\(\int \frac {x}{\sqrt {c+d x} (a-b x^2)^2} \, dx\) [666]

Optimal result

Integrand size = 21, antiderivative size = 184 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=\frac {(c-d x) \sqrt {c+d x}}{2 \left (b c^2-a d^2\right ) \left (a-b x^2\right )}-\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 \sqrt {a} b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{3/2}}+\frac {d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 \sqrt {a} b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.35 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=\frac {1}{4} d \left (\frac {2 (c-d x) \sqrt {c+d x}}{d \left (-b c^2+a d^2\right ) \left (-a+b x^2\right )}-\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} b \left (\sqrt {b} c+\sqrt {a} d\right )^2}+\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a} b \left (\sqrt {b} c-\sqrt {a} d\right )^2}\right ) \] Input:

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

Output:

(d*((2*(c - d*x)*Sqrt[c + d*x])/(d*(-(b*c^2) + a*d^2)*(-a + b*x^2)) - (Sqr 
t[-(b*c) - Sqrt[a]*Sqrt[b]*d]*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqr 
t[c + d*x])/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[a]*b*(Sqrt[b]*c + Sqrt[a]*d)^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[a]*b*(Sqrt[b]*c - Sqrt 
[a]*d)^2)))/4
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.45, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {561, 25, 27, 1492, 27, 1480, 221}

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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {2 \left (\frac {d^4 \int -\frac {2 a b (c-d x)}{d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {d^2 (c-d x) \sqrt {c+d x}}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^2}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 \left (-\frac {d^2 \left (-\frac {d^2 \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 \left (b c^2-a d^2\right )}-\frac {d^2 (c-d x) \sqrt {c+d x}}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^2}\)

Input:

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

Output:

(-2*(-1/4*(d^2*(c - d*x)*Sqrt[c + d*x])/((b*c^2 - a*d^2)*(a - (b*c^2)/d^2 
+ (2*b*c*(c + d*x))/d^2 - (b*(c + d*x)^2)/d^2)) - (d^2*(-1/2*(d*((Sqrt[b]* 
c)/Sqrt[a] + d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d 
]])/(b^(3/4)*Sqrt[Sqrt[b]*c - Sqrt[a]*d]) - ((1 - (Sqrt[b]*c)/(Sqrt[a]*d)) 
*d^2*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*b^(3 
/4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d])))/(4*(b*c^2 - a*d^2))))/d^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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {\sqrt {a b \,d^{2}}\, \left (\frac {\sqrt {d x +c}}{\left (b c +\sqrt {a b \,d^{2}}\right ) \left (-b d x +\sqrt {a b \,d^{2}}\right )}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\sqrt {d x +c}}{\left (-b c +\sqrt {a b \,d^{2}}\right ) \left (b d x +\sqrt {a b \,d^{2}}\right )}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\left (-b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a}\)	\(210\)
derivativedivides	\(2 d^{2} b^{2} \left (\frac {\sqrt {a b \,d^{2}}\, \left (\frac {\sqrt {d x +c}}{2 \left (b c +\sqrt {a b \,d^{2}}\right ) \left (-b \left (d x +c \right )+b c +\sqrt {a b \,d^{2}}\right )}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \left (b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,b^{2} d^{2}}-\frac {\sqrt {a b \,d^{2}}\, \left (\frac {\sqrt {d x +c}}{2 \left (-b c +\sqrt {a b \,d^{2}}\right ) \left (b \left (d x +c \right )-b c +\sqrt {a b \,d^{2}}\right )}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \left (-b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,b^{2} d^{2}}\right )\)	\(260\)
default	\(2 d^{2} b^{2} \left (\frac {\sqrt {a b \,d^{2}}\, \left (\frac {\sqrt {d x +c}}{2 \left (b c +\sqrt {a b \,d^{2}}\right ) \left (-b \left (d x +c \right )+b c +\sqrt {a b \,d^{2}}\right )}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \left (b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,b^{2} d^{2}}-\frac {\sqrt {a b \,d^{2}}\, \left (\frac {\sqrt {d x +c}}{2 \left (-b c +\sqrt {a b \,d^{2}}\right ) \left (b \left (d x +c \right )-b c +\sqrt {a b \,d^{2}}\right )}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \left (-b c +\sqrt {a b \,d^{2}}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 a \,b^{2} d^{2}}\right )\)	\(260\)

Input:

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

Output:

1/4/a*(a*b*d^2)^(1/2)*((d*x+c)^(1/2)/(b*c+(a*b*d^2)^(1/2))/(-b*d*x+(a*b*d^ 
2)^(1/2))+1/(b*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))-(d*x+c)^(1/2)/(-b*c+(a*b* 
d^2)^(1/2))/(b*d*x+(a*b*d^2)^(1/2))-1/(-b*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 
)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2990 vs. \(2 (138) = 276\).

