6*c^6 - a^13...
 

Reduce [B] (verification not implemented)

Time = 3.51 (sec) , antiderivative size = 2568, normalized size of antiderivative = 9.14 \[ \int \frac {\sqrt {c+d x}}{\left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(1/2)/(-b*x^2+a)^3,x)
 

Output:

( - 10*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**4*d**4 + 38*sqrt(a)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c 
)))*a**3*b*c**2*d**2 + 20*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**3*b*d**4*x**2 - 2 
4*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sq 
rt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*c**4 - 76*sqrt(a)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c 
)))*a**2*b**2*c**2*d**2*x**2 - 10*sqrt(a)*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*b**2*d* 
*4*x**4 + 48*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**3*c**4*x**2 + 38*sqrt(a)*sqr 
t(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sq 
rt(a)*d - b*c)))*a*b**3*c**2*d**2*x**4 - 24*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* 
*4*c**4*x**4 + 16*sqrt(b)*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**4*c*d**3 - 12*sqrt(b)*sqr 
t(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sq 
rt(a)*d - b*c)))*a**3*b*c**3*d - 32*sqrt(b)*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**3*b*...