\(\int \frac {1}{x \sqrt {c+d x} (a-b x^2)^3} \, dx\) [723]

Optimal result

Integrand size = 23, antiderivative size = 331 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {b (c-d x) \sqrt {c+d x}}{4 a \left (b c^2-a d^2\right ) \left (a-b x^2\right )^2}-\frac {b \sqrt {c+d x} \left (a d^2 (14 c-13 d x)-b c^2 (8 c-7 d x)\right )}{16 a^2 \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}+\frac {\sqrt [4]{b} \left (32 b c^2-74 \sqrt {a} \sqrt {b} c d+45 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^3 \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {\sqrt [4]{b} \left (32 b c^2+74 \sqrt {a} \sqrt {b} c d+45 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^3 \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:

1/4*b*(-d*x+c)*(d*x+c)^(1/2)/a/(-a*d^2+b*c^2)/(-b*x^2+a)^2-1/16*b*(d*x+c)^ 
(1/2)*(a*d^2*(-13*d*x+14*c)-b*c^2*(-7*d*x+8*c))/a^2/(-a*d^2+b*c^2)^2/(-b*x 
^2+a)-2*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(1/2)+1/32*b^(1/4)*(32*b*c^2- 
74*a^(1/2)*b^(1/2)*c*d+45*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c- 
a^(1/2)*d)^(1/2))/a^3/(b^(1/2)*c-a^(1/2)*d)^(5/2)+1/32*b^(1/4)*(32*b*c^2+7 
4*a^(1/2)*b^(1/2)*c*d+45*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a 
^(1/2)*d)^(1/2))/a^3/(b^(1/2)*c+a^(1/2)*d)^(5/2)
 

Mathematica [A] (verified)

Time = 2.42 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=-\frac {\frac {2 a b \sqrt {c+d x} \left (a^2 d^2 (18 c-17 d x)+b^2 c^2 x^2 (8 c-7 d x)+a b \left (-12 c^3+11 c^2 d x-14 c d^2 x^2+13 d^3 x^3\right )\right )}{\left (b c^2-a d^2\right )^2 \left (a-b x^2\right )^2}+\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (32 b c^2+74 \sqrt {a} \sqrt {b} c d+45 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^3}-\frac {\sqrt {b} \left (32 b c^2-74 \sqrt {a} \sqrt {b} c d+45 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^2 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}+\frac {64 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{32 a^3} \] Input:

Integrate[1/(x*Sqrt[c + d*x]*(a - b*x^2)^3),x]
 

Output:

-1/32*((2*a*b*Sqrt[c + d*x]*(a^2*d^2*(18*c - 17*d*x) + b^2*c^2*x^2*(8*c - 
7*d*x) + a*b*(-12*c^3 + 11*c^2*d*x - 14*c*d^2*x^2 + 13*d^3*x^3)))/((b*c^2 
- a*d^2)^2*(a - b*x^2)^2) + (Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*(32*b*c^2 + 
74*Sqrt[a]*Sqrt[b]*c*d + 45*a*d^2)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d 
]*Sqrt[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[b]*c + Sqrt[a]*d)^3 - (Sq 
rt[b]*(32*b*c^2 - 74*Sqrt[a]*Sqrt[b]*c*d + 45*a*d^2)*ArcTan[(Sqrt[-(b*c) + 
 Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/((Sqrt[b]*c - 
 Sqrt[a]*d)^2*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]) + (64*ArcTanh[Sqrt[c + d*x 
]/Sqrt[c]])/Sqrt[c])/a^3
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(720\) vs. \(2(331)=662\).

Time = 2.17 (sec) , antiderivative size = 720, normalized size of antiderivative = 2.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {561, 25, 27, 1567, 2009}

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.

