\(\int \frac {1}{x^2 (c+d x)^{5/2} (a-b x^2)} \, dx\) [603]

Optimal result

Integrand size = 23, antiderivative size = 258 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\frac {2 d^3}{3 c^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2}}+\frac {4 d^3 \left (2 b c^2-a d^2\right )}{c^3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}-\frac {\sqrt {c+d x}}{a c^3 x}+\frac {5 d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a c^{7/2}}-\frac {b^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a^{3/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {b^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a^{3/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\frac {-\frac {\sqrt {a} \left (3 b^2 c^4 (c+d x)^2-2 a b c^2 d^2 \left (3 c^2+19 c d x+15 d^2 x^2\right )+a^2 d^4 \left (3 c^2+20 c d x+15 d^2 x^2\right )\right )}{c^3 \left (b c^2-a d^2\right )^2 x (c+d x)^{3/2}}-\frac {3 b^{3/2} \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^3}+\frac {3 b^{3/2} \sqrt {-b c+\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^3}+\frac {15 \sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{7/2}}}{3 a^{3/2}} \] Input:

Integrate[1/(x^2*(c + d*x)^(5/2)*(a - b*x^2)),x]
 

Output:

(-((Sqrt[a]*(3*b^2*c^4*(c + d*x)^2 - 2*a*b*c^2*d^2*(3*c^2 + 19*c*d*x + 15* 
d^2*x^2) + a^2*d^4*(3*c^2 + 20*c*d*x + 15*d^2*x^2)))/(c^3*(b*c^2 - a*d^2)^ 
2*x*(c + d*x)^(3/2))) - (3*b^(3/2)*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*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)^3 + (3*b^(3/2)*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*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)^3 + (15*Sqrt[a]*d*ArcTanh[Sqrt[c + d*x]/Sq 
rt[c]])/c^(7/2))/(3*a^(3/2))
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {561, 27, 1610, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

\(\displaystyle 2 d \int \left (\frac {\left (3 b c^2-2 b (c+d x) c+a d^2\right ) b^2}{a \left (a d^2-b c^2\right )^2 \left (-b c^2+2 b (c+d x) c+a d^2-b (c+d x)^2\right )}-\frac {2}{a c^3 d x}-\frac {2 \left (2 b c^2 d^2-a d^4\right )}{c^3 \left (b c^2-a d^2\right )^2 (c+d x)}+\frac {1}{a c^2 d^2 x^2}-\frac {d^2}{c^2 \left (b c^2-a d^2\right ) (c+d x)^2}\right )d\sqrt {c+d x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 d \left (-\frac {b^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a^{3/2} d \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {b^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 a^{3/2} d \left (\sqrt {a} d+\sqrt {b} c\right )^{5/2}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 a c^{7/2}}+\frac {d^2}{3 c^2 (c+d x)^{3/2} \left (b c^2-a d^2\right )}+\frac {2 d^2 \left (2 b c^2-a d^2\right )}{c^3 \sqrt {c+d x} \left (b c^2-a d^2\right )^2}-\frac {\sqrt {c+d x}}{2 a c^3 d x}\right )\)

Input:

Int[1/(x^2*(c + d*x)^(5/2)*(a - b*x^2)),x]
 

Output:

2*d*(d^2/(3*c^2*(b*c^2 - a*d^2)*(c + d*x)^(3/2)) + (2*d^2*(2*b*c^2 - a*d^2 
))/(c^3*(b*c^2 - a*d^2)^2*Sqrt[c + d*x]) - Sqrt[c + d*x]/(2*a*c^3*d*x) + ( 
5*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*a*c^(7/2)) - (b^(7/4)*ArcTanh[(b^(1/4 
)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^(3/2)*d*(Sqrt[b]*c - S 
qrt[a]*d)^(5/2)) + (b^(7/4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c 
 + Sqrt[a]*d]])/(2*a^(3/2)*d*(Sqrt[b]*c + Sqrt[a]*d)^(5/2)))
 

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 1610

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a 
+ b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 
*a*c, 0] && IntegerQ[q] && IntegerQ[m]
 

rule 2009

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

Maple [A] (verified)

