\(\int \frac {x^3}{\sqrt {c+d x} (a-b x^2)^3} \, dx\) [719]

Optimal result

Integrand size = 23, antiderivative size = 289 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {a (c-d x) \sqrt {c+d x}}{4 b \left (b c^2-a d^2\right ) \left (a-b x^2\right )^2}-\frac {\sqrt {c+d x} \left (b c^2 (8 c-9 d x)-a d^2 (2 c-3 d x)\right )}{16 b \left (b c^2-a d^2\right )^2 \left (a-b x^2\right )}+\frac {3 d \left (2 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{32 \sqrt {a} b^{7/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{5/2}}-\frac {3 d \left (2 \sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{32 \sqrt {a} b^{7/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{5/2}} \] Output:

1/4*a*(-d*x+c)*(d*x+c)^(1/2)/b/(-a*d^2+b*c^2)/(-b*x^2+a)^2-1/16*(d*x+c)^(1 
/2)*(b*c^2*(-9*d*x+8*c)-a*d^2*(-3*d*x+2*c))/b/(-a*d^2+b*c^2)^2/(-b*x^2+a)+ 
3/32*d*(2*b^(1/2)*c-a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c-a^ 
(1/2)*d)^(1/2))/a^(1/2)/b^(7/4)/(b^(1/2)*c-a^(1/2)*d)^(5/2)-3/32*d*(2*b^(1 
/2)*c+a^(1/2)*d)*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2) 
)/a^(1/2)/b^(7/4)/(b^(1/2)*c+a^(1/2)*d)^(5/2)
 

Mathematica [A] (verified)

Time = 1.88 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.18 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\frac {\frac {2 b \sqrt {c+d x} \left (b^2 c^2 x^2 (8 c-9 d x)+a^2 d^2 (-2 c+d x)+a b \left (-4 c^3+5 c^2 d x-2 c d^2 x^2+3 d^3 x^3\right )\right )}{\left (b c^2-a d^2\right )^2 \left (a-b x^2\right )^2}+\frac {3 d \left (2 \sqrt {b} c+\sqrt {a} d\right ) \sqrt {-b c-\sqrt {a} \sqrt {b} d} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \left (\sqrt {b} c+\sqrt {a} d\right )^3}+\frac {3 \sqrt {b} d \left (2 \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a} \left (\sqrt {b} c-\sqrt {a} d\right )^2 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}}{32 b^2} \] Input:

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

Output:

((2*b*Sqrt[c + d*x]*(b^2*c^2*x^2*(8*c - 9*d*x) + a^2*d^2*(-2*c + d*x) + a* 
b*(-4*c^3 + 5*c^2*d*x - 2*c*d^2*x^2 + 3*d^3*x^3)))/((b*c^2 - a*d^2)^2*(a - 
 b*x^2)^2) + (3*d*(2*Sqrt[b]*c + 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)])/(Sqrt[a]*(Sqrt[b]*c + Sqrt[a]*d)^3) + (3*Sqrt[b]*d*(2*Sqrt[b]*c 
 - Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqr 
t[b]*c - Sqrt[a]*d)])/(Sqrt[a]*(Sqrt[b]*c - Sqrt[a]*d)^2*Sqrt[-(b*c) + Sqr 
t[a]*Sqrt[b]*d]))/(32*b^2)
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.47, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {561, 25, 27, 1517, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1517

\(\displaystyle -\frac {2 \left (\frac {d^4 \int \frac {2 a \left (2 c \left (a-\frac {4 b c^2}{d^2}\right )-\left (3 a-\frac {8 b c^2}{d^2}\right ) (c+d x)\right )}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{16 a b \left (b c^2-a d^2\right )}-\frac {a d^4 (c-d x) \sqrt {c+d x}}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^4 \int \frac {2 c \left (a-\frac {4 b c^2}{d^2}\right )-\left (3 a-\frac {8 b c^2}{d^2}\right ) (c+d x)}{\left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )^2}d\sqrt {c+d x}}{8 b \left (b c^2-a d^2\right )}-\frac {a d^4 (c-d x) \sqrt {c+d x}}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 1492