\(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^3 \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 561

\(\displaystyle \frac {2 \int \frac {1}{x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \int -\frac {1}{x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -2 \int -\frac {1}{d x \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^3}d\sqrt {c+d x}\)

\(\Big \downarrow \) 1567

\(\displaystyle -2 \int \left (-\frac {b x d^5}{a \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )^3}-\frac {b x d^3}{a^2 \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )^2}-\frac {b x d}{a^3 \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {1}{a^3 x d}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {\sqrt [4]{b} d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{8 a^{5/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\sqrt [4]{b} d \left (2 \sqrt {b} c-5 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{64 a^{5/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}-\frac {\sqrt [4]{b} d \left (5 \sqrt {a} d+2 \sqrt {b} c\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{64 a^{5/2} \left (\sqrt {a} d+\sqrt {b} c\right )^{5/2}}-\frac {\sqrt [4]{b} d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{8 a^{5/2} \left (\sqrt {a} d+\sqrt {b} c\right )^{3/2}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a^3 \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 a^3 \sqrt {\sqrt {a} d+\sqrt {b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {b d^2 \sqrt {c+d x} \left (c \left (11 a d^2+b c^2\right )-(c+d x) \left (5 a d^2+b c^2\right )\right )}{32 a^2 \left (b c^2-a d^2\right )^2 \left (-a d^2+b c^2-2 b c (c+d x)+b (c+d x)^2\right )}+\frac {b d^2 (c-d x) \sqrt {c+d x}}{4 a^2 \left (b c^2-a d^2\right ) \left (-a d^2+b c^2-2 b c (c+d x)+b (c+d x)^2\right )}-\frac {b d^4 (c-d x) \sqrt {c+d x}}{8 a \left (b c^2-a d^2\right ) \left (-a d^2+b c^2-2 b c (c+d x)+b (c+d x)^2\right )^2}\right )\)

Input:

Int[1/(x*Sqrt[c + d*x]*(a - b*x^2)^3),x]
 

Output:

-2*(-1/8*(b*d^4*(c - d*x)*Sqrt[c + d*x])/(a*(b*c^2 - a*d^2)*(b*c^2 - a*d^2 
 - 2*b*c*(c + d*x) + b*(c + d*x)^2)^2) + (b*d^2*(c - d*x)*Sqrt[c + d*x])/( 
4*a^2*(b*c^2 - a*d^2)*(b*c^2 - a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2)) - 
 (b*d^2*Sqrt[c + d*x]*(c*(b*c^2 + 11*a*d^2) - (b*c^2 + 5*a*d^2)*(c + d*x)) 
)/(32*a^2*(b*c^2 - a*d^2)^2*(b*c^2 - a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x) 
^2)) + ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(a^3*Sqrt[c]) + (b^(1/4)*d*(2*Sqrt[b 
]*c - 5*Sqrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a 
]*d]])/(64*a^(5/2)*(Sqrt[b]*c - Sqrt[a]*d)^(5/2)) + (b^(1/4)*d*ArcTanh[(b^ 
(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(8*a^(5/2)*(Sqrt[b]*c - 
 Sqrt[a]*d)^(3/2)) - (b^(1/4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b] 
*c - Sqrt[a]*d]])/(2*a^3*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - (b^(1/4)*d*ArcTanh 
[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(8*a^(5/2)*(Sqrt[b] 
*c + Sqrt[a]*d)^(3/2)) - (b^(1/4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqr 
t[b]*c + Sqrt[a]*d]])/(2*a^3*Sqrt[Sqrt[b]*c + Sqrt[a]*d]) - (b^(1/4)*d*(2* 
Sqrt[b]*c + 5*Sqrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + 
Sqrt[a]*d]])/(64*a^(5/2)*(Sqrt[b]*c + Sqrt[a]*d)^(5/2)))
 

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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1567

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((IntegerQ[p] 
 && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.53

