\(\int \frac {x}{\sqrt {c+d x} (a-b x^2)^3} \, dx\) [721]

Optimal result

Integrand size = 21, antiderivative size = 278 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {(c-d x) \sqrt {c+d x}}{4 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^2}+\frac {d \sqrt {c+d x} \left (b c^2 x-a d (6 c-5 d x)\right )}{16 a \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )}-\frac {d \left (2 \sqrt {b} c-5 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 a^{3/2} b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}+\frac {d \left (2 \sqrt {b} c+5 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 a^{3/2} b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:

1/4*(-d*x+c)*(d*x+c)^(1/2)/(-a*d^2+b*c^2)/(-b*x^2+a)^2+1/16*d*(d*x+c)^(1/2 
)*(b*c^2*x-a*d*(-5*d*x+6*c))/a/(-a*d^2+b*c^2)^2/(-b*x^2+a)-1/32*d*(2*b^(1/ 
2)*c-5*a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2 
))/a^(3/2)/b^(3/4)/(b^(1/2)*c-a^(1/2)*d)^(5/2)+1/32*d*(2*b^(1/2)*c+5*a^(1/ 
2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a^(3/2)/b 
^(3/4)/(b^(1/2)*c+a^(1/2)*d)^(5/2)
 

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.22 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {d \left (-\frac {2 \sqrt {a} \sqrt {c+d x} \left (b^2 c^2 d x^3+a^2 d^2 (10 c-9 d x)+a b \left (-4 c^3+3 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )\right )}{d \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )^2}-\frac {\left (2 \sqrt {b} c+5 \sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{b \left (\sqrt {b} c+\sqrt {a} d\right )^3}-\frac {\left (2 \sqrt {b} c-5 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )^2 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}\right )}{32 a^{3/2}} \] Input:

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

Output:

(d*((-2*Sqrt[a]*Sqrt[c + d*x]*(b^2*c^2*d*x^3 + a^2*d^2*(10*c - 9*d*x) + a* 
b*(-4*c^3 + 3*c^2*d*x - 6*c*d^2*x^2 + 5*d^3*x^3)))/(d*(b*c^2 - a*d^2)^2*(a 
 - b*x^2)^2) - ((2*Sqrt[b]*c + 5*Sqrt[a]*d)*Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]* 
d]*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c + Sq 
rt[a]*d)])/(b*(Sqrt[b]*c + Sqrt[a]*d)^3) - ((2*Sqrt[b]*c - 5*Sqrt[a]*d)*Ar 
cTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a] 
*d)])/(Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)^2*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]) 
))/(32*a^(3/2))
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.56, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {561, 25, 27, 1492, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {2 \left (\frac {d^4 \int -\frac {2 a b (6 c-5 (c+d x))}{d^2 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^2 \int \frac {6 c-5 (c+d x)}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 \left (b c^2-a d^2\right )}-\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {2 \left (-\frac {d^2 \left (\frac {d^4 \int \frac {2 b \left (c \left (b c^2-13 a d^2\right )+\left (b c^2+5 a d^2\right ) (c+d x)\right )}{d^4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (11 a d^2+b c^2\right )-(c+d x) \left (5 a d^2+b c^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 \left (b c^2-a d^2\right )}-\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {d^2 \left (\frac {\int \frac {c \left (b c^2-13 a d^2\right )+\left (b c^2+5 a d^2\right ) (c+d x)}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (11 a d^2+b c^2\right )-(c+d x) \left (5 a d^2+b c^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 \left (b c^2-a d^2\right )}-\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {2 \left (-\frac {d^2 \left (\frac {\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (5 \sqrt {a} d+2 \sqrt {b} c\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}-\frac {\left (2 \sqrt {b} c-5 \sqrt {a} d\right ) \left (\sqrt {a} d+\sqrt {b} c\right )^2 \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}}{2 \sqrt {a} d}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (11 a d^2+b c^2\right )-(c+d x) \left (5 a d^2+b c^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 \left (b c^2-a d^2\right )}-\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 \left (-\frac {d^2 \left (\frac {\frac {d \left (\sqrt {b} c-\sqrt {a} d\right )^2 \left (5 \sqrt {a} d+2 \sqrt {b} c\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 \sqrt {a} b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}-\frac {d \left (2 \sqrt {b} c-5 \sqrt {a} d\right ) \left (\sqrt {a} d+\sqrt {b} c\right )^2 \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 \sqrt {a} b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}}{4 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (c \left (11 a d^2+b c^2\right )-(c+d x) \left (5 a d^2+b c^2\right )\right )}{4 a \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 \left (b c^2-a d^2\right )}-\frac {d^2 (c-d x) \sqrt {c+d x}}{8 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^2}\)