Time = 0.93 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.12

method	result	size
risch	\(-\frac {\sqrt {d x +c}}{a \,c^{3} x}-\frac {d \left (-\frac {2 c^{3} b^{3} \left (-\frac {\left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {2 a \,d^{2} c}{3 \left (a \,d^{2}-b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 a \,d^{2} \left (a \,d^{2}-2 b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {d x +c}}-\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{a \,c^{3}}\)	\(289\)
derivativedivides	\(-2 d^{3} \left (-\frac {-2 a \,d^{2}+4 b \,c^{2}}{c^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {d x +c}}+\frac {1}{3 c^{2} \left (a \,d^{2}-b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}+\frac {b^{3} \left (-\frac {\left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-a \,d^{2}-b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {-\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a \,d^{2} c^{3}}\right )\)	\(293\)
default	\(2 d^{3} \left (-\frac {1}{3 c^{2} \left (a \,d^{2}-b \,c^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 a \,d^{2}-4 b \,c^{2}}{c^{3} \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {d x +c}}-\frac {b^{3} \left (-\frac {\left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-a \,d^{2}-b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {-\frac {\sqrt {d x +c}}{2 d x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a \,d^{2} c^{3}}\right )\)	\(293\)
pseudoelliptic	\(-\frac {-d x \,b^{3} \left (2 c^{\frac {9}{2}} \left (d x +c \right ) \sqrt {a b \,d^{2}}+a \,d^{3} x \,c^{\frac {7}{2}}+\left (b c d x +a \,d^{2}+b \,c^{2}\right ) c^{\frac {9}{2}}\right ) \sqrt {d x +c}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (-d x \,b^{3} \sqrt {d x +c}\, \left (-2 c^{\frac {9}{2}} \left (d x +c \right ) \sqrt {a b \,d^{2}}+a \,d^{3} x \,c^{\frac {7}{2}}+\left (b c d x +a \,d^{2}+b \,c^{2}\right ) c^{\frac {9}{2}}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {a b \,d^{2}}\, \left (-5 d \left (d x +c \right )^{\frac {3}{2}} x \left (a \,d^{2}-b \,c^{2}\right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )-\frac {38 a b \,d^{3} x \,c^{\frac {7}{2}}}{3}+\frac {20 c^{\frac {3}{2}} a^{2} d^{5} x}{3}+5 a^{2} d^{6} x^{2} \sqrt {c}+\left (a \left (-10 b \,x^{2}+a \right ) d^{4}-2 \left (-\frac {b \,x^{2}}{2}+a \right ) b \,c^{2} d^{2}+2 b^{2} c^{3} d x +b^{2} c^{4}\right ) c^{\frac {5}{2}}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, c^{\frac {7}{2}} \sqrt {a b \,d^{2}}\, \left (d x +c \right )^{\frac {3}{2}} \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (a \,d^{2}-b \,c^{2}\right )^{2} a x}\)	\(439\)

Input:

int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-(d*x+c)^(1/2)/a/c^3/x-1/a/c^3*d*(-2*c^3*b^3/(a*d^2-b*c^2)^2*(-1/2*(-a*d^2 
-b*c^2+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*(a*d^2+b*c^ 
2+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)*ar 
ctan(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)))+2/3*a*d^2*c/(a*d^2 
-b*c^2)/(d*x+c)^(3/2)+4*a*d^2*(a*d^2-2*b*c^2)/(a*d^2-b*c^2)^2/(d*x+c)^(1/2 
)-5/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. 5513 vs. \(2 (204) = 408\).

