\(\int \frac {x^3}{(c+d x)^{5/2} (a-b x^2)^2} \, dx\) [680]

Optimal result

Integrand size = 23, antiderivative size = 295 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\frac {2 c^3}{3 \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}+\frac {2 c^2 \left (b c^2+3 a d^2\right )}{\left (b c^2-a d^2\right )^3 \sqrt {c+d x}}+\frac {a \sqrt {c+d x} \left (b c^2 (c-3 d x)+a d^2 (3 c-d x)\right )}{2 \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )}-\frac {\left (4 \sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 b^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right )^{7/2}}-\frac {\left (4 \sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right )^{7/2}} \] Output:

2/3*c^3/(-a*d^2+b*c^2)^2/(d*x+c)^(3/2)+2*c^2*(3*a*d^2+b*c^2)/(-a*d^2+b*c^2 
)^3/(d*x+c)^(1/2)+1/2*a*(d*x+c)^(1/2)*(b*c^2*(-3*d*x+c)+a*d^2*(-d*x+3*c))/ 
(-a*d^2+b*c^2)^3/(-b*x^2+a)-1/4*(4*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))/b^(3/4)/(b^(1/2)*c-a^(1/2)*d)^(7/ 
2)-1/4*(4*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))/b^(3/4)/(b^(1/2)*c+a^(1/2)*d)^(7/2)
 

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.22 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\frac {1}{12} \left (\frac {2 \left (4 b^2 c^4 x^2 (4 c+3 d x)+a^2 d^2 \left (-41 c^3-51 c^2 d x-3 c d^2 x^2+3 d^3 x^3\right )+a b c^2 \left (-19 c^3-9 c^2 d x+47 c d^2 x^2+45 d^3 x^3\right )\right )}{\left (b c^2-a d^2\right )^3 (c+d x)^{3/2} \left (-a+b x^2\right )}-\frac {3 \left (4 \sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right )^3 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}-\frac {3 \left (4 \sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right )^3 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}\right ) \] Input:

Integrate[x^3/((c + d*x)^(5/2)*(a - b*x^2)^2),x]
 

Output:

((2*(4*b^2*c^4*x^2*(4*c + 3*d*x) + a^2*d^2*(-41*c^3 - 51*c^2*d*x - 3*c*d^2 
*x^2 + 3*d^3*x^3) + a*b*c^2*(-19*c^3 - 9*c^2*d*x + 47*c*d^2*x^2 + 45*d^3*x 
^3)))/((b*c^2 - a*d^2)^3*(c + d*x)^(3/2)*(-a + b*x^2)) - (3*(4*Sqrt[b]*c - 
 Sqrt[a]*d)*ArcTan[(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)^3*Sqrt[-(b*c) - Sqrt[ 
a]*Sqrt[b]*d]) - (3*(4*Sqrt[b]*c + Sqrt[a]*d)*ArcTan[(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)^3*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/12
 

Rubi [A] (verified)

Time = 2.60 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.37, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {561, 25, 27, 1673, 27, 2195, 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.

\(\displaystyle \int \frac {x^3}{\left (a-b x^2\right )^2 (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1673

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2195

\(\displaystyle -\frac {2 \left (-\frac {d^4 \int \left (-\frac {4 a b c^3}{\left (b c^2-a d^2\right ) (c+d x)^2}-\frac {4 a b \left (b c^2+3 a d^2\right ) c^2}{\left (b c^2-a d^2\right )^2 (c+d x)}+\frac {a b \left (\left (4 b^2 c^4+15 a b d^2 c^2+a^2 d^4\right ) (c+d x)-4 c \left (b^2 c^4+7 a b d^2 c^2+2 a^2 d^4\right )\right )}{\left (b c^2-a d^2\right )^2 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {a d^4 \sqrt {c+d x} \left (4 c \left (a d^2+b c^2\right )-(c+d x) \left (a d^2+3 b c^2\right )\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^4}\)

\(\Big \downarrow \) 2009

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

Input:

Int[x^3/((c + d*x)^(5/2)*(a - b*x^2)^2),x]
 

Output:

(-2*(-1/4*(a*d^4*Sqrt[c + d*x]*(4*c*(b*c^2 + a*d^2) - (3*b*c^2 + a*d^2)*(c 
 + d*x)))/((b*c^2 - a*d^2)^3*(a - (b*c^2)/d^2 + (2*b*c*(c + d*x))/d^2 - (b 
*(c + d*x)^2)/d^2)) - (d^4*((4*a*b*c^3)/(3*(b*c^2 - a*d^2)*(c + d*x)^(3/2) 
) + (4*a*b*c^2*(b*c^2 + 3*a*d^2))/((b*c^2 - a*d^2)^2*Sqrt[c + d*x]) - (a*b 
^(1/4)*(Sqrt[b]*c + Sqrt[a]*d)*(4*Sqrt[b]*c + Sqrt[a]*d)*ArcTanh[(b^(1/4)* 
Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*(Sqrt[b]*c - Sqrt[a]*d)^(5 
/2)) - (a*b^(1/4)*(Sqrt[b]*c - Sqrt[a]*d)*(4*Sqrt[b]*c - Sqrt[a]*d)*ArcTan 
h[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]*d]])/(2*(Sqrt[b]*c + Sq 
rt[a]*d)^(5/2))))/(4*a*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 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 1673

