\(\int \frac {\sqrt {c+d x}}{x^2 (a-b x^2)^3} \, dx\) [695]

Optimal result

Integrand size = 23, antiderivative size = 327 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^3} \, dx=-\frac {5 \sqrt {c+d x}}{4 a^3 x}+\frac {\sqrt {c+d x}}{4 a x \left (a-b x^2\right )^2}+\frac {b \sqrt {c+d x} \left (10 b c^2 x-a d (c+9 d x)\right )}{16 a^3 \left (b c^2-a d^2\right ) \left (a-b x^2\right )}-\frac {d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {\sqrt [4]{b} \left (60 b c^2-106 \sqrt {a} \sqrt {b} c d+45 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{7/2} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\sqrt [4]{b} \left (60 b c^2+106 \sqrt {a} \sqrt {b} c d+45 a d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{7/2} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}} \] Output:

-5/4*(d*x+c)^(1/2)/a^3/x+1/4*(d*x+c)^(1/2)/a/x/(-b*x^2+a)^2+1/16*b*(d*x+c) 
^(1/2)*(10*b*c^2*x-a*d*(9*d*x+c))/a^3/(-a*d^2+b*c^2)/(-b*x^2+a)-d*arctanh( 
(d*x+c)^(1/2)/c^(1/2))/a^3/c^(1/2)-1/32*b^(1/4)*(60*b*c^2-106*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^(7/2)/(b^(1/2)*c-a^(1/2)*d)^(3/2)+1/32*b^(1/4)*(60*b*c^2+106*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^(7/2)/(b^(1/2)*c+a^(1/2)*d)^(3/2)
 

Mathematica [A] (verified)

Time = 2.75 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {c+d x} \left (-16 a^3 d^2+30 b^3 c^2 x^4-a b^2 x^2 \left (50 c^2+c d x+29 d^2 x^2\right )+a^2 b \left (16 c^2+c d x+49 d^2 x^2\right )\right )}{\left (-b c^2+a d^2\right ) x \left (a-b x^2\right )^2}-\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \left (60 b c^2+106 \sqrt {a} \sqrt {b} c d+45 a d^2\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\left (\sqrt {b} c+\sqrt {a} d\right )^2}-\frac {\sqrt {b} \left (60 b c^2-106 \sqrt {a} \sqrt {b} c d+45 a d^2\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {-b c+\sqrt {a} \sqrt {b} d}}-\frac {32 \sqrt {a} d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{32 a^{7/2}} \] Input:

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

Output:

((2*Sqrt[a]*Sqrt[c + d*x]*(-16*a^3*d^2 + 30*b^3*c^2*x^4 - a*b^2*x^2*(50*c^ 
2 + c*d*x + 29*d^2*x^2) + a^2*b*(16*c^2 + c*d*x + 49*d^2*x^2)))/((-(b*c^2) 
 + a*d^2)*x*(a - b*x^2)^2) - (Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*(60*b*c^2 + 
 106*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]*(60*b*c^2 - 106*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)*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]) - (32*Sqrt[a]*d*ArcTanh[S 
qrt[c + d*x]/Sqrt[c]])/Sqrt[c])/(32*a^(7/2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(777\) vs. \(2(327)=654\).

Time = 2.10 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.38, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {561, 27, 1674, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1674

