\(\int \frac {x^2}{(c+d x)^{5/2} (a+b x^2)} \, dx\) [635]

Optimal result

Integrand size = 22, antiderivative size = 571 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a+b x^2\right )} \, dx=-\frac {2 c^2}{3 d \left (b c^2+a d^2\right ) (c+d x)^{3/2}}+\frac {4 a c d}{\left (b c^2+a d^2\right )^2 \sqrt {c+d x}}+\frac {a d \left (3 b c^2-a d^2-2 \sqrt {b} c \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^{5/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {a d \left (3 b c^2-a d^2-2 \sqrt {b} c \sqrt {b c^2+a d^2}\right ) \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^{5/2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {a d \left (3 b c^2-a d^2+2 \sqrt {b} c \sqrt {b c^2+a d^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^{5/2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.47 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a+b x^2\right )} \, dx=\frac {-2 b c^4+2 a c d^2 (5 c+6 d x)}{3 d \left (b c^2+a d^2\right )^2 (c+d x)^{3/2}}-\frac {i \sqrt {a} \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{\left (\sqrt {b} c+i \sqrt {a} d\right )^2 \sqrt {-b c-i \sqrt {a} \sqrt {b} d}}+\frac {i \sqrt {a} \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{\left (\sqrt {b} c-i \sqrt {a} d\right )^2 \sqrt {-b c+i \sqrt {a} \sqrt {b} d}} \] Input:

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

Output:

(-2*b*c^4 + 2*a*c*d^2*(5*c + 6*d*x))/(3*d*(b*c^2 + a*d^2)^2*(c + d*x)^(3/2 
)) - (I*Sqrt[a]*ArcTan[(Sqrt[-(b*c) - I*Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d*x])/ 
(Sqrt[b]*c + I*Sqrt[a]*d)])/((Sqrt[b]*c + I*Sqrt[a]*d)^2*Sqrt[-(b*c) - I*S 
qrt[a]*Sqrt[b]*d]) + (I*Sqrt[a]*ArcTan[(Sqrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d] 
*Sqrt[c + d*x])/(Sqrt[b]*c - I*Sqrt[a]*d)])/((Sqrt[b]*c - I*Sqrt[a]*d)^2*S 
qrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d])
 

Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.34, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {561, 27, 1610, 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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1610

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*(-1/3*(c^2*d^2)/((b*c^2 + a*d^2)*(c + d*x)^(3/2)) + (2*a*c*d^4)/((b*c^2 
 + a*d^2)^2*Sqrt[c + d*x]) - (a*d^4*(3*b*c^2 - a*d^2 - 2*Sqrt[b]*c*Sqrt[b* 
c^2 + a*d^2])*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]] - Sqrt[2]*b^( 
1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]])/(2*Sqrt[2]*b^( 
1/4)*(b*c^2 + a*d^2)^(5/2)*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]) + (a*d^4 
*(3*b*c^2 - a*d^2 - 2*Sqrt[b]*c*Sqrt[b*c^2 + a*d^2])*ArcTanh[(Sqrt[Sqrt[b] 
*c + Sqrt[b*c^2 + a*d^2]] + Sqrt[2]*b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c 
- Sqrt[b*c^2 + a*d^2]]])/(2*Sqrt[2]*b^(1/4)*(b*c^2 + a*d^2)^(5/2)*Sqrt[Sqr 
t[b]*c - Sqrt[b*c^2 + a*d^2]]) + (a*d^4*(3*b*c^2 - a*d^2 + 2*Sqrt[b]*c*Sqr 
t[b*c^2 + a*d^2])*Log[Sqrt[b*c^2 + a*d^2] - Sqrt[2]*b^(1/4)*Sqrt[Sqrt[b]*c 
 + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] + Sqrt[b]*(c + d*x)])/(4*Sqrt[2]*b^( 
1/4)*(b*c^2 + a*d^2)^(5/2)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]) - (a*d^4 
*(3*b*c^2 - a*d^2 + 2*Sqrt[b]*c*Sqrt[b*c^2 + a*d^2])*Log[Sqrt[b*c^2 + a*d^ 
2] + Sqrt[2]*b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] + 
 Sqrt[b]*(c + d*x)])/(4*Sqrt[2]*b^(1/4)*(b*c^2 + a*d^2)^(5/2)*Sqrt[Sqrt[b] 
*c + Sqrt[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 1610