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

rule 2009

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

rule 2195

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ 
Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; 
FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.43

method	result	size
pseudoelliptic	\(-\frac {7 \left (\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d x +c \right )^{\frac {3}{2}} \left (\frac {\left (a^{2} d^{4}+15 b \,c^{2} d^{2} a +4 b^{2} c^{4}\right ) \sqrt {a b \,d^{2}}}{7}+c a \,d^{2} b \left (a \,d^{2}+\frac {13 b \,c^{2}}{7}\right )\right ) \left (-b \,x^{2}+a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (\left (\frac {\left (-a^{2} d^{4}-15 b \,c^{2} d^{2} a -4 b^{2} c^{4}\right ) \sqrt {a b \,d^{2}}}{7}+c a \,d^{2} b \left (a \,d^{2}+\frac {13 b \,c^{2}}{7}\right )\right ) \left (d x +c \right )^{\frac {3}{2}} \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {82 \left (-\frac {16 x^{2} \left (\frac {3 d x}{4}+c \right ) c^{4} b^{2}}{41}+\frac {19 a \left (-\frac {45}{19} d^{3} x^{3}-\frac {47}{19} c \,d^{2} x^{2}+\frac {9}{19} c^{2} d x +c^{3}\right ) c^{2} b}{41}+a^{2} d^{2} \left (c^{3}-\frac {3}{41} d^{3} x^{3}+\frac {3}{41} c \,d^{2} x^{2}+\frac {51}{41} c^{2} d x \right )\right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}}{21}\right )\right )}{4 \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {a b \,d^{2}}\, \left (d x +c \right )^{\frac {3}{2}} \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (-b \,x^{2}+a \right ) \left (a \,d^{2}-b \,c^{2}\right )^{3}}\)	\(423\)
derivativedivides	\(-\frac {2 c^{2} \left (3 a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}+\frac {2 c^{3}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{2} d^{4}-\frac {3}{4} b \,c^{2} d^{2} a \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (a^{2} c \,d^{4}+a \,c^{3} d^{2} b \right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (\frac {\left (7 a^{2} c \,d^{4} b +13 a \,b^{2} c^{3} d^{2}+\sqrt {a b \,d^{2}}\, a^{2} d^{4}+15 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+4 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-7 a^{2} c \,d^{4} b -13 a \,b^{2} c^{3} d^{2}+\sqrt {a b \,d^{2}}\, a^{2} d^{4}+15 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+4 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}\)	\(431\)
default	\(-\frac {2 c^{2} \left (3 a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}+\frac {2 c^{3}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{2} d^{4}-\frac {3}{4} b \,c^{2} d^{2} a \right ) \left (d x +c \right )^{\frac {3}{2}}+\left (a^{2} c \,d^{4}+a \,c^{3} d^{2} b \right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (\frac {\left (7 a^{2} c \,d^{4} b +13 a \,b^{2} c^{3} d^{2}+\sqrt {a b \,d^{2}}\, a^{2} d^{4}+15 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+4 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\left (-7 a^{2} c \,d^{4} b -13 a \,b^{2} c^{3} d^{2}+\sqrt {a b \,d^{2}}\, a^{2} d^{4}+15 \sqrt {a b \,d^{2}}\, a b \,c^{2} d^{2}+4 \sqrt {a b \,d^{2}}\, b^{2} c^{4}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}\)	\(431\)

Input:

int(x^3/(d*x+c)^(5/2)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-7/4/((b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(a*b*d^2)^(1/2)/(d*x+c)^(3/2)*(((b*c+ 
(a*b*d^2)^(1/2))*b)^(1/2)*(d*x+c)^(3/2)*(1/7*(a^2*d^4+15*a*b*c^2*d^2+4*b^2 
*c^4)*(a*b*d^2)^(1/2)+c*a*d^2*b*(a*d^2+13/7*b*c^2))*(-b*x^2+a)*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)*((1/7*(-a^2*d^4-15*a*b*c^2*d^2-4*b^2*c^4)*(a*b*d^2)^(1/2)+c*a*d^2*b*( 
a*d^2+13/7*b*c^2))*(d*x+c)^(3/2)*(-b*x^2+a)*arctanh(b*(d*x+c)^(1/2)/((b*c+ 
(a*b*d^2)^(1/2))*b)^(1/2))+82/21*(-16/41*x^2*(3/4*d*x+c)*c^4*b^2+19/41*a*( 
-45/19*d^3*x^3-47/19*c*d^2*x^2+9/19*c^2*d*x+c^3)*c^2*b+a^2*d^2*(c^3-3/41*d 
^3*x^3+3/41*c*d^2*x^2+51/41*c^2*d*x))*((b*c+(a*b*d^2)^(1/2))*b)^(1/2)*(a*b 
*d^2)^(1/2)))/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)/(-b*x^2+a)/(a*d^2-b*c^2)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8506 vs. \(2 (239) = 478\).

