\(\int \frac {1}{x^2 \sqrt {c+d x} (a-b x^2)^3} \, dx\) [724]

Optimal result

Integrand size = 23, antiderivative size = 391 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=-\frac {\left (5 b c^2-4 a d^2\right ) \sqrt {c+d x}}{4 a^3 c \left (b c^2-a d^2\right ) x}+\frac {b (c-d x) \sqrt {c+d x}}{4 a \left (b c^2-a d^2\right ) x \left (a-b x^2\right )^2}+\frac {b \sqrt {c+d x} \left (15 a^2 d^3+10 b^2 c^3 x-a b c d (9 c+16 d x)\right )}{16 a^3 \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )}+\frac {d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{3/2}}-\frac {b^{3/4} \left (60 b c^2-134 \sqrt {a} \sqrt {b} c d+77 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {b^{3/4} \left (60 b c^2+134 \sqrt {a} \sqrt {b} c d+77 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{7/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:

-1/4*(-4*a*d^2+5*b*c^2)*(d*x+c)^(1/2)/a^3/c/(-a*d^2+b*c^2)/x+1/4*b*(-d*x+c 
)*(d*x+c)^(1/2)/a/(-a*d^2+b*c^2)/x/(-b*x^2+a)^2+1/16*b*(d*x+c)^(1/2)*(15*a 
^2*d^3+10*b^2*c^3*x-a*b*c*d*(16*d*x+9*c))/a^3/(-a*d^2+b*c^2)^2/(-b*x^2+a)+ 
d*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(3/2)-1/32*b^(3/4)*(60*b*c^2-134*a^ 
(1/2)*b^(1/2)*c*d+77*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/ 
2)*d)^(1/2))/a^(7/2)/(b^(1/2)*c-a^(1/2)*d)^(5/2)+1/32*b^(3/4)*(60*b*c^2+13 
4*a^(1/2)*b^(1/2)*c*d+77*a*d^2)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a 
^(1/2)*d)^(1/2))/a^(7/2)/(b^(1/2)*c+a^(1/2)*d)^(5/2)
 

Mathematica [A] (verified)

Time = 2.21 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {c+d x} \left (16 a^4 d^4+30 b^4 c^4 x^4-a^3 b d^2 \left (32 c^2+19 c d x+32 d^2 x^2\right )-a b^3 c^2 x^2 \left (50 c^2+9 c d x+52 d^2 x^2\right )+a^2 b^2 \left (16 c^4+13 c^3 d x+88 c^2 d^2 x^2+15 c d^3 x^3+16 d^4 x^4\right )\right )}{c \left (b c^2-a d^2\right )^2 x \left (a-b x^2\right )^2}+\frac {b \left (60 b c^2+134 \sqrt {a} \sqrt {b} c d+77 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^2 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {b \left (60 b c^2-134 \sqrt {a} \sqrt {b} c d+77 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right )^2 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}+\frac {32 \sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}}}{32 a^{7/2}} \] Input:

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

Output:

((-2*Sqrt[a]*Sqrt[c + d*x]*(16*a^4*d^4 + 30*b^4*c^4*x^4 - a^3*b*d^2*(32*c^ 
2 + 19*c*d*x + 32*d^2*x^2) - a*b^3*c^2*x^2*(50*c^2 + 9*c*d*x + 52*d^2*x^2) 
 + a^2*b^2*(16*c^4 + 13*c^3*d*x + 88*c^2*d^2*x^2 + 15*c*d^3*x^3 + 16*d^4*x 
^4)))/(c*(b*c^2 - a*d^2)^2*x*(a - b*x^2)^2) + (b*(60*b*c^2 + 134*Sqrt[a]*S 
qrt[b]*c*d + 77*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) - Sq 
rt[a]*Sqrt[b]*d]) - (b*(60*b*c^2 - 134*Sqrt[a]*Sqrt[b]*c*d + 77*a*d^2)*Arc 
Tan[(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]) + (32*Sq 
rt[a]*d*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(3/2))/(32*a^(7/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(857\) vs. \(2(391)=782\).