Input:

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

Output:

(-2*(-1/8*(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)^2) - (d^2*(-1/4*(Sqrt[c + d 
*x]*(c*(b*c^2 + 11*a*d^2) - (b*c^2 + 5*a*d^2)*(c + d*x)))/(a*(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)) + (-1/ 
2*(d*(2*Sqrt[b]*c - 5*Sqrt[a]*d)*(Sqrt[b]*c + Sqrt[a]*d)^2*ArcTanh[(b^(1/4 
)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(Sqrt[a]*b^(3/4)*Sqrt[Sqrt[ 
b]*c - Sqrt[a]*d]) + (d*(Sqrt[b]*c - Sqrt[a]*d)^2*(2*Sqrt[b]*c + 5*Sqrt[a] 
*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*Sqrt[ 
a]*b^(3/4)*Sqrt[Sqrt[b]*c + Sqrt[a]*d]))/(4*a*(b*c^2 - a*d^2))))/(8*(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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(454\) vs. \(2(218)=436\).

Time = 1.03 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.64

method	result	size
derivativedivides	\(-2 d^{4} b^{3} \left (-\frac {\frac {-\frac {\left (2 b c +5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{2} \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right )}+\frac {\left (2 b c +7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{2} \left (b c +\sqrt {a b \,d^{2}}\right )}}{{\left (-d x +\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}+\frac {\left (2 b c +5 \sqrt {a b \,d^{2}}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 \sqrt {a b \,d^{2}}\, a \,b^{2} d^{2}}+\frac {\frac {-\frac {\left (2 b c -5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{2} \left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right )}+\frac {\left (2 b c -7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{2} \left (b c -\sqrt {a b \,d^{2}}\right )}}{{\left (-d x -\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}-\frac {\left (-2 b c +5 \sqrt {a b \,d^{2}}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 \sqrt {a b \,d^{2}}\, a \,b^{2} d^{2}}\right )\)	\(455\)
default	\(2 d^{4} b^{3} \left (\frac {\frac {-\frac {\left (2 b c +5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{2} \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right )}+\frac {\left (2 b c +7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{2} \left (b c +\sqrt {a b \,d^{2}}\right )}}{{\left (-d x +\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}+\frac {\left (2 b c +5 \sqrt {a b \,d^{2}}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 \sqrt {a b \,d^{2}}\, a \,b^{2} d^{2}}-\frac {\frac {-\frac {\left (2 b c -5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 b^{2} \left (a \,d^{2}+b \,c^{2}-2 \sqrt {a b \,d^{2}}\, c \right )}+\frac {\left (2 b c -7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{4 b^{2} \left (b c -\sqrt {a b \,d^{2}}\right )}}{{\left (-d x -\frac {\sqrt {a b \,d^{2}}}{b}\right )}^{2}}-\frac {\left (-2 b c +5 \sqrt {a b \,d^{2}}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4 b \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}}{16 \sqrt {a b \,d^{2}}\, a \,b^{2} d^{2}}\right )\)	\(455\)
pseudoelliptic	\(-\frac {d^{2} b^{3} \left (\frac {\left (d x +c \right )^{\frac {3}{2}} c}{\left (b d x +\sqrt {a b \,d^{2}}\right )^{2} b \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {a b \,d^{2}}}-\frac {5 \left (d x +c \right )^{\frac {3}{2}}}{2 \left (b d x +\sqrt {a b \,d^{2}}\right )^{2} b^{2} \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right )}-\frac {\sqrt {d x +c}\, c}{\left (b d x +\sqrt {a b \,d^{2}}\right )^{2} b \left (-b c +\sqrt {a b \,d^{2}}\right ) \sqrt {a b \,d^{2}}}+\frac {7 \sqrt {d x +c}}{2 \left (b d x +\sqrt {a b \,d^{2}}\right )^{2} b^{2} \left (-b c +\sqrt {a b \,d^{2}}\right )}+\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right ) c}{\left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, b^{2}}-\frac {5 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b^{3} \left (-a \,d^{2}-b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\frac {-\frac {\left (2 b c +5 \sqrt {a b \,d^{2}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c}+\frac {\left (2 b c +7 \sqrt {a b \,d^{2}}\right ) \sqrt {d x +c}}{b c +\sqrt {a b \,d^{2}}}}{\left (-b d x +\sqrt {a b \,d^{2}}\right )^{2}}+\frac {\left (2 b c +5 \sqrt {a b \,d^{2}}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{b \left (a \,d^{2}+b \,c^{2}+2 \sqrt {a b \,d^{2}}\, c \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{2 \sqrt {a b \,d^{2}}\, b^{2}}\right )}{16 a}\)	\(580\)