Time = 0.15 (sec) , antiderivative size = 2990, normalized size of antiderivative = 16.25 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*((a*b*c^2 - a^2*d^2 - (b^2*c^2 - a*b*d^2)*x^2)*sqrt((b*c^3*d^2 + 3*a*c 
*d^4 + (a*b^4*c^6 - 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 - a^4*b*d^6)*sqr 
t((9*b^2*c^4*d^6 + 6*a*b*c^2*d^8 + a^2*d^10)/(a*b^9*c^12 - 6*a^2*b^8*c^10* 
d^2 + 15*a^3*b^7*c^8*d^4 - 20*a^4*b^6*c^6*d^6 + 15*a^5*b^5*c^4*d^8 - 6*a^6 
*b^4*c^2*d^10 + a^7*b^3*d^12)))/(a*b^4*c^6 - 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2 
*c^2*d^4 - a^4*b*d^6))*log((3*b*c^2*d^4 + a*d^6)*sqrt(d*x + c) + (6*a*b^2* 
c^3*d^4 + 2*a^2*b*c*d^6 - (a*b^6*c^8 - 2*a^2*b^5*c^6*d^2 + 2*a^4*b^3*c^2*d 
^6 - a^5*b^2*d^8)*sqrt((9*b^2*c^4*d^6 + 6*a*b*c^2*d^8 + a^2*d^10)/(a*b^9*c 
^12 - 6*a^2*b^8*c^10*d^2 + 15*a^3*b^7*c^8*d^4 - 20*a^4*b^6*c^6*d^6 + 15*a^ 
5*b^5*c^4*d^8 - 6*a^6*b^4*c^2*d^10 + a^7*b^3*d^12)))*sqrt((b*c^3*d^2 + 3*a 
*c*d^4 + (a*b^4*c^6 - 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 - a^4*b*d^6)*s 
qrt((9*b^2*c^4*d^6 + 6*a*b*c^2*d^8 + a^2*d^10)/(a*b^9*c^12 - 6*a^2*b^8*c^1 
0*d^2 + 15*a^3*b^7*c^8*d^4 - 20*a^4*b^6*c^6*d^6 + 15*a^5*b^5*c^4*d^8 - 6*a 
^6*b^4*c^2*d^10 + a^7*b^3*d^12)))/(a*b^4*c^6 - 3*a^2*b^3*c^4*d^2 + 3*a^3*b 
^2*c^2*d^4 - a^4*b*d^6))) - (a*b*c^2 - a^2*d^2 - (b^2*c^2 - a*b*d^2)*x^2)* 
sqrt((b*c^3*d^2 + 3*a*c*d^4 + (a*b^4*c^6 - 3*a^2*b^3*c^4*d^2 + 3*a^3*b^2*c 
^2*d^4 - a^4*b*d^6)*sqrt((9*b^2*c^4*d^6 + 6*a*b*c^2*d^8 + a^2*d^10)/(a*b^9 
*c^12 - 6*a^2*b^8*c^10*d^2 + 15*a^3*b^7*c^8*d^4 - 20*a^4*b^6*c^6*d^6 + 15* 
a^5*b^5*c^4*d^8 - 6*a^6*b^4*c^2*d^10 + a^7*b^3*d^12)))/(a*b^4*c^6 - 3*a^2* 
b^3*c^4*d^2 + 3*a^3*b^2*c^2*d^4 - a^4*b*d^6))*log((3*b*c^2*d^4 + a*d^6)...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 732 vs. \(2 (138) = 276\).