Time = 2.30 (sec) , antiderivative size = 857, normalized size of antiderivative = 2.19, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1567

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 d \left (-\frac {b \sqrt {c+d x} \left (b c^2-b (c+d x) c+a d^2\right ) d^2}{8 a^2 \left (b c^2-a d^2\right ) \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )^2}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 a^3 c^{3/2}}+\frac {b \sqrt {c+d x} \left (b c^2-b (c+d x) c+a d^2\right )}{4 a^3 \left (b c^2-a d^2\right ) \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}+\frac {b \sqrt {c+d x} \left (\left (3 b c^2-7 a d^2\right ) \left (2 b c^2+a d^2\right )-6 b c \left (b c^2-2 a d^2\right ) (c+d x)\right )}{32 a^3 \left (b c^2-a d^2\right )^2 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}-\frac {3 b^{3/4} \left (4 b c^2-10 \sqrt {a} \sqrt {b} d c+7 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{64 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2} d}-\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a^{7/2} \sqrt {\sqrt {b} c-\sqrt {a} d} d}-\frac {b^{3/4} \left (2 \sqrt {b} c-3 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{8 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2} d}+\frac {b^{3/4} \left (2 \sqrt {b} c+3 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{8 a^{7/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2} d}+\frac {3 b^{3/4} \left (4 b c^2+10 \sqrt {a} \sqrt {b} d c+7 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{64 a^{7/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2} d}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{2 a^{7/2} \sqrt {\sqrt {b} c+\sqrt {a} d} d}-\frac {\sqrt {c+d x}}{2 a^3 c x d}\right )\)

Input:

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

Output:

2*d*(-1/2*Sqrt[c + d*x]/(a^3*c*d*x) - (b*d^2*Sqrt[c + d*x]*(b*c^2 + a*d^2 
- b*c*(c + d*x)))/(8*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)^2) + (b*Sqrt[c + d*x]*(b*c^2 + a*d^2 - b*c*(c + d*x)))/(4 
*a^3*(b*c^2 - a*d^2)*(b*c^2 - a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2)) + 
(b*Sqrt[c + d*x]*((3*b*c^2 - 7*a*d^2)*(2*b*c^2 + a*d^2) - 6*b*c*(b*c^2 - 2 
*a*d^2)*(c + d*x)))/(32*a^3*(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]]/(2*a^3*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^(7/2)*d*(Sqrt[b]*c - Sqrt[a]*d)^(3/2)) - (b^ 
(3/4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^( 
7/2)*d*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - (3*b^(3/4)*(4*b*c^2 - 10*Sqrt[a]*Sqr 
t[b]*c*d + 7*a*d^2)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[ 
a]*d]])/(64*a^(7/2)*d*(Sqrt[b]*c - Sqrt[a]*d)^(5/2)) + (b^(3/4)*ArcTanh[(b 
^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*a^(7/2)*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^(7/2)*d*(Sqrt[b]*c + S 
qrt[a]*d)^(3/2)) + (3*b^(3/4)*(4*b*c^2 + 10*Sqrt[a]*Sqrt[b]*c*d + 7*a*d^2) 
*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(64*a^(7/2) 
*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 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 = 0.90 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.37