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a 
+ b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 
*a*c, 0] && IntegerQ[q] && IntegerQ[m]
 

rule 2009

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

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 809, normalized size of antiderivative = 1.42

method	result	size
pseudoelliptic	\(-\frac {\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \left (\left (a \,d^{2} \sqrt {b}-2 b c \sqrt {a \,d^{2}+b \,c^{2}}-3 b^{\frac {3}{2}} c^{2}\right ) \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-a \,b^{\frac {3}{2}} c \,d^{2}+2 b^{2} c^{2} \sqrt {a \,d^{2}+b \,c^{2}}+3 b^{\frac {5}{2}} c^{3}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}-\frac {\left (d x +c \right )^{\frac {3}{2}} \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \left (\left (a \,d^{2} \sqrt {b}-2 b c \sqrt {a \,d^{2}+b \,c^{2}}-3 b^{\frac {3}{2}} c^{2}\right ) \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-a \,b^{\frac {3}{2}} c \,d^{2}+2 b^{2} c^{2} \sqrt {a \,d^{2}+b \,c^{2}}+3 b^{\frac {5}{2}} c^{3}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}-\frac {10 \left (a \,d^{2} \left (\frac {6 d x}{5}+c \right ) b^{\frac {3}{2}}-\frac {b^{\frac {5}{2}} c^{3}}{5}\right ) c \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}{3}+d^{2} \left (d x +c \right )^{\frac {3}{2}} \left (\arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )-\arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )\right ) \left (b^{\frac {3}{2}} a \,d^{2}+2 b^{2} \sqrt {a \,d^{2}+b \,c^{2}}\, c -3 c^{2} b^{\frac {5}{2}}\right ) a}{\left (a \,d^{2}+b \,c^{2}\right )^{\frac {5}{2}} b^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, d}\)	\(809\)
derivativedivides	\(\text {Expression too large to display}\)	\(2557\)
default	\(\text {Expression too large to display}\)	\(2557\)

Input:

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

Output:

-(1/4*(d*x+c)^(3/2)*(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/ 
2)-2*b*c)^(1/2)*((a*d^2*b^(1/2)-2*b*c*(a*d^2+b*c^2)^(1/2)-3*b^(3/2)*c^2)*( 
(a*d^2+b*c^2)*b)^(1/2)-a*b^(3/2)*c*d^2+2*b^2*c^2*(a*d^2+b*c^2)^(1/2)+3*b^( 
5/2)*c^3)*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)*ln(b^(1/2)*(d*x+c)-(d*x+ 
c)^(1/2)*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)+(a*d^2+b*c^2)^(1/2))-1/4* 
(d*x+c)^(3/2)*(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b 
*c)^(1/2)*((a*d^2*b^(1/2)-2*b*c*(a*d^2+b*c^2)^(1/2)-3*b^(3/2)*c^2)*((a*d^2 
+b*c^2)*b)^(1/2)-a*b^(3/2)*c*d^2+2*b^2*c^2*(a*d^2+b*c^2)^(1/2)+3*b^(5/2)*c 
^3)*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)*ln(b^(1/2)*(d*x+c)+(d*x+c)^(1/ 
2)*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)+(a*d^2+b*c^2)^(1/2))-10/3*(a*d^ 
2*(6/5*d*x+c)*b^(3/2)-1/5*b^(5/2)*c^3)*c*(a*d^2+b*c^2)^(1/2)*(4*(a*d^2+b*c 
^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)+d^2*(d*x+c)^(3/2) 
*(arctan((-2*b^(1/2)*(d*x+c)^(1/2)+(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2) 
)/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2))-a 
rctan((2*b^(1/2)*(d*x+c)^(1/2)+(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2))/(4 
*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)))*(b^( 
3/2)*a*d^2+2*b^2*(a*d^2+b*c^2)^(1/2)*c-3*c^2*b^(5/2))*a)/(a*d^2+b*c^2)^(5/ 
2)/b^(3/2)/(d*x+c)^(3/2)/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b 
)^(1/2)-2*b*c)^(1/2)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5099 vs. \(2 (468) = 936\).

Time = 0.20 (sec) , antiderivative size = 5099, normalized size of antiderivative = 8.93 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1271 vs. \(2 (468) = 936\).