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 d \left (-\frac {\sqrt [4]{b} \left (-18 \sqrt {a} \sqrt {b} c d+5 a d^2+12 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{64 a^{7/2} d \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {\sqrt [4]{b} \left (18 \sqrt {a} \sqrt {b} c d+5 a d^2+12 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{64 a^{7/2} d \left (\sqrt {a} d+\sqrt {b} c\right )^{3/2}}-\frac {\sqrt [4]{b} \left (2 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{8 a^{7/2} d \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\sqrt [4]{b} \sqrt {\sqrt {b} c-\sqrt {a} d} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a^{7/2} d}+\frac {\sqrt [4]{b} \left (\sqrt {a} d+2 \sqrt {b} c\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{8 a^{7/2} d \sqrt {\sqrt {a} d+\sqrt {b} c}}+\frac {\sqrt [4]{b} \sqrt {\sqrt {a} d+\sqrt {b} c} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 a^{7/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 a^3 \sqrt {c}}-\frac {b d x \sqrt {c+d x}}{4 a^3 \left (-a d^2+b c^2-2 b c (c+d x)+b (c+d x)^2\right )}+\frac {b \sqrt {c+d x} \left (2 c \left (3 b c^2-2 a d^2\right )-(c+d x) \left (6 b c^2-5 a d^2\right )\right )}{32 a^3 \left (b c^2-a d^2\right ) \left (-a d^2+b c^2-2 b c (c+d x)+b (c+d x)^2\right )}-\frac {\sqrt {c+d x}}{2 a^3 d x}+\frac {b d^3 x \sqrt {c+d x}}{8 a^2 \left (-a d^2+b c^2-2 b c (c+d x)+b (c+d x)^2\right )^2}\right )\)

Input:

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

Output:

2*d*(-1/2*Sqrt[c + d*x]/(a^3*d*x) + (b*d^3*x*Sqrt[c + d*x])/(8*a^2*(b*c^2 
- a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2)^2) - (b*d*x*Sqrt[c + d*x])/(4*a 
^3*(b*c^2 - a*d^2 - 2*b*c*(c + d*x) + b*(c + d*x)^2)) + (b*Sqrt[c + d*x]*( 
2*c*(3*b*c^2 - 2*a*d^2) - (6*b*c^2 - 5*a*d^2)*(c + d*x)))/(32*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)) - ArcTanh[Sqrt[ 
c + d*x]/Sqrt[c]]/(2*a^3*Sqrt[c]) - (b^(1/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]*A 
rcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a^(7/2)*d) 
 - (b^(1/4)*(2*Sqrt[b]*c - Sqrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt 
[Sqrt[b]*c - Sqrt[a]*d]])/(8*a^(7/2)*d*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - (b^( 
1/4)*(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]])/(64*a^(7/2)*d*(Sqrt[b]*c - Sqrt[a]* 
d)^(3/2)) + (b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d]*ArcTanh[(b^(1/4)*Sqrt[c + 
 d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*a^(7/2)*d) + (b^(1/4)*(2*Sqrt[b]*c 
 + Sqrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]] 
)/(8*a^(7/2)*d*Sqrt[Sqrt[b]*c + Sqrt[a]*d]) + (b^(1/4)*(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]])/(64*a^(7/2)*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 1674