method	result	size
pseudoelliptic	\(-\frac {15 \left (d \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-a \,d^{2}+b \,c^{2}\right ) \left (\frac {c \left (10 a \,d^{2}-7 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{30}+b^{2} c^{4}-\frac {131 b \,c^{2} d^{2} a}{60}+\frac {77 a^{2} d^{4}}{60}\right ) x \,b^{2} \left (-b \,x^{2}+a \right )^{2} c^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (d \left (-a \,d^{2}+b \,c^{2}\right ) x \left (-\frac {c \left (10 a \,d^{2}-7 b \,c^{2}\right ) \sqrt {a b \,d^{2}}}{30}+b^{2} c^{4}-\frac {131 b \,c^{2} d^{2} a}{60}+\frac {77 a^{2} d^{4}}{60}\right ) b^{2} \left (-b \,x^{2}+a \right )^{2} c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {8 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d x c \left (-b \,x^{2}+a \right )^{2} \left (a \,d^{2}-b \,c^{2}\right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\left (-a \,d^{2}+b \,c^{2}\right ) \left (\frac {15 b^{4} c^{4} x^{4}}{8}-\frac {25 x^{2} \left (\frac {26}{25} d^{2} x^{2}+\frac {9}{50} c d x +c^{2}\right ) a \,c^{2} b^{3}}{8}+a^{2} \left (c^{4}+\frac {13}{16} c^{3} d x +\frac {11}{2} d^{2} c^{2} x^{2}+\frac {15}{16} c \,d^{3} x^{3}+d^{4} x^{4}\right ) b^{2}-2 d^{2} a^{3} \left (d^{2} x^{2}+\frac {19}{32} c d x +c^{2}\right ) b +a^{4} d^{4}\right ) \sqrt {d x +c}\, c^{\frac {3}{2}}\right ) \sqrt {a b \,d^{2}}}{15}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{8 \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, x \left (a \,d^{2}-b \,c^{2}\right )^{3} a^{3} \left (-b \,x^{2}+a \right )^{2} c^{\frac {5}{2}}}\)	\(536\)
risch	\(-\frac {\sqrt {d x +c}}{c \,a^{3} x}-\frac {d \left (2 b c \left (\frac {-\frac {b^{2} c \left (10 a \,d^{2}-7 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{16 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {3 \left (5 a^{2} d^{4}+17 b \,c^{2} d^{2} a -14 b^{2} c^{4}\right ) b \left (d x +c \right )^{\frac {5}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {3 b c \left (a^{2} d^{4}+10 b \,c^{2} d^{2} a -7 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}-\frac {\left (19 a^{2} d^{4}+15 b \,c^{2} d^{2} a -14 b^{2} c^{4}\right ) \sqrt {d x +c}}{32 \left (a \,d^{2}-b \,c^{2}\right )}}{\left (b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {b \left (-\frac {\left (77 a^{2} d^{4}-131 b \,c^{2} d^{2} a +60 b^{2} c^{4}-20 \sqrt {a b \,d^{2}}\, a c \,d^{2}+14 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-77 a^{2} d^{4}+131 b \,c^{2} d^{2} a -60 b^{2} c^{4}-20 \sqrt {a b \,d^{2}}\, a c \,d^{2}+14 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}\right )-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{a^{3} c}\)	\(566\)
default	\(2 d^{7} \left (\frac {-\frac {\sqrt {d x +c}}{2 c d x}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a^{3} d^{6}}+\frac {b \left (\frac {\frac {b^{2} c \left (10 a \,d^{2}-7 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{16 a^{2} d^{4}-32 b \,c^{2} d^{2} a +16 b^{2} c^{4}}-\frac {3 \left (5 a^{2} d^{4}+17 b \,c^{2} d^{2} a -14 b^{2} c^{4}\right ) b \left (d x +c \right )^{\frac {5}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {3 b c \left (a^{2} d^{4}+10 b \,c^{2} d^{2} a -7 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\left (19 a^{2} d^{4}+15 b \,c^{2} d^{2} a -14 b^{2} c^{4}\right ) \sqrt {d x +c}}{32 a \,d^{2}-32 b \,c^{2}}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (\frac {\left (77 a^{2} d^{4}-131 b \,c^{2} d^{2} a +60 b^{2} c^{4}+20 \sqrt {a b \,d^{2}}\, a c \,d^{2}-14 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-77 a^{2} d^{4}+131 b \,c^{2} d^{2} a -60 b^{2} c^{4}+20 \sqrt {a b \,d^{2}}\, a c \,d^{2}-14 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}\right )}{a^{3} d^{6}}\right )\)	\(574\)
derivativedivides	\(-2 d^{7} \left (-\frac {-\frac {\sqrt {d x +c}}{2 c d x}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a^{3} d^{6}}-\frac {b \left (\frac {\frac {b^{2} c \left (10 a \,d^{2}-7 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{16 a^{2} d^{4}-32 b \,c^{2} d^{2} a +16 b^{2} c^{4}}-\frac {3 \left (5 a^{2} d^{4}+17 b \,c^{2} d^{2} a -14 b^{2} c^{4}\right ) b \left (d x +c \right )^{\frac {5}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {3 b c \left (a^{2} d^{4}+10 b \,c^{2} d^{2} a -7 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{16 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\left (19 a^{2} d^{4}+15 b \,c^{2} d^{2} a -14 b^{2} c^{4}\right ) \sqrt {d x +c}}{32 a \,d^{2}-32 b \,c^{2}}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (\frac {\left (77 a^{2} d^{4}-131 b \,c^{2} d^{2} a +60 b^{2} c^{4}+20 \sqrt {a b \,d^{2}}\, a c \,d^{2}-14 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-77 a^{2} d^{4}+131 b \,c^{2} d^{2} a -60 b^{2} c^{4}+20 \sqrt {a b \,d^{2}}\, a c \,d^{2}-14 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}\right )}{a^{3} d^{6}}\right )\)	\(576\)