Time = 0.21 (sec) , antiderivative size = 732, normalized size of antiderivative = 3.98 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=\frac {\frac {{\left ({\left (b c^{2} d - a d^{3}\right )}^{2} a d^{3} {\left | b \right |} - 2 \, {\left (\sqrt {a b} b c^{3} d^{3} - \sqrt {a b} a c d^{5}\right )} {\left | b c^{2} d - a d^{3} \right |} {\left | b \right |} + {\left (b^{3} c^{6} d^{3} - 2 \, a b^{2} c^{4} d^{5} + a^{2} b c^{2} d^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c^{3} - a b c d^{2} + \sqrt {{\left (b^{2} c^{3} - a b c d^{2}\right )}^{2} - {\left (b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} {\left (b^{2} c^{2} - a b d^{2}\right )}}}{b^{2} c^{2} - a b d^{2}}}}\right )}{{\left (a b^{3} c^{4} d - 2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b d^{5} - \sqrt {a b} b^{3} c^{5} + 2 \, \sqrt {a b} a b^{2} c^{3} d^{2} - \sqrt {a b} a^{2} b c d^{4}\right )} \sqrt {-b^{2} c - \sqrt {a b} b d} {\left | b c^{2} d - a d^{3} \right |}} + \frac {{\left ({\left (b c^{2} d - a d^{3}\right )}^{2} a d^{3} {\left | b \right |} + 2 \, {\left (\sqrt {a b} b c^{3} d^{3} - \sqrt {a b} a c d^{5}\right )} {\left | b c^{2} d - a d^{3} \right |} {\left | b \right |} + {\left (b^{3} c^{6} d^{3} - 2 \, a b^{2} c^{4} d^{5} + a^{2} b c^{2} d^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c^{3} - a b c d^{2} - \sqrt {{\left (b^{2} c^{3} - a b c d^{2}\right )}^{2} - {\left (b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} {\left (b^{2} c^{2} - a b d^{2}\right )}}}{b^{2} c^{2} - a b d^{2}}}}\right )}{{\left (a b^{3} c^{4} d - 2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b d^{5} + \sqrt {a b} b^{3} c^{5} - 2 \, \sqrt {a b} a b^{2} c^{3} d^{2} + \sqrt {a b} a^{2} b c d^{4}\right )} \sqrt {-b^{2} c + \sqrt {a b} b d} {\left | b c^{2} d - a d^{3} \right |}} + \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} d^{3} - 2 \, \sqrt {d x + c} c d^{3}\right )}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} - a d^{2}\right )} {\left (b c^{2} - a d^{2}\right )}}}{4 \, d} \] Input:

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

Output:

1/4*(((b*c^2*d - a*d^3)^2*a*d^3*abs(b) - 2*(sqrt(a*b)*b*c^3*d^3 - sqrt(a*b 
)*a*c*d^5)*abs(b*c^2*d - a*d^3)*abs(b) + (b^3*c^6*d^3 - 2*a*b^2*c^4*d^5 + 
a^2*b*c^2*d^7)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^2*c^3 - a*b*c*d^2 + s 
qrt((b^2*c^3 - a*b*c*d^2)^2 - (b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)*(b^2*c^2 
 - a*b*d^2)))/(b^2*c^2 - a*b*d^2)))/((a*b^3*c^4*d - 2*a^2*b^2*c^2*d^3 + a^ 
3*b*d^5 - sqrt(a*b)*b^3*c^5 + 2*sqrt(a*b)*a*b^2*c^3*d^2 - sqrt(a*b)*a^2*b* 
c*d^4)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b*c^2*d - a*d^3)) + ((b*c^2*d - a* 
d^3)^2*a*d^3*abs(b) + 2*(sqrt(a*b)*b*c^3*d^3 - sqrt(a*b)*a*c*d^5)*abs(b*c^ 
2*d - a*d^3)*abs(b) + (b^3*c^6*d^3 - 2*a*b^2*c^4*d^5 + a^2*b*c^2*d^7)*abs( 
b))*arctan(sqrt(d*x + c)/sqrt(-(b^2*c^3 - a*b*c*d^2 - sqrt((b^2*c^3 - a*b* 
c*d^2)^2 - (b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)*(b^2*c^2 - a*b*d^2)))/(b^2* 
c^2 - a*b*d^2)))/((a*b^3*c^4*d - 2*a^2*b^2*c^2*d^3 + a^3*b*d^5 + sqrt(a*b) 
*b^3*c^5 - 2*sqrt(a*b)*a*b^2*c^3*d^2 + sqrt(a*b)*a^2*b*c*d^4)*sqrt(-b^2*c 
+ sqrt(a*b)*b*d)*abs(b*c^2*d - a*d^3)) + 2*((d*x + c)^(3/2)*d^3 - 2*sqrt(d 
*x + c)*c*d^3)/(((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 - a*d^2)*(b*c^2 - 
 a*d^2)))/d
 