method	result	size
pseudoelliptic	\(\frac {\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-a \,d^{2}+b \,c^{2}\right ) \sqrt {c}\, b \left (\frac {\left (45 a^{2} d^{4}-71 b \,c^{2} d^{2} a +32 b^{2} c^{4}\right ) \sqrt {a b \,d^{2}}}{16}-\frac {5 a \,b^{2} c^{3} d^{2}}{8}+a^{2} c \,d^{4} b \right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (-a \,d^{2}+b \,c^{2}\right ) \sqrt {c}\, \left (\frac {\left (45 a^{2} d^{4}-71 b \,c^{2} d^{2} a +32 b^{2} c^{4}\right ) \sqrt {a b \,d^{2}}}{16}+\frac {5 a \,b^{2} c^{3} d^{2}}{8}-a^{2} c \,d^{4} b \right ) b \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (8 \left (-b \,x^{2}+a \right )^{2} \left (a \,d^{2}-b \,c^{2}\right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\frac {\left (13 a b \,d^{3} x^{3}-7 b^{2} c^{2} d \,x^{3}-14 a b c \,d^{2} x^{2}+8 b^{2} c^{3} x^{2}-17 a^{2} d^{3} x +11 a b \,c^{2} d x +18 a^{2} c \,d^{2}-12 a b \,c^{3}\right ) \left (a \,d^{2}-b \,c^{2}\right ) \sqrt {c}\, b a \sqrt {d x +c}}{4}\right ) \sqrt {a b \,d^{2}}}{2}\right )}{2 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {c}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} a^{3} \left (a \,d^{2}-b \,c^{2}\right )^{3}}\)	\(507\)
derivativedivides	\(-2 d^{6} \left (\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3} d^{6} \sqrt {c}}+\frac {b \left (\frac {\frac {a b \,d^{2} \left (13 a \,d^{2}-7 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}-\frac {\left (53 a \,d^{2}-29 b \,c^{2}\right ) a b c \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {a \,d^{2} \left (17 a^{2} d^{4}-78 b \,c^{2} d^{2} a +37 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {5 \left (7 a \,d^{2}-3 b \,c^{2}\right ) a \,d^{2} c \sqrt {d x +c}}{32 \left (a \,d^{2}-b \,c^{2}\right )}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (\frac {\left (16 a^{2} c \,d^{4} b -10 a \,b^{2} c^{3} d^{2}+45 \sqrt {a b \,d^{2}}\, a^{2} d^{4}-71 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-16 a^{2} c \,d^{4} b +10 a \,b^{2} c^{3} d^{2}+45 \sqrt {a b \,d^{2}}\, a^{2} d^{4}-71 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}\right )}{a^{3} d^{6}}\right )\)	\(579\)
default	\(2 d^{6} \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3} d^{6} \sqrt {c}}-\frac {b \left (\frac {\frac {a b \,d^{2} \left (13 a \,d^{2}-7 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}-\frac {\left (53 a \,d^{2}-29 b \,c^{2}\right ) a b c \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {a \,d^{2} \left (17 a^{2} d^{4}-78 b \,c^{2} d^{2} a +37 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {5 \left (7 a \,d^{2}-3 b \,c^{2}\right ) a \,d^{2} c \sqrt {d x +c}}{32 \left (a \,d^{2}-b \,c^{2}\right )}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (\frac {\left (16 a^{2} c \,d^{4} b -10 a \,b^{2} c^{3} d^{2}+45 \sqrt {a b \,d^{2}}\, a^{2} d^{4}-71 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-16 a^{2} c \,d^{4} b +10 a \,b^{2} c^{3} d^{2}+45 \sqrt {a b \,d^{2}}\, a^{2} d^{4}-71 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+32 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}\right )}{a^{3} d^{6}}\right )\)	\(581\)

Input:

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

Output:

1/2/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/c^(1/2)*(((b*c+(a*b*d^2)^(1/2))*b)^(1/ 
2)*(-a*d^2+b*c^2)*c^(1/2)*b*(1/16*(45*a^2*d^4-71*a*b*c^2*d^2+32*b^2*c^4)*( 
a*b*d^2)^(1/2)-5/8*a*b^2*c^3*d^2+a^2*c*d^4*b)*(-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) 
*((-a*d^2+b*c^2)*c^(1/2)*(1/16*(45*a^2*d^4-71*a*b*c^2*d^2+32*b^2*c^4)*(a*b 
*d^2)^(1/2)+5/8*a*b^2*c^3*d^2-a^2*c*d^4*b)*b*(-b*x^2+a)^2*arctanh(b*(d*x+c 
)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+1/2*((b*c+(a*b*d^2)^(1/2))*b)^(1/ 
2)*(8*(-b*x^2+a)^2*(a*d^2-b*c^2)^3*arctanh((d*x+c)^(1/2)/c^(1/2))+1/4*(13* 
a*b*d^3*x^3-7*b^2*c^2*d*x^3-14*a*b*c*d^2*x^2+8*b^2*c^3*x^2-17*a^2*d^3*x+11 
*a*b*c^2*d*x+18*a^2*c*d^2-12*a*b*c^3)*(a*d^2-b*c^2)*c^(1/2)*b*a*(d*x+c)^(1 
/2))*(a*b*d^2)^(1/2)))/(a*b*d^2)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(- 
b*x^2+a)^2/a^3/(a*d^2-b*c^2)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6829 vs. \(2 (272) = 544\).