Input:

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

Output:

-15/8/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(a*b*d^2)^(1/2)*(d*((b*c+(a*b*d^2)^ 
(1/2))*b)^(1/2)*(-a*d^2+b*c^2)*(1/30*c*(10*a*d^2-7*b*c^2)*(a*b*d^2)^(1/2)+ 
b^2*c^4-131/60*b*c^2*d^2*a+77/60*a^2*d^4)*x*b^2*(-b*x^2+a)^2*c^(5/2)*arcta 
n(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))+(d*(-a*d^2+b*c^2)*x*(- 
1/30*c*(10*a*d^2-7*b*c^2)*(a*b*d^2)^(1/2)+b^2*c^4-131/60*b*c^2*d^2*a+77/60 
*a^2*d^4)*b^2*(-b*x^2+a)^2*c^(5/2)*arctanh(b*(d*x+c)^(1/2)/((b*c+(a*b*d^2) 
^(1/2))*b)^(1/2))-8/15*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(d*x*c*(-b*x^2+a)^2 
*(a*d^2-b*c^2)^3*arctanh((d*x+c)^(1/2)/c^(1/2))+(-a*d^2+b*c^2)*(15/8*b^4*c 
^4*x^4-25/8*x^2*(26/25*d^2*x^2+9/50*c*d*x+c^2)*a*c^2*b^3+a^2*(c^4+13/16*c^ 
3*d*x+11/2*d^2*c^2*x^2+15/16*c*d^3*x^3+d^4*x^4)*b^2-2*d^2*a^3*(d^2*x^2+19/ 
32*c*d*x+c^2)*b+a^4*d^4)*(d*x+c)^(1/2)*c^(3/2))*(a*b*d^2)^(1/2))*((-b*c+(a 
*b*d^2)^(1/2))*b)^(1/2))/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/x/(a*d^2-b*c^2)^3 
/a^3/(-b*x^2+a)^2/c^(5/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7005 vs. \(2 (324) = 648\).

Time = 80.32 (sec) , antiderivative size = 14019, normalized size of antiderivative = 35.85 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(1/x^2/(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^2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1676 vs. \(2 (324) = 648\).

Time = 0.36 (sec) , antiderivative size = 1676, normalized size of antiderivative = 4.29 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/32*(2*(a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5)^2*(7*b^2*c^3*d - 10*a 
*b*c*d^3)*abs(b) + (46*sqrt(a*b)*a^2*b^4*c^8*d - 203*sqrt(a*b)*a^3*b^3*c^6 
*d^3 + 345*sqrt(a*b)*a^4*b^2*c^4*d^5 - 265*sqrt(a*b)*a^5*b*c^2*d^7 + 77*sq 
rt(a*b)*a^6*d^9)*abs(a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5)*abs(b) - ( 
60*a^5*b^7*c^13*d - 371*a^6*b^6*c^11*d^3 + 961*a^7*b^5*c^9*d^5 - 1334*a^8* 
b^4*c^7*d^7 + 1046*a^9*b^3*c^5*d^9 - 439*a^10*b^2*c^3*d^11 + 77*a^11*b*c*d 
^13)*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*(2*(a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5)^2*(7*b^2*c^ 
3*d - 10*a*b*c*d^3)*abs(b) - (46*sqrt(a*b)*a^2*b^4*c^8*d - 203*sqrt(a*b)*a 
^3*b^3*c^6*d^3 + 345*sqrt(a*b)*a^4*b^2*c^4*d^5 - 265*sqrt(a*b)*a^5*b*c^2*d 
^7 + 77*sqrt(a*b)*a^6*d^9)*abs(a^3*b^2*c^4*d - 2*a^4*b*c^2*d^3 + a^5*d^5)* 
abs(b) - (60*a^5*b^7*c^13*d - 371*a^6*b^6*c^11*d^3 + 961*a^7*b^5*c^9*d^5 - 
 1334*a^8*b^4*c^7*d^7 + 1046*a^9*b^3*c^5*d^9 - 439*a^10*b^2*c^3*d^11 + ...
 