Time = 16.30 (sec) , antiderivative size = 11035, normalized size of antiderivative = 42.77 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )} \, dx=- \int \frac {1}{- a c^{2} x^{2} \sqrt {c + d x} - 2 a c d x^{3} \sqrt {c + d x} - a d^{2} x^{4} \sqrt {c + d x} + b c^{2} x^{4} \sqrt {c + d x} + 2 b c d x^{5} \sqrt {c + d x} + b d^{2} x^{6} \sqrt {c + d x}}\, dx \] Input:

integrate(1/x**2/(d*x+c)**(5/2)/(-b*x**2+a),x)
 

Output:

-Integral(1/(-a*c**2*x**2*sqrt(c + d*x) - 2*a*c*d*x**3*sqrt(c + d*x) - a*d 
**2*x**4*sqrt(c + d*x) + b*c**2*x**4*sqrt(c + d*x) + 2*b*c*d*x**5*sqrt(c + 
 d*x) + b*d**2*x**6*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )} {\left (d x + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:

integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a),x, algorithm="maxima")
 

Output:

-integrate(1/((b*x^2 - a)*(d*x + c)^(5/2)*x^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1296 vs. \(2 (204) = 408\).

Time = 0.25 (sec) , antiderivative size = 1296, normalized size of antiderivative = 5.02 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a),x, algorithm="giac")
 

Output:

(2*(a*b^2*c^4*d - 2*a^2*b*c^2*d^3 + a^3*d^5)^2*sqrt(-b^2*c - sqrt(a*b)*b*d 
)*b*c*d*abs(b) - (3*sqrt(a*b)*b^3*c^6*d - 5*sqrt(a*b)*a*b^2*c^4*d^3 + sqrt 
(a*b)*a^2*b*c^2*d^5 + sqrt(a*b)*a^3*d^7)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs( 
a*b^2*c^4*d - 2*a^2*b*c^2*d^3 + a^3*d^5)*abs(b) + (a*b^6*c^11*d - 3*a^2*b^ 
5*c^9*d^3 + 2*a^3*b^4*c^7*d^5 + 2*a^4*b^3*c^5*d^7 - 3*a^5*b^2*c^3*d^9 + a^ 
6*b*c*d^11)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b))*arctan(sqrt(d*x + c)/sqrt 
(-(a*b^3*c^5 - 2*a^2*b^2*c^3*d^2 + a^3*b*c*d^4 + sqrt((a*b^3*c^5 - 2*a^2*b 
^2*c^3*d^2 + a^3*b*c*d^4)^2 - (a*b^3*c^6 - 3*a^2*b^2*c^4*d^2 + 3*a^3*b*c^2 
*d^4 - a^4*d^6)*(a*b^3*c^4 - 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)))/(a*b^3*c^4 - 
 2*a^2*b^2*c^2*d^2 + a^3*b*d^4)))/((sqrt(a*b)*a*b^5*c^10 - 5*sqrt(a*b)*a^2 
*b^4*c^8*d^2 + 10*sqrt(a*b)*a^3*b^3*c^6*d^4 - 10*sqrt(a*b)*a^4*b^2*c^4*d^6 
 + 5*sqrt(a*b)*a^5*b*c^2*d^8 - sqrt(a*b)*a^6*d^10)*abs(a*b^2*c^4*d - 2*a^2 
*b*c^2*d^3 + a^3*d^5)) - (2*(a*b^2*c^4*d - 2*a^2*b*c^2*d^3 + a^3*d^5)^2*sq 
rt(-b^2*c + sqrt(a*b)*b*d)*sqrt(a*b)*c*d*abs(b) + (3*a*b^3*c^6*d - 5*a^2*b 
^2*c^4*d^3 + a^3*b*c^2*d^5 + a^4*d^7)*sqrt(-b^2*c + sqrt(a*b)*b*d)*abs(a*b 
^2*c^4*d - 2*a^2*b*c^2*d^3 + a^3*d^5)*abs(b) + (sqrt(a*b)*a*b^5*c^11*d - 3 
*sqrt(a*b)*a^2*b^4*c^9*d^3 + 2*sqrt(a*b)*a^3*b^3*c^7*d^5 + 2*sqrt(a*b)*a^4 
*b^2*c^5*d^7 - 3*sqrt(a*b)*a^5*b*c^3*d^9 + sqrt(a*b)*a^6*c*d^11)*sqrt(-b^2 
*c + sqrt(a*b)*b*d)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b^3*c^5 - 2*a^2* 
b^2*c^3*d^2 + a^3*b*c*d^4 - sqrt((a*b^3*c^5 - 2*a^2*b^2*c^3*d^2 + a^3*b...
 

Mupad [B] (verification not implemented)

Time = 13.48 (sec) , antiderivative size = 32968, normalized size of antiderivative = 127.78 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

int(1/(x^2*(a - b*x^2)*(c + d*x)^(5/2)),x)
 

Output:

- atan(((5*a^3*b^15*c^19*d - a^7*b^7*(845*b^4*c^11*d^9 - 400*a^4*c^3*d^17 
+ 1712*a*b^3*c^9*d^11 + 2500*a^3*b*c^5*d^15 - 4001*a^2*b^2*c^7*d^13) + 75* 
a^12*b^6*c*d^19 + 25*a^4*b^14*c^17*d^3 + 631*a^5*b^13*c^15*d^5 + 1968*a^6* 
b^12*c^13*d^7)*(a^4*b^15*c^22*((a^2*d^5*(a^7*b^7)^(1/2) - a^3*b^6*c^5 - 5* 
a^5*b^4*c*d^4 - 10*a^4*b^5*c^3*d^2 + 5*b^2*c^4*d*(a^7*b^7)^(1/2) + 10*a*b* 
c^2*d^3*(a^7*b^7)^(1/2))/(a^11*d^10 - a^6*b^5*c^10 - 5*a^10*b*c^2*d^8 + 5* 
a^7*b^4*c^8*d^2 - 10*a^8*b^3*c^6*d^4 + 10*a^9*b^2*c^4*d^6))^(1/2)*(c + d*x 
)^(1/2)*2i - a^7*b^14*c^27*((a^2*d^5*(a^7*b^7)^(1/2) - a^3*b^6*c^5 - 5*a^5 
*b^4*c*d^4 - 10*a^4*b^5*c^3*d^2 + 5*b^2*c^4*d*(a^7*b^7)^(1/2) + 10*a*b*c^2 
*d^3*(a^7*b^7)^(1/2))/(a^11*d^10 - a^6*b^5*c^10 - 5*a^10*b*c^2*d^8 + 5*a^7 
*b^4*c^8*d^2 - 10*a^8*b^3*c^6*d^4 + 10*a^9*b^2*c^4*d^6))^(3/2)*(c + d*x)^( 
1/2)*5i + a^10*b^13*c^32*((a^2*d^5*(a^7*b^7)^(1/2) - a^3*b^6*c^5 - 5*a^5*b 
^4*c*d^4 - 10*a^4*b^5*c^3*d^2 + 5*b^2*c^4*d*(a^7*b^7)^(1/2) + 10*a*b*c^2*d 
^3*(a^7*b^7)^(1/2))/(a^11*d^10 - a^6*b^5*c^10 - 5*a^10*b*c^2*d^8 + 5*a^7*b 
^4*c^8*d^2 - 10*a^8*b^3*c^6*d^4 + 10*a^9*b^2*c^4*d^6))^(5/2)*(c + d*x)^(1/ 
2)*3i + a^15*b^4*d^22*((a^2*d^5*(a^7*b^7)^(1/2) - a^3*b^6*c^5 - 5*a^5*b^4* 
c*d^4 - 10*a^4*b^5*c^3*d^2 + 5*b^2*c^4*d*(a^7*b^7)^(1/2) + 10*a*b*c^2*d^3* 
(a^7*b^7)^(1/2))/(a^11*d^10 - a^6*b^5*c^10 - 5*a^10*b*c^2*d^8 + 5*a^7*b^4* 
c^8*d^2 - 10*a^8*b^3*c^6*d^4 + 10*a^9*b^2*c^4*d^6))^(1/2)*(c + d*x)^(1/2)* 
25i - a^23*c^6*d^26*((a^2*d^5*(a^7*b^7)^(1/2) - a^3*b^6*c^5 - 5*a^5*b^4...
 

Reduce [F]

\[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )} \, dx=\int \frac {1}{x^{2} \left (d x +c \right )^{\frac {5}{2}} \left (-b \,x^{2}+a \right )}d x \] Input:

int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a),x)
 

Output:

int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a),x)