\(\displaystyle -\frac {2 \left (\frac {d^4 \left (\frac {d^4 \int \frac {6 a b \left (c \left (5 b c^2-a d^2\right )-\left (3 b c^2-a d^2\right ) (c+d x)\right )}{d^4 \left (-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a\right )}d\sqrt {c+d x}}{8 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (c \left (17 b c^2-5 a d^2\right )-3 (c+d x) \left (3 b c^2-a d^2\right )\right )}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 b \left (b c^2-a d^2\right )}-\frac {a d^4 (c-d x) \sqrt {c+d x}}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {d^4 \left (\frac {3 \int \frac {c \left (5 b c^2-a d^2\right )-\left (3 b c^2-a d^2\right ) (c+d x)}{-\frac {b c^2}{d^2}+\frac {2 b (c+d x) c}{d^2}-\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{4 \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (c \left (17 b c^2-5 a d^2\right )-3 (c+d x) \left (3 b c^2-a d^2\right )\right )}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 b \left (b c^2-a d^2\right )}-\frac {a d^4 (c-d x) \sqrt {c+d x}}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {2 \left (\frac {d^4 \left (\frac {3 \left (-\frac {1}{2} \left (\frac {2 b^{3/2} c^3}{\sqrt {a} d}-a d^2+3 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}-\frac {1}{2} \left (-\frac {2 b^{3/2} c^3}{\sqrt {a} d}-a d^2+3 b c^2\right ) \int \frac {1}{\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )}{d^2}-\frac {b (c+d x)}{d^2}}d\sqrt {c+d x}\right )}{4 \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (c \left (17 b c^2-5 a d^2\right )-3 (c+d x) \left (3 b c^2-a d^2\right )\right )}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 b \left (b c^2-a d^2\right )}-\frac {a d^4 (c-d x) \sqrt {c+d x}}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {2 \left (\frac {d^4 \left (\frac {3 \left (-\frac {d^2 \left (\frac {2 b^{3/2} c^3}{\sqrt {a} d}-a d^2+3 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 b^{3/4} \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {d^2 \left (-\frac {2 b^{3/2} c^3}{\sqrt {a} d}-a d^2+3 b c^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 b^{3/4} \sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{4 \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (c \left (17 b c^2-5 a d^2\right )-3 (c+d x) \left (3 b c^2-a d^2\right )\right )}{4 \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{8 b \left (b c^2-a d^2\right )}-\frac {a d^4 (c-d x) \sqrt {c+d x}}{8 b \left (b c^2-a d^2\right ) \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )^2}\right )}{d^4}\)

Input:

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

Output:

(-2*(-1/8*(a*d^4*(c - d*x)*Sqrt[c + d*x])/(b*(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^4*((Sqrt[c + d* 
x]*(c*(17*b*c^2 - 5*a*d^2) - 3*(3*b*c^2 - a*d^2)*(c + d*x)))/(4*(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)) + ( 
3*(-1/2*(d^2*(3*b*c^2 + (2*b^(3/2)*c^3)/(Sqrt[a]*d) - a*d^2)*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]) - (d^2*(3*b*c^2 - (2*b^(3/2)*c^3)/(Sqrt[a]*d) - a*d^2)*ArcTan 
h[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*b^(3/4)*Sqrt[Sq 
rt[b]*c + Sqrt[a]*d])))/(4*(b*c^2 - a*d^2))))/(8*b*(b*c^2 - a*d^2))))/d^4
 