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

rule 2009

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

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(-\frac {15 \left (d x \left (\left (-\frac {23 c^{\frac {5}{2}} b}{30}+\frac {3 a \,d^{2} \sqrt {c}}{4}\right ) \sqrt {a b \,d^{2}}+\left (b \,c^{2}-\frac {61 a \,d^{2}}{60}\right ) b \,c^{\frac {3}{2}}\right ) b \left (-b \,x^{2}+a \right )^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\left (d x \left (\left (\frac {23 c^{\frac {5}{2}} b}{30}-\frac {3 a \,d^{2} \sqrt {c}}{4}\right ) \sqrt {a b \,d^{2}}+\left (b \,c^{2}-\frac {61 a \,d^{2}}{60}\right ) b \,c^{\frac {3}{2}}\right ) b \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )-\frac {8 \left (-d x \left (-b \,x^{2}+a \right )^{2} \left (a \,d^{2}-b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\left (-d^{2} \left (\frac {29}{16} b^{2} x^{4}-\frac {49}{16} a b \,x^{2}+a^{2}\right ) a \sqrt {c}+\left (\frac {15 b^{2} c \,x^{4}}{8}-\frac {25 x^{2} a \left (\frac {d x}{50}+c \right ) b}{8}+a^{2} \left (c +\frac {d x}{16}\right )\right ) b \,c^{\frac {3}{2}}\right ) \sqrt {d x +c}\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{15}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\right )}{8 \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {c}\, \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right )^{2} \left (a \,d^{2}-b \,c^{2}\right ) a^{3} x}\)	\(420\)
risch	\(-\frac {\sqrt {d x +c}}{a^{3} x}-\frac {d \left (2 b \left (\frac {\frac {b \left (13 a \,d^{2}-14 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 a \,d^{2}-32 b \,c^{2}}-\frac {\left (19 a \,d^{2}-21 b \,c^{2}\right ) b c \left (d x +c \right )^{\frac {5}{2}}}{16 \left (a \,d^{2}-b \,c^{2}\right )}-\frac {\left (17 a^{2} d^{4}-55 b \,c^{2} d^{2} a +42 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {\left (8 a \,d^{2}-7 b \,c^{2}\right ) c \sqrt {d x +c}}{16}}{\left (b \left (d x +c \right )^{2}-2 b c \left (d x +c \right )-a \,d^{2}+b \,c^{2}\right )^{2}}+\frac {b \left (-\frac {\left (61 a b c \,d^{2}-60 c^{3} b^{2}+45 \sqrt {a b \,d^{2}}\, a \,d^{2}-46 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (-61 a b c \,d^{2}+60 c^{3} b^{2}+45 \sqrt {a b \,d^{2}}\, a \,d^{2}-46 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a \,d^{2}-32 b \,c^{2}}\right )+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{a^{3}}\)	\(446\)
default	\(2 d^{7} \left (\frac {b \left (\frac {-\frac {b \left (13 a \,d^{2}-14 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {\left (19 a \,d^{2}-21 b \,c^{2}\right ) b c \left (d x +c \right )^{\frac {5}{2}}}{16 a \,d^{2}-16 b \,c^{2}}+\frac {\left (17 a^{2} d^{4}-55 b \,c^{2} d^{2} a +42 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 a \,d^{2}-32 b \,c^{2}}-\frac {\left (8 a \,d^{2}-7 b \,c^{2}\right ) c \sqrt {d x +c}}{16}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (-\frac {\left (-61 a b c \,d^{2}+60 c^{3} b^{2}-45 \sqrt {a b \,d^{2}}\, a \,d^{2}+46 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (61 a b c \,d^{2}-60 c^{3} b^{2}-45 \sqrt {a b \,d^{2}}\, a \,d^{2}+46 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a \,d^{2}-32 b \,c^{2}}\right )}{a^{3} d^{6}}+\frac {-\frac {\sqrt {d x +c}}{2 d x}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{6}}\right )\)	\(459\)
derivativedivides	\(-2 d^{7} \left (-\frac {b \left (\frac {-\frac {b \left (13 a \,d^{2}-14 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 \left (a \,d^{2}-b \,c^{2}\right )}+\frac {\left (19 a \,d^{2}-21 b \,c^{2}\right ) b c \left (d x +c \right )^{\frac {5}{2}}}{16 a \,d^{2}-16 b \,c^{2}}+\frac {\left (17 a^{2} d^{4}-55 b \,c^{2} d^{2} a +42 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 a \,d^{2}-32 b \,c^{2}}-\frac {\left (8 a \,d^{2}-7 b \,c^{2}\right ) c \sqrt {d x +c}}{16}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {b \left (-\frac {\left (-61 a b c \,d^{2}+60 c^{3} b^{2}-45 \sqrt {a b \,d^{2}}\, a \,d^{2}+46 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (61 a b c \,d^{2}-60 c^{3} b^{2}-45 \sqrt {a b \,d^{2}}\, a \,d^{2}+46 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 a \,d^{2}-32 b \,c^{2}}\right )}{a^{3} d^{6}}-\frac {-\frac {\sqrt {d x +c}}{2 d x}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{6}}\right )\)	\(461\)

Input:

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

Output:

-15/8/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/c^(1/2)/(a*b*d^2)^(1/2)*(d*x*((-23/ 
30*c^(5/2)*b+3/4*a*d^2*c^(1/2))*(a*b*d^2)^(1/2)+(b*c^2-61/60*a*d^2)*b*c^(3 
/2))*b*(-b*x^2+a)^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))+(d*x*((23/30*c^(5/2)*b-3/4*a*d^2*c^(1/2 
))*(a*b*d^2)^(1/2)+(b*c^2-61/60*a*d^2)*b*c^(3/2))*b*(-b*x^2+a)^2*arctanh(b 
*(d*x+c)^(1/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))-8/15*(-d*x*(-b*x^2+a)^2*(a 
*d^2-b*c^2)*arctanh((d*x+c)^(1/2)/c^(1/2))+(-d^2*(29/16*b^2*x^4-49/16*a*b* 
x^2+a^2)*a*c^(1/2)+(15/8*b^2*c*x^4-25/8*x^2*a*(1/50*d*x+c)*b+a^2*(c+1/16*d 
*x))*b*c^(3/2))*(d*x+c)^(1/2))*(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))/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(- 
b*x^2+a)^2/(a*d^2-b*c^2)/a^3/x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4772 vs. \(2 (259) = 518\).

Time = 12.11 (sec) , antiderivative size = 9553, normalized size of antiderivative = 29.21 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (259) = 518\).

Time = 0.35 (sec) , antiderivative size = 1105, normalized size of antiderivative = 3.38 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/32*((a^3*b*c^2*d - a^4*d^3)^2*(46*b*c^2*d - 45*a*d^3)*abs(b) + 2*(7*sqr 
t(a*b)*a^2*b^2*c^5*d - 15*sqrt(a*b)*a^3*b*c^3*d^3 + 8*sqrt(a*b)*a^4*c*d^5) 
*abs(a^3*b*c^2*d - a^4*d^3)*abs(b) - (60*a^5*b^4*c^8*d - 181*a^6*b^3*c^6*d 
^3 + 182*a^7*b^2*c^4*d^5 - 61*a^8*b*c^2*d^7)*abs(b))*arctan(sqrt(d*x + c)/ 
sqrt(-(a^3*b^2*c^3 - a^4*b*c*d^2 + sqrt((a^3*b^2*c^3 - a^4*b*c*d^2)^2 - (a 
^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)*(a^3*b^2*c^2 - a^4*b*d^2)))/(a^3*b 
^2*c^2 - a^4*b*d^2)))/((a^6*b^2*c^4*d - 2*a^7*b*c^2*d^3 + a^8*d^5 - sqrt(a 
*b)*a^5*b^2*c^5 + 2*sqrt(a*b)*a^6*b*c^3*d^2 - sqrt(a*b)*a^7*c*d^4)*sqrt(-b 
^2*c - sqrt(a*b)*b*d)*abs(a^3*b*c^2*d - a^4*d^3)) - 1/32*((a^3*b*c^2*d - a 
^4*d^3)^2*(46*b*c^2*d - 45*a*d^3)*abs(b) - 2*(7*sqrt(a*b)*a^2*b^2*c^5*d - 
15*sqrt(a*b)*a^3*b*c^3*d^3 + 8*sqrt(a*b)*a^4*c*d^5)*abs(a^3*b*c^2*d - a^4* 
d^3)*abs(b) - (60*a^5*b^4*c^8*d - 181*a^6*b^3*c^6*d^3 + 182*a^7*b^2*c^4*d^ 
5 - 61*a^8*b*c^2*d^7)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a^3*b^2*c^3 - a^ 
4*b*c*d^2 - sqrt((a^3*b^2*c^3 - a^4*b*c*d^2)^2 - (a^3*b^2*c^4 - 2*a^4*b*c^ 
2*d^2 + a^5*d^4)*(a^3*b^2*c^2 - a^4*b*d^2)))/(a^3*b^2*c^2 - a^4*b*d^2)))/( 
(a^6*b^2*c^4*d - 2*a^7*b*c^2*d^3 + a^8*d^5 + sqrt(a*b)*a^5*b^2*c^5 - 2*sqr 
t(a*b)*a^6*b*c^3*d^2 + sqrt(a*b)*a^7*c*d^4)*sqrt(-b^2*c + sqrt(a*b)*b*d)*a 
bs(a^3*b*c^2*d - a^4*d^3)) + d*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c 
)) - 1/16*(14*(d*x + c)^(7/2)*b^3*c^2*d - 42*(d*x + c)^(5/2)*b^3*c^3*d + 4 
2*(d*x + c)^(3/2)*b^3*c^4*d - 14*sqrt(d*x + c)*b^3*c^5*d - 13*(d*x + c)...
 

Mupad [B] (verification not implemented)

Time = 11.89 (sec) , antiderivative size = 13877, normalized size of antiderivative = 42.44 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(d*atan(((d*(((c + d*x)^(1/2)*(5137425*a^4*b^5*d^16 + 12960000*b^9*c^8*d^8 
 - 40176000*a*b^8*c^6*d^10 + 45974448*a^2*b^7*c^4*d^12 - 23850328*a^3*b^6* 
c^2*d^14))/(32768*(a^12*d^4 + a^10*b^2*c^4 - 2*a^11*b*c^2*d^2)) - (d*(((51 
8261*a^10*b^5*c*d^15)/1024 - (19575*a^7*b^8*c^7*d^9)/32 + (223599*a^8*b^7* 
c^5*d^11)/128 - (840477*a^9*b^6*c^3*d^13)/512)/(a^15*d^4 + a^13*b^2*c^4 - 
2*a^14*b*c^2*d^2) - (d*(((c + d*x)^(1/2)*(22140928*a^10*b^5*c*d^14 - 36864 
000*a^7*b^8*c^7*d^8 + 96411648*a^8*b^7*c^5*d^10 - 81725440*a^9*b^6*c^3*d^1 
2))/(32768*(a^12*d^4 + a^10*b^2*c^4 - 2*a^11*b*c^2*d^2)) + (d*((256*a^17*b 
^4*d^15 - 312*a^14*b^7*c^6*d^9 + 888*a^15*b^6*c^4*d^11 - 832*a^16*b^5*c^2* 
d^13)/(a^15*d^4 + a^13*b^2*c^4 - 2*a^14*b*c^2*d^2) - (d*(c + d*x)^(1/2)*(1 
6777216*a^17*b^4*d^14 - 25165824*a^14*b^7*c^6*d^8 + 67108864*a^15*b^6*c^4* 
d^10 - 58720256*a^16*b^5*c^2*d^12))/(65536*a^3*c^(1/2)*(a^12*d^4 + a^10*b^ 
2*c^4 - 2*a^11*b*c^2*d^2))))/(2*a^3*c^(1/2))))/(2*a^3*c^(1/2))))/(2*a^3*c^ 
(1/2)))*1i)/(2*a^3*c^(1/2)) + (d*(((c + d*x)^(1/2)*(5137425*a^4*b^5*d^16 + 
 12960000*b^9*c^8*d^8 - 40176000*a*b^8*c^6*d^10 + 45974448*a^2*b^7*c^4*d^1 
2 - 23850328*a^3*b^6*c^2*d^14))/(32768*(a^12*d^4 + a^10*b^2*c^4 - 2*a^11*b 
*c^2*d^2)) + (d*(((518261*a^10*b^5*c*d^15)/1024 - (19575*a^7*b^8*c^7*d^9)/ 
32 + (223599*a^8*b^7*c^5*d^11)/128 - (840477*a^9*b^6*c^3*d^13)/512)/(a^15* 
d^4 + a^13*b^2*c^4 - 2*a^14*b*c^2*d^2) + (d*(((c + d*x)^(1/2)*(22140928*a^ 
10*b^5*c*d^14 - 36864000*a^7*b^8*c^7*d^8 + 96411648*a^8*b^7*c^5*d^10 - ...
 

Reduce [B] (verification not implemented)

Time = 26.59 (sec) , antiderivative size = 3157, normalized size of antiderivative = 9.65 \[ \int \frac {\sqrt {c+d x}}{x^2 \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^(1/2)/x^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**4*x + 214*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**2*x + 180*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**3*b*c*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**2*c**5*x - 428*sqrt(a)*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**2*b**2*c**3*d**2*x**3 - 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**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**3*c**5*x**3 
 + 214*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**3*d**2*x**5 - 120*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)))*b**4*c**5*x**5 + 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**4*c**2 
*d**3*x - 28*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*c**4*d*x - 64*sqrt(b)*sqrt 
(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*...