\(\int \frac {1}{x^2 \sqrt {c+d x} (a-b x^2)^2} \, dx\) [669]

Optimal result

Integrand size = 23, antiderivative size = 298 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=-\frac {\left (3 b c^2-2 a d^2\right ) \sqrt {c+d x}}{2 a^2 c \left (b c^2-a d^2\right ) x}+\frac {b (c-d x) \sqrt {c+d x}}{2 a \left (b c^2-a d^2\right ) x \left (a-b x^2\right )}+\frac {d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2 c^{3/2}}-\frac {b^{3/4} \left (6 \sqrt {b} c-7 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 a^{5/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {b^{3/4} \left (6 \sqrt {b} c+7 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 a^{5/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}} \] Output:

-1/2*(-2*a*d^2+3*b*c^2)*(d*x+c)^(1/2)/a^2/c/(-a*d^2+b*c^2)/x+1/2*b*(-d*x+c 
)*(d*x+c)^(1/2)/a/(-a*d^2+b*c^2)/x/(-b*x^2+a)+d*arctanh((d*x+c)^(1/2)/c^(1 
/2))/a^2/c^(3/2)-1/4*b^(3/4)*(6*b^(1/2)*c-7*a^(1/2)*d)*arctanh(b^(1/4)*(d* 
x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a^(5/2)/(b^(1/2)*c-a^(1/2)*d)^(3/2 
)+1/4*b^(3/4)*(6*b^(1/2)*c+7*a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^( 
1/2)*c+a^(1/2)*d)^(1/2))/a^(5/2)/(b^(1/2)*c+a^(1/2)*d)^(3/2)
 

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c+d x} \left (2 a^2 d^2+3 b^2 c^2 x^2-a b \left (2 c^2+c d x+2 d^2 x^2\right )\right )}{c \left (b c^2-a d^2\right ) x \left (-a+b x^2\right )}+\frac {b \left (6 \sqrt {b} c+7 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {b \left (6 \sqrt {b} c-7 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d}}+\frac {4 \sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}}}{4 a^{5/2}} \] Input:

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

Output:

((-2*Sqrt[a]*Sqrt[c + d*x]*(2*a^2*d^2 + 3*b^2*c^2*x^2 - a*b*(2*c^2 + c*d*x 
 + 2*d^2*x^2)))/(c*(b*c^2 - a*d^2)*x*(-a + b*x^2)) + (b*(6*Sqrt[b]*c + 7*S 
qrt[a]*d)*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)*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d 
]) - (b*(6*Sqrt[b]*c - 7*Sqrt[a]*d)*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)*Sqrt[ 
-(b*c) + Sqrt[a]*Sqrt[b]*d]) + (4*Sqrt[a]*d*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] 
)/c^(3/2))/(4*a^(5/2))
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.55, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {561, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1567

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*d*(-1/2*Sqrt[c + d*x]/(a^2*c*d*x) + (b*Sqrt[c + d*x]*(b*c^2 + a*d^2 - b* 
c*(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)) + ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(2*a^2*c^(3/2)) - (b^(3/4)* 
(2*Sqrt[b]*c - 3*Sqrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c 
 - Sqrt[a]*d]])/(8*a^(5/2)*d*(Sqrt[b]*c - Sqrt[a]*d)^(3/2)) - (b^(3/4)*Arc 
Tanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^(5/2)*d*Sq 
rt[Sqrt[b]*c - Sqrt[a]*d]) + (b^(3/4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt 
[Sqrt[b]*c + Sqrt[a]*d]])/(2*a^(5/2)*d*Sqrt[Sqrt[b]*c + Sqrt[a]*d]) + (b^( 
3/4)*(2*Sqrt[b]*c + 3*Sqrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt 
[b]*c + Sqrt[a]*d]])/(8*a^(5/2)*d*(Sqrt[b]*c + Sqrt[a]*d)^(3/2)))
 

Defintions of rubi rules used

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 = 0.54 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {\sqrt {d x +c}}{c \,a^{2} x}-\frac {d \left (2 b c \left (\frac {-\frac {b c \left (d x +c \right )^{\frac {3}{2}}}{4 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d x +c}}{4 a \,d^{2}-4 b \,c^{2}}}{b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}}+\frac {b \left (-\frac {\left (7 a \,d^{2}-6 b \,c^{2}-\sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-7 a \,d^{2}+6 b \,c^{2}-\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,d^{2}-4 b \,c^{2}}\right )-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{c \,a^{2}}\)	\(317\)
derivativedivides	\(2 d^{5} \left (\frac {-\frac {\sqrt {d x +c}}{2 c d x}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a^{2} d^{4}}+\frac {b \left (\frac {-\frac {b c \left (d x +c \right )^{\frac {3}{2}}}{4 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d x +c}}{4 a \,d^{2}-4 b \,c^{2}}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (\frac {\left (7 a \,d^{2}-6 b \,c^{2}+\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-7 a \,d^{2}+6 b \,c^{2}+\sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,d^{2}-4 b \,c^{2}}\right )}{a^{2} d^{4}}\right )\)	\(323\)
default	\(2 d^{5} \left (\frac {-\frac {\sqrt {d x +c}}{2 c d x}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a^{2} d^{4}}+\frac {b \left (\frac {-\frac {b c \left (d x +c \right )^{\frac {3}{2}}}{4 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d x +c}}{4 a \,d^{2}-4 b \,c^{2}}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (\frac {\left (7 a \,d^{2}-6 b \,c^{2}+\sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-7 a \,d^{2}+6 b \,c^{2}+\sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,d^{2}-4 b \,c^{2}}\right )}{a^{2} d^{4}}\right )\)	\(323\)
pseudoelliptic	\(-\frac {3 \left (d x \left (b \,c^{\frac {9}{2}}-\frac {c^{\frac {7}{2}} \sqrt {a b \,d^{2}}}{6}-\frac {7 a \,d^{2} c^{\frac {5}{2}}}{6}\right ) b^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (d x \,b^{2} \left (b \,c^{\frac {9}{2}}+\frac {c^{\frac {7}{2}} \sqrt {a b \,d^{2}}}{6}-\frac {7 a \,d^{2} c^{\frac {5}{2}}}{6}\right ) \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {2 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (d x c \left (-b \,x^{2}+a \right ) \left (a \,d^{2}-b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\sqrt {d x +c}\, \left (\frac {a b d x \,c^{\frac {5}{2}}}{2}-a \,d^{2} \left (-b \,x^{2}+a \right ) c^{\frac {3}{2}}+\left (-\frac {3 b \,x^{2}}{2}+a \right ) b \,c^{\frac {7}{2}}\right )\right )}{3}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{2 c^{\frac {5}{2}} \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right ) \left (a \,d^{2}-b \,c^{2}\right ) a^{2} x}\)	\(361\)

Input:

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

Output:

-1/c/a^2*(d*x+c)^(1/2)/x-1/c/a^2*d*(2*b*c*((-1/4*b*c/(a*d^2-b*c^2)*(d*x+c) 
^(3/2)+1/4*(a*d^2+b*c^2)/(a*d^2-b*c^2)*(d*x+c)^(1/2))/(b*(d*x+c)^2-2*b*c*( 
d*x+c)-a*d^2+b*c^2)+1/4/(a*d^2-b*c^2)*b*(-1/2*(7*a*d^2-6*b*c^2-(a*b*d^2)^( 
1/2)*c)/(a*b*d^2)^(1/2)/((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))+1/2*(-7*a*d^2+6*b*c^2-(a*b*d^2)^(1/ 
2)*c)/(a*b*d^2)^(1/2)/((-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/c^(1/2)*arctanh((d*x+c)^(1/2)/c^ 
(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3965 vs. \(2 (234) = 468\).

Time = 6.84 (sec) , antiderivative size = 7939, normalized size of antiderivative = 26.64 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (234) = 468\).

Time = 0.24 (sec) , antiderivative size = 1042, normalized size of antiderivative = 3.50 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*((a^2*b*c^2*d - a^3*d^3)^2*sqrt(-b^2*c - sqrt(a*b)*b*d)*sqrt(a*b)*c*d* 
abs(b) - (5*a^2*b^2*c^4*d - 12*a^3*b*c^2*d^3 + 7*a^4*d^5)*sqrt(-b^2*c - sq 
rt(a*b)*b*d)*abs(-a^2*b*c^2*d + a^3*d^3)*abs(b) - (6*sqrt(a*b)*a^3*b^3*c^7 
*d - 19*sqrt(a*b)*a^4*b^2*c^5*d^3 + 20*sqrt(a*b)*a^5*b*c^3*d^5 - 7*sqrt(a* 
b)*a^6*c*d^7)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b))*arctan(sqrt(d*x + c)/sq 
rt(-(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^4*c^6 - 3*a^5*b^3*c^4*d^2 + 3*a^6*b^2*c^2*d^4 
- a^7*b*d^6)*abs(-a^2*b*c^2*d + a^3*d^3)) + 1/4*((a^2*b*c^2*d - a^3*d^3)^2 
*b*c*d*abs(b) + (5*sqrt(a*b)*a*b^2*c^4*d - 12*sqrt(a*b)*a^2*b*c^2*d^3 + 7* 
sqrt(a*b)*a^3*d^5)*abs(-a^2*b*c^2*d + a^3*d^3)*abs(b) - (6*a^3*b^4*c^7*d - 
 19*a^4*b^3*c^5*d^3 + 20*a^5*b^2*c^3*d^5 - 7*a^6*b*c*d^7)*abs(b))*arctan(s 
qrt(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^2*c^4*d - 2*a^5*b*c^2*d^3 + a^6 
*d^5 + sqrt(a*b)*a^3*b^2*c^5 - 2*sqrt(a*b)*a^4*b*c^3*d^2 + sqrt(a*b)*a^5*c 
*d^4)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(-a^2*b*c^2*d + a^3*d^3)) - 1/2*(3*( 
d*x + c)^(5/2)*b^2*c^2*d - 6*(d*x + c)^(3/2)*b^2*c^3*d + 3*sqrt(d*x + c)*b 
^2*c^4*d - 2*(d*x + c)^(5/2)*a*b*d^3 + 3*(d*x + c)^(3/2)*a*b*c*d^3 - 3*sqr 
t(d*x + c)*a*b*c^2*d^3 + 2*sqrt(d*x + c)*a^2*d^5)/((a^2*b*c^3 - a^3*c*d...
 