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]
 

rule 1517

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x 
_Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + 
c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + 
c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* 
a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* 
(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p 
 + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] 
 + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* 
x^2, x], 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] && IGtQ[q, 1] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.19

method	result	size
pseudoelliptic	\(-\frac {-\frac {3 d^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (a \,d^{2}-3 b \,c^{2}\right ) \sqrt {a b \,d^{2}}-2 c^{3} b^{2}\right ) \left (-b \,x^{2}+a \right )^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\frac {3 d^{2} \left (\left (a \,d^{2}-3 b \,c^{2}\right ) \sqrt {a b \,d^{2}}+2 c^{3} b^{2}\right ) \left (-b \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}+\left (-4 x^{2} \left (-\frac {9 d x}{8}+c \right ) c^{2} b^{2}+2 a \left (-\frac {3}{4} d^{3} x^{3}+\frac {1}{2} c \,d^{2} x^{2}-\frac {5}{4} c^{2} d x +c^{3}\right ) b +a^{2} d^{2} \left (-\frac {d x}{2}+c \right )\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {d x +c}\, \sqrt {a b \,d^{2}}\right )}{8 \sqrt {a b \,d^{2}}\, \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, b \left (-b \,x^{2}+a \right )^{2}}\)	\(345\)
derivativedivides	\(-2 d^{2} \left (\frac {-\frac {3 \left (a \,d^{2}-3 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\left (11 a \,d^{2}-35 b \,c^{2}\right ) c \left (d x +c \right )^{\frac {5}{2}}}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}-\frac {\left (a^{2} d^{4}+18 b \,c^{2} d^{2} a -43 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\left (3 a \,d^{2}+17 b \,c^{2}\right ) c \sqrt {d x +c}}{32 \left (a \,d^{2}-b \,c^{2}\right ) b}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}+\frac {\frac {3 \left (2 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {3 \left (-2 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{64 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}}{a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}}\right )\)	\(461\)
default	\(2 d^{2} \left (-\frac {-\frac {3 \left (a \,d^{2}-3 b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\left (11 a \,d^{2}-35 b \,c^{2}\right ) c \left (d x +c \right )^{\frac {5}{2}}}{32 a^{2} d^{4}-64 b \,c^{2} d^{2} a +32 b^{2} c^{4}}-\frac {\left (a^{2} d^{4}+18 b \,c^{2} d^{2} a -43 b^{2} c^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{32 b \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}+\frac {\left (3 a \,d^{2}+17 b \,c^{2}\right ) c \sqrt {d x +c}}{32 \left (a \,d^{2}-b \,c^{2}\right ) b}}{\left (-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {3 \left (\frac {\left (2 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-2 c^{3} b^{2}-\sqrt {a b \,d^{2}}\, a \,d^{2}+3 \sqrt {a b \,d^{2}}\, b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{32 \left (a^{2} d^{4}-2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}\right )\)	\(462\)

Input:

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

Output:

-1/8/(a*b*d^2)^(1/2)*(-3/4*d^2*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*((a*d^2-3*b 
*c^2)*(a*b*d^2)^(1/2)-2*c^3*b^2)*(-b*x^2+a)^2*arctan(b*(d*x+c)^(1/2)/((-b* 
c+(a*b*d^2)^(1/2))*b)^(1/2))+((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(3/4*d^2*((a 
*d^2-3*b*c^2)*(a*b*d^2)^(1/2)+2*c^3*b^2)*(-b*x^2+a)^2*arctanh(b*(d*x+c)^(1 
/2)/((b*c+(a*b*d^2)^(1/2))*b)^(1/2))+(-4*x^2*(-9/8*d*x+c)*c^2*b^2+2*a*(-3/ 
4*d^3*x^3+1/2*c*d^2*x^2-5/4*c^2*d*x+c^3)*b+a^2*d^2*(-1/2*d*x+c))*((b*c+(a* 
b*d^2)^(1/2))*b)^(1/2)*(d*x+c)^(1/2)*(a*b*d^2)^(1/2)))/(a*d^2-b*c^2)^2/((b 
*c+(a*b*d^2)^(1/2))*b)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/b/(-b*x^2+a) 
^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6041 vs. \(2 (232) = 464\).

Time = 2.29 (sec) , antiderivative size = 6041, normalized size of antiderivative = 20.90 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-integrate(x^3/((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. 1500 vs. \(2 (232) = 464\).