Mupad [B] (verification not implemented)

Time = 14.78 (sec) , antiderivative size = 20668, normalized size of antiderivative = 52.86 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((((((16777216*a^19*b^4*c*d^19 - 4718592*a^14*b^9*c^11*d^9 + 306708 
48*a^15*b^8*c^9*d^11 - 77332480*a^16*b^7*c^7*d^13 + 98304000*a^17*b^6*c^5* 
d^15 - 63700992*a^18*b^5*c^3*d^17)/(65536*(a^13*b^4*c^10 + a^17*c^2*d^8 - 
4*a^16*b*c^4*d^6 - 4*a^14*b^3*c^8*d^2 + 6*a^15*b^2*c^6*d^4)) - ((c + d*x)^ 
(1/2)*((5929*a^4*d^9*(a^15*b^3)^(1/2) - 3600*a^7*b^6*c^9 - 9009*a^11*b^2*c 
*d^8 + 17204*a^8*b^5*c^7*d^2 - 31909*a^9*b^4*c^5*d^4 + 27170*a^10*b^3*c^3* 
d^6 + 1920*b^4*c^8*d*(a^15*b^3)^(1/2) - 9456*a*b^3*c^6*d^3*(a^15*b^3)^(1/2 
) - 16694*a^3*b*c^2*d^7*(a^15*b^3)^(1/2) + 18445*a^2*b^2*c^4*d^5*(a^15*b^3 
)^(1/2))/(4096*(a^19*d^10 - a^14*b^5*c^10 - 5*a^18*b*c^2*d^8 + 5*a^15*b^4* 
c^8*d^2 - 10*a^16*b^3*c^6*d^4 + 10*a^17*b^2*c^4*d^6)))^(1/2)*(25165824*a^1 
4*b^9*c^12*d^8 - 117440512*a^15*b^8*c^10*d^10 + 218103808*a^16*b^7*c^8*d^1 
2 - 201326592*a^17*b^6*c^6*d^14 + 92274688*a^18*b^5*c^4*d^16 - 16777216*a^ 
19*b^4*c^2*d^18))/(32768*(a^10*b^4*c^10 + a^14*c^2*d^8 - 4*a^13*b*c^4*d^6 
- 4*a^11*b^3*c^8*d^2 + 6*a^12*b^2*c^6*d^4)))*((5929*a^4*d^9*(a^15*b^3)^(1/ 
2) - 3600*a^7*b^6*c^9 - 9009*a^11*b^2*c*d^8 + 17204*a^8*b^5*c^7*d^2 - 3190 
9*a^9*b^4*c^5*d^4 + 27170*a^10*b^3*c^3*d^6 + 1920*b^4*c^8*d*(a^15*b^3)^(1/ 
2) - 9456*a*b^3*c^6*d^3*(a^15*b^3)^(1/2) - 16694*a^3*b*c^2*d^7*(a^15*b^3)^ 
(1/2) + 18445*a^2*b^2*c^4*d^5*(a^15*b^3)^(1/2))/(4096*(a^19*d^10 - a^14*b^ 
5*c^10 - 5*a^18*b*c^2*d^8 + 5*a^15*b^4*c^8*d^2 - 10*a^16*b^3*c^6*d^4 + 10* 
a^17*b^2*c^4*d^6)))^(1/2) + ((c + d*x)^(1/2)*(2097152*a^12*b^5*c*d^18 +...
 

Reduce [B] (verification not implemented)

Time = 27.32 (sec) , antiderivative size = 4012, normalized size of antiderivative = 10.26 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(194*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*b*c**3*d**4*x - 290*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)))*a**3*b**2*c**5*d**2*x - 388*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 
**2*c**3*d**4*x**3 + 120*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**3*c**7*x + 58 
0*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**3*c**5*d**2*x**3 + 194*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**3*c**3*d**4*x**5 - 240*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**4*c**7*x**3 - 290*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**4*c**5*d**2*x**5 + 
 120*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**5*c**7*x**5 + 154*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**5*c**2*d**5*x - 222*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*b*c**4*d** 
3*x - 308*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/...