Time = 47.57 (sec) , antiderivative size = 13667, normalized size of antiderivative = 41.29 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )}^{3} \sqrt {d x + c} x} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1614 vs. \(2 (272) = 544\).

Time = 0.34 (sec) , antiderivative size = 1614, normalized size of antiderivative = 4.88 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/32*((a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5)^2*(32*b^2*c^4 - 71*a*b* 
c^2*d^2 + 45*a^2*d^4)*abs(b) - (32*sqrt(a*b)*a^2*b^4*c^9 - 145*sqrt(a*b)*a 
^3*b^3*c^7*d^2 + 255*sqrt(a*b)*a^4*b^2*c^5*d^4 - 203*sqrt(a*b)*a^5*b*c^3*d 
^6 + 61*sqrt(a*b)*a^6*c*d^8)*abs(a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5 
)*abs(b) - 2*(5*a^6*b^6*c^12*d^2 - 28*a^7*b^5*c^10*d^4 + 62*a^8*b^4*c^8*d^ 
6 - 68*a^9*b^3*c^6*d^8 + 37*a^10*b^2*c^4*d^10 - 8*a^11*b*c^2*d^12)*abs(b)) 
*arctan(sqrt(d*x + c)/sqrt(-(a^3*b^3*c^5 - 2*a^4*b^2*c^3*d^2 + a^5*b*c*d^4 
 + sqrt((a^3*b^3*c^5 - 2*a^4*b^2*c^3*d^2 + a^5*b*c*d^4)^2 - (a^3*b^3*c^6 - 
 3*a^4*b^2*c^4*d^2 + 3*a^5*b*c^2*d^4 - a^6*d^6)*(a^3*b^3*c^4 - 2*a^4*b^2*c 
^2*d^2 + a^5*b*d^4)))/(a^3*b^3*c^4 - 2*a^4*b^2*c^2*d^2 + a^5*b*d^4)))/((a^ 
6*b^4*c^8*d - 4*a^7*b^3*c^6*d^3 + 6*a^8*b^2*c^4*d^5 - 4*a^9*b*c^2*d^7 + a^ 
10*d^9 - sqrt(a*b)*a^5*b^4*c^9 + 4*sqrt(a*b)*a^6*b^3*c^7*d^2 - 6*sqrt(a*b) 
*a^7*b^2*c^5*d^4 + 4*sqrt(a*b)*a^8*b*c^3*d^6 - sqrt(a*b)*a^9*c*d^8)*sqrt(- 
b^2*c - sqrt(a*b)*b*d)*abs(a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5)) - 1 
/32*((a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5)^2*(32*b^2*c^4 - 71*a*b*c^ 
2*d^2 + 45*a^2*d^4)*abs(b) + (32*sqrt(a*b)*a^2*b^4*c^9 - 145*sqrt(a*b)*a^3 
*b^3*c^7*d^2 + 255*sqrt(a*b)*a^4*b^2*c^5*d^4 - 203*sqrt(a*b)*a^5*b*c^3*d^6 
 + 61*sqrt(a*b)*a^6*c*d^8)*abs(a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5)* 
abs(b) - 2*(5*a^6*b^6*c^12*d^2 - 28*a^7*b^5*c^10*d^4 + 62*a^8*b^4*c^8*d^6 
- 68*a^9*b^3*c^6*d^8 + 37*a^10*b^2*c^4*d^10 - 8*a^11*b*c^2*d^12)*abs(b)...
 

Mupad [B] (verification not implemented)

Time = 13.47 (sec) , antiderivative size = 19134, normalized size of antiderivative = 57.81 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((5*(3*b^2*c^3*d^2 - 7*a*b*c*d^4)*(c + d*x)^(1/2))/(16*a^2*(a*d^2 - b*c^2) 
) + ((c + d*x)^(3/2)*(17*a^2*b*d^6 + 37*b^3*c^4*d^2 - 78*a*b^2*c^2*d^4))/( 
16*a^2*(a*d^2 - b*c^2)^2) + (b*(7*b^2*c^2*d^2 - 13*a*b*d^4)*(c + d*x)^(7/2 
))/(16*a^2*(a*d^2 - b*c^2)^2) - (b*(29*b^2*c^3*d^2 - 53*a*b*c*d^4)*(c + d* 
x)^(5/2))/(16*a^2*(a*d^2 - b*c^2)^2))/(b^2*(c + d*x)^4 + a^2*d^4 + b^2*c^4 
 + (6*b^2*c^2 - 2*a*b*d^2)*(c + d*x)^2 - (4*b^2*c^3 - 4*a*b*c*d^2)*(c + d* 
x) - 4*b^2*c*(c + d*x)^3 - 2*a*b*c^2*d^2) - atan(((((17473888*a^9*b^5*c*d^ 
16 + 3145728*a^5*b^9*c^9*d^8 - 17657856*a^6*b^8*c^7*d^10 + 39419008*a^7*b^ 
7*c^5*d^12 - 41288096*a^8*b^6*c^3*d^14)/(32768*(a^14*d^8 + a^10*b^4*c^8 - 
4*a^13*b*c^2*d^6 - 4*a^11*b^3*c^6*d^2 + 6*a^12*b^2*c^4*d^4)) + (((16777216 
*a^16*b^4*d^18 - 12582912*a^11*b^9*c^10*d^8 + 64880640*a^12*b^8*c^8*d^10 - 
 134348800*a^13*b^7*c^6*d^12 + 141164544*a^14*b^6*c^4*d^14 - 75890688*a^15 
*b^5*c^2*d^16)/(32768*(a^14*d^8 + a^10*b^4*c^8 - 4*a^13*b*c^2*d^6 - 4*a^11 
*b^3*c^6*d^2 + 6*a^12*b^2*c^4*d^4)) - ((c + d*x)^(1/2)*(-(1024*a^6*b^5*c^9 
 + 2025*a^4*d^9*(a^13*b)^(1/2) + 384*b^4*c^8*d*(a^13*b)^(1/2) - 5084*a^7*b 
^4*c^7*d^2 + 10045*a^8*b^3*c^5*d^4 - 9306*a^9*b^2*c^3*d^6 + 3465*a^10*b*c* 
d^8 + 4429*a^2*b^2*c^4*d^5*(a^13*b)^(1/2) - 2000*a*b^3*c^6*d^3*(a^13*b)^(1 
/2) - 4694*a^3*b*c^2*d^7*(a^13*b)^(1/2))/(4096*(a^17*d^10 - a^12*b^5*c^10 
- 5*a^16*b*c^2*d^8 + 5*a^13*b^4*c^8*d^2 - 10*a^14*b^3*c^6*d^4 + 10*a^15*b^ 
2*c^4*d^6)))^(1/2)*(16777216*a^17*b^4*d^18 - 25165824*a^12*b^9*c^10*d^8...
 

Reduce [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 3792, normalized size of antiderivative = 11.46 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 90*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*c*d**5 + 110*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*c**3*d**3 + 180*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*c*d**5*x* 
*2 - 44*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*b**2*c**5*d - 220*sqrt(a)*sqrt(sq 
rt(b)*sqrt(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**3*d**3*x**2 - 90*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*b**2*c*d**5*x**4 + 88*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**5*d*x**2 + 1 
10*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s 
qrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**3*c**3*d**3*x**4 - 44*sqrt(a)*sqrt(sqr 
t(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**5*d*x**4 - 122*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*a 
tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**4*c**2*d 
**4 + 162*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sq 
rt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**3*b*c**4*d**2 + 244*sqrt(b)*sqrt( 
sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*s...