Time = 0.32 (sec) , antiderivative size = 1500, normalized size of antiderivative = 5.19 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

3/32*((b^3*c^4*d - 2*a*b^2*c^2*d^3 + a^2*b*d^5)^2*(3*sqrt(a*b)*a*b*c^2*d^2 
 - sqrt(a*b)*a^2*d^4)*abs(b) - (5*a*b^5*c^7*d^2 - 11*a^2*b^4*c^5*d^4 + 7*a 
^3*b^3*c^3*d^6 - a^4*b^2*c*d^8)*abs(b^3*c^4*d - 2*a*b^2*c^2*d^3 + a^2*b*d^ 
5)*abs(b) + 2*(sqrt(a*b)*b^8*c^12*d^2 - 4*sqrt(a*b)*a*b^7*c^10*d^4 + 6*sqr 
t(a*b)*a^2*b^6*c^8*d^6 - 4*sqrt(a*b)*a^3*b^5*c^6*d^8 + sqrt(a*b)*a^4*b^4*c 
^4*d^10)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^4*c^5 - 2*a*b^3*c^3*d^2 + a 
^2*b^2*c*d^4 + sqrt((b^4*c^5 - 2*a*b^3*c^3*d^2 + a^2*b^2*c*d^4)^2 - (b^4*c 
^6 - 3*a*b^3*c^4*d^2 + 3*a^2*b^2*c^2*d^4 - a^3*b*d^6)*(b^4*c^4 - 2*a*b^3*c 
^2*d^2 + a^2*b^2*d^4)))/(b^4*c^4 - 2*a*b^3*c^2*d^2 + a^2*b^2*d^4)))/((a*b^ 
8*c^9 - 4*a^2*b^7*c^7*d^2 + 6*a^3*b^6*c^5*d^4 - 4*a^4*b^5*c^3*d^6 + a^5*b^ 
4*c*d^8 - sqrt(a*b)*a*b^7*c^8*d + 4*sqrt(a*b)*a^2*b^6*c^6*d^3 - 6*sqrt(a*b 
)*a^3*b^5*c^4*d^5 + 4*sqrt(a*b)*a^4*b^4*c^2*d^7 - sqrt(a*b)*a^5*b^3*d^9)*s 
qrt(-b^2*c - sqrt(a*b)*b*d)*abs(b^3*c^4*d - 2*a*b^2*c^2*d^3 + a^2*b*d^5)) 
- 3/32*((b^3*c^4*d - 2*a*b^2*c^2*d^3 + a^2*b*d^5)^2*(3*sqrt(a*b)*a*b*c^2*d 
^2 - sqrt(a*b)*a^2*d^4)*abs(b) + (5*a*b^5*c^7*d^2 - 11*a^2*b^4*c^5*d^4 + 7 
*a^3*b^3*c^3*d^6 - a^4*b^2*c*d^8)*abs(b^3*c^4*d - 2*a*b^2*c^2*d^3 + a^2*b* 
d^5)*abs(b) + 2*(sqrt(a*b)*b^8*c^12*d^2 - 4*sqrt(a*b)*a*b^7*c^10*d^4 + 6*s 
qrt(a*b)*a^2*b^6*c^8*d^6 - 4*sqrt(a*b)*a^3*b^5*c^6*d^8 + sqrt(a*b)*a^4*b^4 
*c^4*d^10)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^4*c^5 - 2*a*b^3*c^3*d^2 + 
 a^2*b^2*c*d^4 - sqrt((b^4*c^5 - 2*a*b^3*c^3*d^2 + a^2*b^2*c*d^4)^2 - (...
 