Time = 5.13 (sec) , antiderivative size = 8506, normalized size of antiderivative = 28.83 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^3/(d*x+c)^(5/2)/(-b*x^2+a)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**3/(d*x+c)**(5/2)/(-b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b x^{2} - a\right )}^{2} {\left (d x + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^3/(d*x+c)^(5/2)/(-b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate(x^3/((b*x^2 - a)^2*(d*x + c)^(5/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1988 vs. \(2 (239) = 478\).

Time = 0.43 (sec) , antiderivative size = 1988, normalized size of antiderivative = 6.74 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^3/(d*x+c)^(5/2)/(-b*x^2+a)^2,x, algorithm="giac")
 

Output:

-1/4*((b^3*c^6*d - 3*a*b^2*c^4*d^3 + 3*a^2*b*c^2*d^5 - a^3*d^7)^2*(4*sqrt( 
a*b)*b^2*c^4 + 15*sqrt(a*b)*a*b*c^2*d^2 + sqrt(a*b)*a^2*d^4)*abs(b) - 4*(b 
^6*c^11 + 4*a*b^5*c^9*d^2 - 16*a^2*b^4*c^7*d^4 + 14*a^3*b^3*c^5*d^6 - a^4* 
b^2*c^3*d^8 - 2*a^5*b*c*d^10)*abs(b^3*c^6*d - 3*a*b^2*c^4*d^3 + 3*a^2*b*c^ 
2*d^5 - a^3*d^7)*abs(b) + (13*sqrt(a*b)*b^8*c^16*d^2 - 71*sqrt(a*b)*a*b^7* 
c^14*d^4 + 153*sqrt(a*b)*a^2*b^6*c^12*d^6 - 155*sqrt(a*b)*a^3*b^5*c^10*d^8 
 + 55*sqrt(a*b)*a^4*b^4*c^8*d^10 + 27*sqrt(a*b)*a^5*b^3*c^6*d^12 - 29*sqrt 
(a*b)*a^6*b^2*c^4*d^14 + 7*sqrt(a*b)*a^7*b*c^2*d^16)*abs(b))*arctan(sqrt(d 
*x + c)/sqrt(-(b^4*c^7 - 3*a*b^3*c^5*d^2 + 3*a^2*b^2*c^3*d^4 - a^3*b*c*d^6 
 + sqrt((b^4*c^7 - 3*a*b^3*c^5*d^2 + 3*a^2*b^2*c^3*d^4 - a^3*b*c*d^6)^2 - 
(b^4*c^8 - 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^4 - 4*a^3*b*c^2*d^6 + a^4*d^8 
)*(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^6 - 
 3*a*b^3*c^4*d^2 + 3*a^2*b^2*c^2*d^4 - a^3*b*d^6)))/((b^8*c^13 - 6*a*b^7*c 
^11*d^2 + 15*a^2*b^6*c^9*d^4 - 20*a^3*b^5*c^7*d^6 + 15*a^4*b^4*c^5*d^8 - 6 
*a^5*b^3*c^3*d^10 + a^6*b^2*c*d^12 - sqrt(a*b)*b^7*c^12*d + 6*sqrt(a*b)*a* 
b^6*c^10*d^3 - 15*sqrt(a*b)*a^2*b^5*c^8*d^5 + 20*sqrt(a*b)*a^3*b^4*c^6*d^7 
 - 15*sqrt(a*b)*a^4*b^3*c^4*d^9 + 6*sqrt(a*b)*a^5*b^2*c^2*d^11 - sqrt(a*b) 
*a^6*b*d^13)*sqrt(-b^2*c - sqrt(a*b)*b*d)*abs(b^3*c^6*d - 3*a*b^2*c^4*d^3 
+ 3*a^2*b*c^2*d^5 - a^3*d^7)) + 1/4*((b^3*c^6*d - 3*a*b^2*c^4*d^3 + 3*a^2* 
b*c^2*d^5 - a^3*d^7)^2*(4*sqrt(a*b)*b^2*c^4 + 15*sqrt(a*b)*a*b*c^2*d^2 ...
 

Mupad [B] (verification not implemented)

Time = 13.73 (sec) , antiderivative size = 12396, normalized size of antiderivative = 42.02 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(x^3/((a - b*x^2)^2*(c + d*x)^(5/2)),x)
 

Output:

((2*c^3)/(3*(a*d^2 - b*c^2)) + ((c + d*x)^3*(a^2*d^4 + 4*b^2*c^4 + 15*a*b* 
c^2*d^2))/(2*(a*d^2 - b*c^2)^3) - (2*(c + d*x)^2*(5*b^2*c^5 + 3*a^2*c*d^4 
+ 22*a*b*c^3*d^2))/(3*(a*d^2 - b*c^2)^3) - (2*c^2*(9*a*d^2 + b*c^2)*(c + d 
*x))/(3*(a*d^2 - b*c^2)^2))/((a*d^2 - b*c^2)*(c + d*x)^(3/2) - b*(c + d*x) 
^(7/2) + 2*b*c*(c + d*x)^(5/2)) - atan(((((16*b^6*c^9 + a^4*d^9*(a*b^3)^(1 
/2) + 393*a*b^5*c^7*d^2 + 15*a^4*b^2*c*d^8 + 120*b^4*c^8*d*(a*b^3)^(1/2) + 
 861*a^2*b^4*c^5*d^4 + 315*a^3*b^3*c^3*d^6 + 651*a^2*b^2*c^4*d^5*(a*b^3)^( 
1/2) + 735*a*b^3*c^6*d^3*(a*b^3)^(1/2) + 93*a^3*b*c^2*d^7*(a*b^3)^(1/2))/( 
64*(b^10*c^14 - a^7*b^3*d^14 - 7*a*b^9*c^12*d^2 + 21*a^2*b^8*c^10*d^4 - 35 
*a^3*b^7*c^8*d^6 + 35*a^4*b^6*c^6*d^8 - 21*a^5*b^5*c^4*d^10 + 7*a^6*b^4*c^ 
2*d^12)))^(1/2)*((c + d*x)^(1/2)*((16*b^6*c^9 + a^4*d^9*(a*b^3)^(1/2) + 39 
3*a*b^5*c^7*d^2 + 15*a^4*b^2*c*d^8 + 120*b^4*c^8*d*(a*b^3)^(1/2) + 861*a^2 
*b^4*c^5*d^4 + 315*a^3*b^3*c^3*d^6 + 651*a^2*b^2*c^4*d^5*(a*b^3)^(1/2) + 7 
35*a*b^3*c^6*d^3*(a*b^3)^(1/2) + 93*a^3*b*c^2*d^7*(a*b^3)^(1/2))/(64*(b^10 
*c^14 - a^7*b^3*d^14 - 7*a*b^9*c^12*d^2 + 21*a^2*b^8*c^10*d^4 - 35*a^3*b^7 
*c^8*d^6 + 35*a^4*b^6*c^6*d^8 - 21*a^5*b^5*c^4*d^10 + 7*a^6*b^4*c^2*d^12)) 
)^(1/2)*(2048*a*b^19*c^31*d^2 - 2048*a^16*b^4*c*d^32 - 30720*a^2*b^18*c^29 
*d^4 + 215040*a^3*b^17*c^27*d^6 - 931840*a^4*b^16*c^25*d^8 + 2795520*a^5*b 
^15*c^23*d^10 - 6150144*a^6*b^14*c^21*d^12 + 10250240*a^7*b^13*c^19*d^14 - 
 13178880*a^8*b^12*c^17*d^16 + 13178880*a^9*b^11*c^15*d^18 - 10250240*a...
 

Reduce [B] (verification not implemented)

Time = 1.16 (sec) , antiderivative size = 4492, normalized size of antiderivative = 15.23 \[ \int \frac {x^3}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^3/(d*x+c)^(5/2)/(-b*x^2+a)^2,x)
 

Output:

( - 6*sqrt(a)*sqrt(c + d*x)*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*c*d**5 - 6*sqrt(a)*sq 
rt(c + d*x)*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*d**6*x - 132*sqrt(a)*sqrt(c + d*x)*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*c**3*d**3 - 132*sqrt(a)*sqrt(c + d*x)*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*c**2*d**4*x + 6*sqrt(a)*sqrt(c + d*x)*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*c*d**5*x**2 + 6*sqrt(a)*sqrt(c + d*x)*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*d* 
*6*x**3 - 102*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sq 
rt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**2*c**5*d - 10 
2*sqrt(a)*sqrt(c + d*x)*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**2*c**4*d**2*x + 132*sqrt( 
a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqr 
t(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**2*c**3*d**3*x**2 + 132*sqrt(a)*s 
qrt(c + d*x)*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**2*c**2*d**4*x**3 + 102*sqrt(a)*sqrt( 
c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*...