Input:

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

Output:

-2*d^4*b^3*(-1/16/(a*b*d^2)^(1/2)/a/b^2/d^2*((-1/4/b^2/(a*d^2+b*c^2+2*(a*b 
*d^2)^(1/2)*c)*(2*b*c+5*(a*b*d^2)^(1/2))*(d*x+c)^(3/2)+1/4/b^2*(2*b*c+7*(a 
*b*d^2)^(1/2))/(b*c+(a*b*d^2)^(1/2))*(d*x+c)^(1/2))/(-d*x+(a*b*d^2)^(1/2)/ 
b)^2+1/4*(2*b*c+5*(a*b*d^2)^(1/2))/b/(a*d^2+b*c^2+2*(a*b*d^2)^(1/2)*c)/((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)))+1/16/(a*b*d^2)^(1/2)/a/b^2/d^2*((-1/4/b^2/(a*d^2+b*c^2-2*(a*b 
*d^2)^(1/2)*c)*(2*b*c-5*(a*b*d^2)^(1/2))*(d*x+c)^(3/2)+1/4/b^2*(2*b*c-7*(a 
*b*d^2)^(1/2))/(b*c-(a*b*d^2)^(1/2))*(d*x+c)^(1/2))/(-d*x-(a*b*d^2)^(1/2)/ 
b)^2-1/4*(-2*b*c+5*(a*b*d^2)^(1/2))/b/(-a*d^2-b*c^2+2*(a*b*d^2)^(1/2)*c)/( 
(-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. 5629 vs. \(2 (220) = 440\).

Time = 0.92 (sec) , antiderivative size = 5629, normalized size of antiderivative = 20.25 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1482 vs. \(2 (220) = 440\).

Time = 0.31 (sec) , antiderivative size = 1482, normalized size of antiderivative = 5.33 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/32*(((a*b^2*c^4*d - 2*a^2*b*c^2*d^3 + a^3*d^5)^2*(b*c^2*d^3 + 5*a*d^5)* 
abs(b) + (sqrt(a*b)*b^3*c^7*d^3 - 15*sqrt(a*b)*a*b^2*c^5*d^5 + 27*sqrt(a*b 
)*a^2*b*c^3*d^7 - 13*sqrt(a*b)*a^3*c*d^9)*abs(a*b^2*c^4*d - 2*a^2*b*c^2*d^ 
3 + a^3*d^5)*abs(b) - 2*(a*b^6*c^12*d^3 - 8*a^2*b^5*c^10*d^5 + 22*a^3*b^4* 
c^8*d^7 - 28*a^4*b^3*c^6*d^9 + 17*a^5*b^2*c^4*d^11 - 4*a^6*b*c^2*d^13)*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)))/((a^2*b^5* 
c^8*d - 4*a^3*b^4*c^6*d^3 + 6*a^4*b^3*c^4*d^5 - 4*a^5*b^2*c^2*d^7 + a^6*b* 
d^9 - sqrt(a*b)*a*b^5*c^9 + 4*sqrt(a*b)*a^2*b^4*c^7*d^2 - 6*sqrt(a*b)*a^3* 
b^3*c^5*d^4 + 4*sqrt(a*b)*a^4*b^2*c^3*d^6 - sqrt(a*b)*a^5*b*c*d^8)*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)) + ((a* 
b^2*c^4*d - 2*a^2*b*c^2*d^3 + a^3*d^5)^2*(b*c^2*d^3 + 5*a*d^5)*abs(b) - (s 
qrt(a*b)*b^3*c^7*d^3 - 15*sqrt(a*b)*a*b^2*c^5*d^5 + 27*sqrt(a*b)*a^2*b*c^3 
*d^7 - 13*sqrt(a*b)*a^3*c*d^9)*abs(a*b^2*c^4*d - 2*a^2*b*c^2*d^3 + a^3*d^5 
)*abs(b) - 2*(a*b^6*c^12*d^3 - 8*a^2*b^5*c^10*d^5 + 22*a^3*b^4*c^8*d^7 - 2 
8*a^4*b^3*c^6*d^9 + 17*a^5*b^2*c^4*d^11 - 4*a^6*b*c^2*d^13)*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...
 

Mupad [B] (verification not implemented)

Time = 11.99 (sec) , antiderivative size = 8623, normalized size of antiderivative = 31.02 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((53248*a^6*b^3*c*d^10 - 4096*a^3*b^6*c^7*d^4 + 61440*a^4*b^5*c^5*d 
^6 - 110592*a^5*b^4*c^3*d^8)/(4096*(a^7*d^8 + a^3*b^4*c^8 - 4*a^6*b*c^2*d^ 
6 - 4*a^4*b^3*c^6*d^2 + 6*a^5*b^2*c^4*d^4)) - ((c + d*x)^(1/2)*((105*a^6*b 
^2*c*d^8 - 25*a^2*d^9*(a^9*b^3)^(1/2) + 4*a^3*b^5*c^7*d^2 - 35*a^4*b^4*c^5 
*d^4 + 70*a^5*b^3*c^3*d^6 + 35*b^2*c^4*d^5*(a^9*b^3)^(1/2) - 154*a*b*c^2*d 
^7*(a^9*b^3)^(1/2))/(4096*(a^6*b^8*c^10 - a^11*b^3*d^10 - 5*a^7*b^7*c^8*d^ 
2 + 10*a^8*b^6*c^6*d^4 - 10*a^9*b^5*c^4*d^6 + 5*a^10*b^4*c^2*d^8)))^(1/2)* 
(4096*a^7*b^4*c*d^10 + 4096*a^3*b^8*c^9*d^2 - 16384*a^4*b^7*c^7*d^4 + 2457 
6*a^5*b^6*c^5*d^6 - 16384*a^6*b^5*c^3*d^8))/(64*(a^6*d^8 + a^2*b^4*c^8 - 4 
*a^5*b*c^2*d^6 - 4*a^3*b^3*c^6*d^2 + 6*a^4*b^2*c^4*d^4)))*((105*a^6*b^2*c* 
d^8 - 25*a^2*d^9*(a^9*b^3)^(1/2) + 4*a^3*b^5*c^7*d^2 - 35*a^4*b^4*c^5*d^4 
+ 70*a^5*b^3*c^3*d^6 + 35*b^2*c^4*d^5*(a^9*b^3)^(1/2) - 154*a*b*c^2*d^7*(a 
^9*b^3)^(1/2))/(4096*(a^6*b^8*c^10 - a^11*b^3*d^10 - 5*a^7*b^7*c^8*d^2 + 1 
0*a^8*b^6*c^6*d^4 - 10*a^9*b^5*c^4*d^6 + 5*a^10*b^4*c^2*d^8)))^(1/2) + ((c 
 + d*x)^(1/2)*(25*a^3*b^2*d^10 + 4*b^5*c^6*d^4 - 31*a*b^4*c^4*d^6 + 74*a^2 
*b^3*c^2*d^8))/(64*(a^6*d^8 + a^2*b^4*c^8 - 4*a^5*b*c^2*d^6 - 4*a^3*b^3*c^ 
6*d^2 + 6*a^4*b^2*c^4*d^4)))*((105*a^6*b^2*c*d^8 - 25*a^2*d^9*(a^9*b^3)^(1 
/2) + 4*a^3*b^5*c^7*d^2 - 35*a^4*b^4*c^5*d^4 + 70*a^5*b^3*c^3*d^6 + 35*b^2 
*c^4*d^5*(a^9*b^3)^(1/2) - 154*a*b*c^2*d^7*(a^9*b^3)^(1/2))/(4096*(a^6*b^8 
*c^10 - a^11*b^3*d^10 - 5*a^7*b^7*c^8*d^2 + 10*a^8*b^6*c^6*d^4 - 10*a^9...
 

Reduce [B] (verification not implemented)

Time = 3.58 (sec) , antiderivative size = 2666, normalized size of antiderivative = 9.59 \[ \int \frac {x}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 10*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*d**5 - 18*sqrt(a)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c 
)))*a**3*b*c**2*d**3 + 20*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*d**5*x**2 + 4 
*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqr 
t(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*c**4*d + 36*sqrt(a)*sqrt(sqrt(b)*sq 
rt(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**2*d**3*x**2 - 10*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*a 
tan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*d 
**5*x**4 - 8*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**4*d*x**2 - 18*sqrt(a)*s 
qrt(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**2*d**3*x**4 + 4*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 
**4*c**4*d*x**4 - 26*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*d**4 + 2*sqrt(b)*s 
qrt(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 + 52*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*...