Mupad [B] (verification not implemented)

Time = 10.60 (sec) , antiderivative size = 9025, normalized size of antiderivative = 31.23 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((3*(20480*a*b^7*c^7*d^4 - 4096*a^4*b^4*c*d^10 - 45056*a^2*b^6*c^5* 
d^6 + 28672*a^3*b^5*c^3*d^8))/(4096*(b^6*c^8 + a^4*b^2*d^8 - 4*a*b^5*c^6*d 
^2 + 6*a^2*b^4*c^4*d^4 - 4*a^3*b^3*c^2*d^6)) - ((c + d*x)^(1/2)*((9*(a^3*d 
^9*(a^3*b^7)^(1/2) + 4*a*b^7*c^7*d^2 + a^4*b^4*c*d^8 + 21*a^2*b^6*c^5*d^4 
- 10*a^3*b^5*c^3*d^6 + 16*b^3*c^6*d^3*(a^3*b^7)^(1/2) + 5*a*b^2*c^4*d^5*(a 
^3*b^7)^(1/2) - 6*a^2*b*c^2*d^7*(a^3*b^7)^(1/2)))/(4096*(a^2*b^12*c^10 - a 
^7*b^7*d^10 - 5*a^3*b^11*c^8*d^2 + 10*a^4*b^10*c^6*d^4 - 10*a^5*b^9*c^4*d^ 
6 + 5*a^6*b^8*c^2*d^8)))^(1/2)*(4096*a*b^8*c^9*d^2 + 4096*a^5*b^4*c*d^10 - 
 16384*a^2*b^7*c^7*d^4 + 24576*a^3*b^6*c^5*d^6 - 16384*a^4*b^5*c^3*d^8))/( 
64*(a^4*d^8 + b^4*c^8 - 4*a*b^3*c^6*d^2 - 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4* 
d^4)))*((9*(a^3*d^9*(a^3*b^7)^(1/2) + 4*a*b^7*c^7*d^2 + a^4*b^4*c*d^8 + 21 
*a^2*b^6*c^5*d^4 - 10*a^3*b^5*c^3*d^6 + 16*b^3*c^6*d^3*(a^3*b^7)^(1/2) + 5 
*a*b^2*c^4*d^5*(a^3*b^7)^(1/2) - 6*a^2*b*c^2*d^7*(a^3*b^7)^(1/2)))/(4096*( 
a^2*b^12*c^10 - a^7*b^7*d^10 - 5*a^3*b^11*c^8*d^2 + 10*a^4*b^10*c^6*d^4 - 
10*a^5*b^9*c^4*d^6 + 5*a^6*b^8*c^2*d^8)))^(1/2) + ((c + d*x)^(1/2)*(9*a^3* 
d^10 + 36*b^3*c^6*d^4 + 81*a*b^2*c^4*d^6 - 54*a^2*b*c^2*d^8))/(64*(a^4*d^8 
 + b^4*c^8 - 4*a*b^3*c^6*d^2 - 4*a^3*b*c^2*d^6 + 6*a^2*b^2*c^4*d^4)))*((9* 
(a^3*d^9*(a^3*b^7)^(1/2) + 4*a*b^7*c^7*d^2 + a^4*b^4*c*d^8 + 21*a^2*b^6*c^ 
5*d^4 - 10*a^3*b^5*c^3*d^6 + 16*b^3*c^6*d^3*(a^3*b^7)^(1/2) + 5*a*b^2*c^4* 
d^5*(a^3*b^7)^(1/2) - 6*a^2*b*c^2*d^7*(a^3*b^7)^(1/2)))/(4096*(a^2*b^12...
 

Reduce [B] (verification not implemented)

Time = 3.84 (sec) , antiderivative size = 2686, normalized size of antiderivative = 9.29 \[ \int \frac {x^3}{\sqrt {c+d x} \left (a-b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s 
qrt(sqrt(b)*sqrt(a)*d - b*c)))*a**4*d**5 - 18*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**2*d**3 - 12*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 - 12*sq 
rt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(s 
qrt(b)*sqrt(a)*d - b*c)))*a**2*b**2*c**4*d + 36*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**2*d**3*x**2 + 6*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*d**5* 
x**4 + 24*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sq 
rt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**3*c**4*d*x**2 - 18*sqrt(a)*sqrt 
(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqr 
t(a)*d - b*c)))*a*b**3*c**2*d**3*x**4 - 12*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 + 6*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 - 30*sqrt(b)*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)))*a**3*b*c**3*d**2 - 12*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...