Mupad [B] (verification not implemented)

Time = 11.26 (sec) , antiderivative size = 12556, normalized size of antiderivative = 42.13 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((((((4096*a^13*b^4*c*d^15 - 1536*a^10*b^7*c^7*d^9 + 6144*a^11*b^6* 
c^5*d^11 - 8704*a^12*b^5*c^3*d^13)/(16*(a^8*b^2*c^6 + a^10*c^2*d^4 - 2*a^9 
*b*c^4*d^2)) - ((c + d*x)^(1/2)*((49*a^2*d^5*(a^11*b^3)^(1/2) - 36*a^5*b^4 
*c^5 - 63*a^7*b^2*c*d^4 + 95*a^6*b^3*c^3*d^2 + 24*b^2*c^4*d*(a^11*b^3)^(1/ 
2) - 69*a*b*c^2*d^3*(a^11*b^3)^(1/2))/(64*(a^13*d^6 - a^10*b^3*c^6 - 3*a^1 
2*b*c^2*d^4 + 3*a^11*b^2*c^4*d^2)))^(1/2)*(6144*a^10*b^7*c^8*d^8 - 16384*a 
^11*b^6*c^6*d^10 + 14336*a^12*b^5*c^4*d^12 - 4096*a^13*b^4*c^2*d^14))/(8*( 
a^6*b^2*c^6 + a^8*c^2*d^4 - 2*a^7*b*c^4*d^2)))*((49*a^2*d^5*(a^11*b^3)^(1/ 
2) - 36*a^5*b^4*c^5 - 63*a^7*b^2*c*d^4 + 95*a^6*b^3*c^3*d^2 + 24*b^2*c^4*d 
*(a^11*b^3)^(1/2) - 69*a*b*c^2*d^3*(a^11*b^3)^(1/2))/(64*(a^13*d^6 - a^10* 
b^3*c^6 - 3*a^12*b*c^2*d^4 + 3*a^11*b^2*c^4*d^2)))^(1/2) + ((c + d*x)^(1/2 
)*(512*a^8*b^5*c*d^14 + 5760*a^5*b^8*c^7*d^8 - 14304*a^6*b^7*c^5*d^10 + 86 
08*a^7*b^6*c^3*d^12))/(8*(a^6*b^2*c^6 + a^8*c^2*d^4 - 2*a^7*b*c^4*d^2)))*( 
(49*a^2*d^5*(a^11*b^3)^(1/2) - 36*a^5*b^4*c^5 - 63*a^7*b^2*c*d^4 + 95*a^6* 
b^3*c^3*d^2 + 24*b^2*c^4*d*(a^11*b^3)^(1/2) - 69*a*b*c^2*d^3*(a^11*b^3)^(1 
/2))/(64*(a^13*d^6 - a^10*b^3*c^6 - 3*a^12*b*c^2*d^4 + 3*a^11*b^2*c^4*d^2) 
))^(1/2) - (896*a^8*b^5*d^15 - 4320*a^5*b^8*c^6*d^9 + 12408*a^6*b^7*c^4*d^ 
11 - 9600*a^7*b^6*c^2*d^13)/(16*(a^8*b^2*c^6 + a^10*c^2*d^4 - 2*a^9*b*c^4* 
d^2)))*((49*a^2*d^5*(a^11*b^3)^(1/2) - 36*a^5*b^4*c^5 - 63*a^7*b^2*c*d^4 + 
 95*a^6*b^3*c^3*d^2 + 24*b^2*c^4*d*(a^11*b^3)^(1/2) - 69*a*b*c^2*d^3*(a...
 

Reduce [B] (verification not implemented)

Time = 168.94 (sec) , antiderivative size = 1742, normalized size of antiderivative = 5.85 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(16*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*c**3*d**2*x - 12*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**2*c**5*x - 16*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((s 
qrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**2*c**3*d**2* 
x**3 + 12*sqrt(a)*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)))*b**3*c**5*x**3 + 14*sqrt(b)*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)))*a**3*c**2*d**3*x - 10*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*b*c**4* 
d*x - 14*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqr 
t(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b*c**2*d**3*x**3 + 10*sqrt(b)*sq 
rt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*s 
qrt(a)*d - b*c)))*a*b**2*c**4*d*x**3 + 8*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*b* 
c**3*d**2*x - 6*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**2*c**5*x - 8*sqrt(a)*sqrt(s 
qrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqr 
t(c + d*x))*a*b**2*c**3*d**2*x**3 + 6*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**3*c*...