Mupad [B] (verification not implemented)

Time = 1.36 (sec) , antiderivative size = 4625, normalized size of antiderivative = 25.14 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((128*a*b^4*c^3*d^4 - 128*a^2*b^3*c*d^6)/(8*(a^2*d^4 + b^2*c^4 - 2* 
a*b*c^2*d^2)) + ((c + d*x)^(1/2)*(64*a*b^6*c^5*d^2 + 64*a^3*b^4*c*d^6 - 12 
8*a^2*b^5*c^3*d^4)*((a*d^5*(a^3*b^3)^(1/2) + a*b^3*c^3*d^2 + 3*a^2*b^2*c*d 
^4 + 3*b*c^2*d^3*(a^3*b^3)^(1/2))/(64*(a^2*b^6*c^6 - a^5*b^3*d^6 - 3*a^3*b 
^5*c^4*d^2 + 3*a^4*b^4*c^2*d^4)))^(1/2))/(a^2*d^4 + b^2*c^4 - 2*a*b*c^2*d^ 
2))*((a*d^5*(a^3*b^3)^(1/2) + a*b^3*c^3*d^2 + 3*a^2*b^2*c*d^4 + 3*b*c^2*d^ 
3*(a^3*b^3)^(1/2))/(64*(a^2*b^6*c^6 - a^5*b^3*d^6 - 3*a^3*b^5*c^4*d^2 + 3* 
a^4*b^4*c^2*d^4)))^(1/2) - ((a*b^2*d^6 + b^3*c^2*d^4)*(c + d*x)^(1/2))/(a^ 
2*d^4 + b^2*c^4 - 2*a*b*c^2*d^2))*((a*d^5*(a^3*b^3)^(1/2) + a*b^3*c^3*d^2 
+ 3*a^2*b^2*c*d^4 + 3*b*c^2*d^3*(a^3*b^3)^(1/2))/(64*(a^2*b^6*c^6 - a^5*b^ 
3*d^6 - 3*a^3*b^5*c^4*d^2 + 3*a^4*b^4*c^2*d^4)))^(1/2)*1i - (((128*a*b^4*c 
^3*d^4 - 128*a^2*b^3*c*d^6)/(8*(a^2*d^4 + b^2*c^4 - 2*a*b*c^2*d^2)) - ((c 
+ d*x)^(1/2)*(64*a*b^6*c^5*d^2 + 64*a^3*b^4*c*d^6 - 128*a^2*b^5*c^3*d^4)*( 
(a*d^5*(a^3*b^3)^(1/2) + a*b^3*c^3*d^2 + 3*a^2*b^2*c*d^4 + 3*b*c^2*d^3*(a^ 
3*b^3)^(1/2))/(64*(a^2*b^6*c^6 - a^5*b^3*d^6 - 3*a^3*b^5*c^4*d^2 + 3*a^4*b 
^4*c^2*d^4)))^(1/2))/(a^2*d^4 + b^2*c^4 - 2*a*b*c^2*d^2))*((a*d^5*(a^3*b^3 
)^(1/2) + a*b^3*c^3*d^2 + 3*a^2*b^2*c*d^4 + 3*b*c^2*d^3*(a^3*b^3)^(1/2))/( 
64*(a^2*b^6*c^6 - a^5*b^3*d^6 - 3*a^3*b^5*c^4*d^2 + 3*a^4*b^4*c^2*d^4)))^( 
1/2) + ((a*b^2*d^6 + b^3*c^2*d^4)*(c + d*x)^(1/2))/(a^2*d^4 + b^2*c^4 - 2* 
a*b*c^2*d^2))*((a*d^5*(a^3*b^3)^(1/2) + a*b^3*c^3*d^2 + 3*a^2*b^2*c*d^4...
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 989, normalized size of antiderivative = 5.38 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*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*d**3 - 2*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*c**2*d + 2*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*d**3*x**2 + 2*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**2*c**2*d*x**2 - 4*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**2*c*d 
**2 + 4*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*b*c*d**2*x**2 - 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*d**3 - 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*c**2*d + 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*d**3*x**2 + 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**2*c**2*d*x**2 + 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*d**3 + sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(s 
qrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*a*b*c**2*d - sqrt(a) 
*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt...