\(\int \frac {x^2}{(c+d x)^{5/2} (a-b x^2)^2} \, dx\) [681]

Optimal result

Integrand size = 23, antiderivative size = 308 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=-\frac {2 c^2 d}{3 \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}-\frac {4 c d \left (b c^2+a d^2\right )}{\left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {\sqrt {c+d x} \left (a^2 d^3-b^2 c^3 x+3 a b c d (c-d x)\right )}{2 \left (b c^2-a d^2\right )^3 \left (a-b x^2\right )}+\frac {\left (2 \sqrt {b} c+3 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{4 \sqrt {a} \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right )^{7/2}}-\frac {\left (2 \sqrt {b} c-3 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{4 \sqrt {a} \sqrt [4]{b} \left (\sqrt {b} c+\sqrt {a} d\right )^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.16 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\frac {1}{12} \left (\frac {2 a^2 d^3 \left (23 c^2+30 c d x+3 d^2 x^2\right )-2 b^2 c^3 x \left (3 c^2+34 c d x+27 d^2 x^2\right )+2 a b c d \left (37 c^3+33 c^2 d x-29 c d^2 x^2-33 d^3 x^3\right )}{\left (b c^2-a d^2\right )^3 (c+d x)^{3/2} \left (-a+b x^2\right )}-\frac {3 \left (2 \sqrt {b} c-3 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \left (\sqrt {b} c+\sqrt {a} d\right )^3 \sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {3 \left (2 \sqrt {b} c+3 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {a} \left (\sqrt {b} c-\sqrt {a} d\right )^3 \sqrt {-b c+\sqrt {a} \sqrt {b} d}}\right ) \] Input:

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

Output:

((2*a^2*d^3*(23*c^2 + 30*c*d*x + 3*d^2*x^2) - 2*b^2*c^3*x*(3*c^2 + 34*c*d* 
x + 27*d^2*x^2) + 2*a*b*c*d*(37*c^3 + 33*c^2*d*x - 29*c*d^2*x^2 - 33*d^3*x 
^3))/((b*c^2 - a*d^2)^3*(c + d*x)^(3/2)*(-a + b*x^2)) - (3*(2*Sqrt[b]*c - 
3*Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) - Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt 
[b]*c + Sqrt[a]*d)])/(Sqrt[a]*(Sqrt[b]*c + Sqrt[a]*d)^3*Sqrt[-(b*c) - Sqrt 
[a]*Sqrt[b]*d]) + (3*(2*Sqrt[b]*c + 3*Sqrt[a]*d)*ArcTan[(Sqrt[-(b*c) + Sqr 
t[a]*Sqrt[b]*d]*Sqrt[c + d*x])/(Sqrt[b]*c - Sqrt[a]*d)])/(Sqrt[a]*(Sqrt[b] 
*c - Sqrt[a]*d)^3*Sqrt[-(b*c) + Sqrt[a]*Sqrt[b]*d]))/12
 

Rubi [A] (verified)

Time = 2.66 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.39, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {561, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1673

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2195

\(\displaystyle \frac {2 \left (-\frac {d^4 \int \left (-\frac {4 a b c^2}{\left (b c^2-a d^2\right ) (c+d x)^2}-\frac {8 a b \left (b c^2+a d^2\right ) c}{\left (b c^2-a d^2\right )^2 (c+d x)}+\frac {a b \left (-11 b^2 c^4-26 a b d^2 c^2+b \left (9 b c^2+11 a d^2\right ) (c+d x) c-3 a^2 d^4\right )}{\left (b c^2-a d^2\right )^2 \left (b c^2-2 b (c+d x) c-a d^2+b (c+d x)^2\right )}\right )d\sqrt {c+d x}}{4 a b \left (b c^2-a d^2\right )}-\frac {d^2 \sqrt {c+d x} \left (a^2 d^4-b c (c+d x) \left (3 a d^2+b c^2\right )+6 a b c^2 d^2+b^2 c^4\right )}{4 \left (b c^2-a d^2\right )^3 \left (a-\frac {b c^2}{d^2}+\frac {2 b c (c+d x)}{d^2}-\frac {b (c+d x)^2}{d^2}\right )}\right )}{d^3}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*(-1/4*(d^2*Sqrt[c + d*x]*(b^2*c^4 + 6*a*b*c^2*d^2 + a^2*d^4 - b*c*(b*c^ 
2 + 3*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^2)/(3*(b*c^2 - a*d^2)* 
(c + d*x)^(3/2)) + (8*a*b*c*(b*c^2 + a*d^2))/((b*c^2 - a*d^2)^2*Sqrt[c + d 
*x]) - (Sqrt[a]*b^(3/4)*(Sqrt[b]*c + Sqrt[a]*d)*(2*Sqrt[b]*c + 3*Sqrt[a]*d 
)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*d*(Sqrt 
[b]*c - Sqrt[a]*d)^(5/2)) + (Sqrt[a]*b^(3/4)*(2*Sqrt[b]*c - 3*Sqrt[a]*d)*( 
Sqrt[b]*c - Sqrt[a]*d)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sq 
rt[a]*d]])/(2*d*(Sqrt[b]*c + Sqrt[a]*d)^(5/2))))/(4*a*b*(b*c^2 - a*d^2)))) 
/d^3
 

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 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 = 0.81 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(2 d \left (\frac {\frac {\left (-\frac {3}{4} a b c \,d^{2}-\frac {1}{4} c^{3} b^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {1}{4} a^{2} d^{4}+\frac {3}{2} b \,c^{2} d^{2} a +\frac {1}{4} b^{2} c^{4}\right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (-\frac {\left (-3 a^{2} d^{4}-15 b \,c^{2} d^{2} a -2 b^{2} c^{4}+11 \sqrt {a b \,d^{2}}\, a c \,d^{2}+9 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (3 a^{2} d^{4}+15 b \,c^{2} d^{2} a +2 b^{2} c^{4}+11 \sqrt {a b \,d^{2}}\, a c \,d^{2}+9 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}-\frac {c^{2}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 c \left (a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}\right )\)	\(399\)
default	\(2 d \left (\frac {\frac {\left (-\frac {3}{4} a b c \,d^{2}-\frac {1}{4} c^{3} b^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}+\left (\frac {1}{4} a^{2} d^{4}+\frac {3}{2} b \,c^{2} d^{2} a +\frac {1}{4} b^{2} c^{4}\right ) \sqrt {d x +c}}{-b \left (d x +c \right )^{2}+2 b c \left (d x +c \right )+a \,d^{2}-b \,c^{2}}+\frac {b \left (-\frac {\left (-3 a^{2} d^{4}-15 b \,c^{2} d^{2} a -2 b^{2} c^{4}+11 \sqrt {a b \,d^{2}}\, a c \,d^{2}+9 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}+\frac {\left (3 a^{2} d^{4}+15 b \,c^{2} d^{2} a +2 b^{2} c^{4}+11 \sqrt {a b \,d^{2}}\, a c \,d^{2}+9 \sqrt {a b \,d^{2}}\, b \,c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}}{\left (a \,d^{2}-b \,c^{2}\right )^{3}}-\frac {c^{2}}{3 \left (a \,d^{2}-b \,c^{2}\right )^{2} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 c \left (a \,d^{2}+b \,c^{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {d x +c}}\right )\)	\(399\)
pseudoelliptic	\(\frac {\frac {3 d b \left (d x +c \right )^{\frac {3}{2}} \left (\left (\frac {11}{3} a \,d^{2} c +3 b \,c^{3}\right ) \sqrt {a b \,d^{2}}+a^{2} d^{4}+5 b \,c^{2} d^{2} a +\frac {2 b^{2} c^{4}}{3}\right ) \left (-b \,x^{2}+a \right ) \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{4}+\frac {3 \left (d \left (\left (-\frac {11}{3} a \,d^{2} c -3 b \,c^{3}\right ) \sqrt {a b \,d^{2}}+a^{2} d^{4}+5 b \,c^{2} d^{2} a +\frac {2 b^{2} c^{4}}{3}\right ) b \left (d x +c \right )^{\frac {3}{2}} \left (-b \,x^{2}+a \right ) \operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )+\frac {46 \left (\frac {3 a^{2} x^{2} d^{5}}{23}+\frac {30 x \left (-\frac {11 b \,x^{2}}{10}+a \right ) a c \,d^{4}}{23}+a \,c^{2} \left (-\frac {29 b \,x^{2}}{23}+a \right ) d^{3}+\frac {33 x b \,c^{3} \left (-\frac {9 b \,x^{2}}{11}+a \right ) d^{2}}{23}+\frac {37 \left (-\frac {34 b \,x^{2}}{37}+a \right ) b \,c^{4} d}{23}-\frac {3 c^{5} b^{2} x}{23}\right ) \sqrt {a b \,d^{2}}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}{9}\right ) \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}{4}}{\left (-b \,x^{2}+a \right ) \left (a \,d^{2}-b \,c^{2}\right )^{3} \sqrt {a b \,d^{2}}\, \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}\, \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}\, \left (d x +c \right )^{\frac {3}{2}}}\)	\(410\)

Input:

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

Output:

2*d*(1/(a*d^2-b*c^2)^3*(((-3/4*a*b*c*d^2-1/4*c^3*b^2)*(d*x+c)^(3/2)+(1/4*a 
^2*d^4+3/2*b*c^2*d^2*a+1/4*b^2*c^4)*(d*x+c)^(1/2))/(-b*(d*x+c)^2+2*b*c*(d* 
x+c)+a*d^2-b*c^2)+1/4*b*(-1/2*(-3*a^2*d^4-15*b*c^2*d^2*a-2*b^2*c^4+11*(a*b 
*d^2)^(1/2)*a*c*d^2+9*(a*b*d^2)^(1/2)*b*c^3)/(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) 
)+1/2*(3*a^2*d^4+15*b*c^2*d^2*a+2*b^2*c^4+11*(a*b*d^2)^(1/2)*a*c*d^2+9*(a* 
b*d^2)^(1/2)*b*c^3)/(a*b*d^2)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2)*arcta 
n(b*(d*x+c)^(1/2)/((-b*c+(a*b*d^2)^(1/2))*b)^(1/2))))-1/3*c^2/(a*d^2-b*c^2 
)^2/(d*x+c)^(3/2)+2*c*(a*d^2+b*c^2)/(a*d^2-b*c^2)^3/(d*x+c)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8457 vs. \(2 (248) = 496\).

Time = 4.46 (sec) , antiderivative size = 8457, normalized size of antiderivative = 27.46 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate(x^2/((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. 2028 vs. \(2 (248) = 496\).

Time = 0.41 (sec) , antiderivative size = 2028, normalized size of antiderivative = 6.58 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(3*((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*(9*sq 
rt(a*b)*a*b*c^3*d^3 + 11*sqrt(a*b)*a^2*c*d^5)*abs(b) - (11*a*b^5*c^10*d^3 
- 7*a^2*b^4*c^8*d^5 - 42*a^3*b^3*c^6*d^7 + 58*a^4*b^2*c^4*d^9 - 17*a^5*b*c 
^2*d^11 - 3*a^6*d^13)*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) + (2*sqrt(a*b)*b^8*c^17*d^3 + 3*sqrt(a*b)*a*b^7*c^15*d^5 - 
 57*sqrt(a*b)*a^2*b^6*c^13*d^7 + 167*sqrt(a*b)*a^3*b^5*c^11*d^9 - 225*sqrt 
(a*b)*a^4*b^4*c^9*d^11 + 153*sqrt(a*b)*a^5*b^3*c^7*d^13 - 43*sqrt(a*b)*a^6 
*b^2*c^5*d^15 - 3*sqrt(a*b)*a^7*b*c^3*d^17 + 3*sqrt(a*b)*a^8*c*d^19)*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)))/((a*b^ 
7*c^13 - 6*a^2*b^6*c^11*d^2 + 15*a^3*b^5*c^9*d^4 - 20*a^4*b^4*c^7*d^6 + 15 
*a^5*b^3*c^5*d^8 - 6*a^6*b^2*c^3*d^10 + a^7*b*c*d^12 - sqrt(a*b)*a*b^6*c^1 
2*d + 6*sqrt(a*b)*a^2*b^5*c^10*d^3 - 15*sqrt(a*b)*a^3*b^4*c^8*d^5 + 20*sqr 
t(a*b)*a^4*b^3*c^6*d^7 - 15*sqrt(a*b)*a^5*b^2*c^4*d^9 + 6*sqrt(a*b)*a^6*b* 
c^2*d^11 - sqrt(a*b)*a^7*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)) - 3*((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*(9*sqrt(a*b)*a*b*c^3*d^3 + 11*sq...
 

Mupad [B] (verification not implemented)

Time = 13.32 (sec) , antiderivative size = 12473, normalized size of antiderivative = 40.50 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((-(4*a*b^5*c^9 - 9*a^4*d^9*(a^3*b)^(1/2) - 40*b^4*c^8*d*(a^3*b)^(1/ 
2) + 177*a^2*b^4*c^7*d^2 + 749*a^3*b^3*c^5*d^4 + 595*a^4*b^2*c^3*d^6 + 75* 
a^5*b*c*d^8 - 819*a^2*b^2*c^4*d^5*(a^3*b)^(1/2) - 455*a*b^3*c^6*d^3*(a^3*b 
)^(1/2) - 277*a^3*b*c^2*d^7*(a^3*b)^(1/2))/(64*(a^9*b*d^14 - a^2*b^8*c^14 
+ 7*a^3*b^7*c^12*d^2 - 21*a^4*b^6*c^10*d^4 + 35*a^5*b^5*c^8*d^6 - 35*a^6*b 
^4*c^6*d^8 + 21*a^7*b^3*c^4*d^10 - 7*a^8*b^2*c^2*d^12)))^(1/2)*((c + d*x)^ 
(1/2)*(-(4*a*b^5*c^9 - 9*a^4*d^9*(a^3*b)^(1/2) - 40*b^4*c^8*d*(a^3*b)^(1/2 
) + 177*a^2*b^4*c^7*d^2 + 749*a^3*b^3*c^5*d^4 + 595*a^4*b^2*c^3*d^6 + 75*a 
^5*b*c*d^8 - 819*a^2*b^2*c^4*d^5*(a^3*b)^(1/2) - 455*a*b^3*c^6*d^3*(a^3*b) 
^(1/2) - 277*a^3*b*c^2*d^7*(a^3*b)^(1/2))/(64*(a^9*b*d^14 - a^2*b^8*c^14 + 
 7*a^3*b^7*c^12*d^2 - 21*a^4*b^6*c^10*d^4 + 35*a^5*b^5*c^8*d^6 - 35*a^6*b^ 
4*c^6*d^8 + 21*a^7*b^3*c^4*d^10 - 7*a^8*b^2*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 - 6150 
144*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^10*b^10*c^13*d^20 + 61 
50144*a^11*b^9*c^11*d^22 - 2795520*a^12*b^8*c^9*d^24 + 931840*a^13*b^7*c^7 
*d^26 - 215040*a^14*b^6*c^5*d^28 + 30720*a^15*b^5*c^3*d^30) - 768*a^15*b^3 
*d^31 - 2816*a*b^17*c^28*d^3 + 27136*a^2*b^16*c^26*d^5 - 106752*a^3*b^15*c 
^24*d^7 + 189440*a^4*b^14*c^22*d^9 + 19712*a^5*b^13*c^20*d^11 - 895488*...
 

Reduce [B] (verification not implemented)

Time = 1.16 (sec) , antiderivative size = 4564, normalized size of antiderivative = 14.82 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a-b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(84*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*b*c**2*d**4 + 84*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*b*c*d**5*x + 144*sqrt(a)*sqrt(c + 
 d*x)*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**2 + 144*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**2*c**3*d**3*x - 84*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**2*c**2*d**4*x**2 - 84*sqrt(a)*sqrt(c + d*x)*sqrt(sq 
rt(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**5*x**3 + 12*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**3*c**6 + 12*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**3* 
c**5*d*x - 144*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((s 
qrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*a*b**3*c**4*d**2* 
x**2 - 144*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**3*c**3*d**3*x**3 
 - 12*sqrt(a)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c ...