Time = 0.27 (sec) , antiderivative size = 1271, normalized size of antiderivative = 2.23 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*(3*(2*(b^2*c^4*d + 2*a*b*c^2*d^3 + a^2*d^5)^2*sqrt(-a*b)*a*c*d^3*abs( 
b) - (3*a*b^3*c^6*d^3 + 5*a^2*b^2*c^4*d^5 + a^3*b*c^2*d^7 - a^4*d^9)*abs(b 
^2*c^4*d + 2*a*b*c^2*d^3 + a^2*d^5)*abs(b) - (sqrt(-a*b)*b^5*c^11*d^3 + 3* 
sqrt(-a*b)*a*b^4*c^9*d^5 + 2*sqrt(-a*b)*a^2*b^3*c^7*d^7 - 2*sqrt(-a*b)*a^3 
*b^2*c^5*d^9 - 3*sqrt(-a*b)*a^4*b*c^3*d^11 - sqrt(-a*b)*a^5*c*d^13)*abs(b) 
)*arctan(sqrt(d*x + c)/sqrt(-(b^3*c^5 + 2*a*b^2*c^3*d^2 + a^2*b*c*d^4 + sq 
rt((b^3*c^5 + 2*a*b^2*c^3*d^2 + a^2*b*c*d^4)^2 - (b^3*c^6 + 3*a*b^2*c^4*d^ 
2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*(b^3*c^4 + 2*a*b^2*c^2*d^2 + a^2*b*d^4)))/( 
b^3*c^4 + 2*a*b^2*c^2*d^2 + a^2*b*d^4)))/((b^5*c^9 + 4*a*b^4*c^7*d^2 + 6*a 
^2*b^3*c^5*d^4 + 4*a^3*b^2*c^3*d^6 + a^4*b*c*d^8 - sqrt(-a*b)*b^4*c^8*d - 
4*sqrt(-a*b)*a*b^3*c^6*d^3 - 6*sqrt(-a*b)*a^2*b^2*c^4*d^5 - 4*sqrt(-a*b)*a 
^3*b*c^2*d^7 - sqrt(-a*b)*a^4*d^9)*sqrt(-b^2*c - sqrt(-a*b)*b*d)*abs(b^2*c 
^4*d + 2*a*b*c^2*d^3 + a^2*d^5)) - 3*(2*(b^2*c^4*d + 2*a*b*c^2*d^3 + a^2*d 
^5)^2*sqrt(-a*b)*a*c*d^3*abs(b) + (3*a*b^3*c^6*d^3 + 5*a^2*b^2*c^4*d^5 + a 
^3*b*c^2*d^7 - a^4*d^9)*abs(b^2*c^4*d + 2*a*b*c^2*d^3 + a^2*d^5)*abs(b) - 
(sqrt(-a*b)*b^5*c^11*d^3 + 3*sqrt(-a*b)*a*b^4*c^9*d^5 + 2*sqrt(-a*b)*a^2*b 
^3*c^7*d^7 - 2*sqrt(-a*b)*a^3*b^2*c^5*d^9 - 3*sqrt(-a*b)*a^4*b*c^3*d^11 - 
sqrt(-a*b)*a^5*c*d^13)*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(b^3*c^5 + 2*a*b 
^2*c^3*d^2 + a^2*b*c*d^4 - sqrt((b^3*c^5 + 2*a*b^2*c^3*d^2 + a^2*b*c*d^4)^ 
2 - (b^3*c^6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*(b^3*c^4 + ...
 

Mupad [B] (verification not implemented)

Time = 10.52 (sec) , antiderivative size = 7718, normalized size of antiderivative = 13.52 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((c + d*x)^(1/2)*(320*a^4*b^9*c^12*d^6 - 16*a^2*b^11*c^16*d^2 - 16*a 
^10*b^3*d^18 + 1024*a^5*b^8*c^10*d^8 + 1440*a^6*b^7*c^8*d^10 + 1024*a^7*b^ 
6*c^6*d^12 + 320*a^8*b^5*c^4*d^14) + (-(a*b^3*c^5 + a^2*d^5*(-a^3*b)^(1/2) 
 + 5*b^2*c^4*d*(-a^3*b)^(1/2) - 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c*d^4 - 10*a* 
b*c^2*d^3*(-a^3*b)^(1/2))/(4*(b^6*c^10 + a^5*b*d^10 + 5*a*b^5*c^8*d^2 + 10 
*a^2*b^4*c^6*d^4 + 10*a^3*b^3*c^4*d^6 + 5*a^4*b^2*c^2*d^8)))^(1/2)*(96*a^2 
*b^12*c^18*d^3 - (c + d*x)^(1/2)*(-(a*b^3*c^5 + a^2*d^5*(-a^3*b)^(1/2) + 5 
*b^2*c^4*d*(-a^3*b)^(1/2) - 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c*d^4 - 10*a*b*c^ 
2*d^3*(-a^3*b)^(1/2))/(4*(b^6*c^10 + a^5*b*d^10 + 5*a*b^5*c^8*d^2 + 10*a^2 
*b^4*c^6*d^4 + 10*a^3*b^3*c^4*d^6 + 5*a^4*b^2*c^2*d^8)))^(1/2)*(64*a*b^14* 
c^21*d^2 + 64*a^11*b^4*c*d^22 + 640*a^2*b^13*c^19*d^4 + 2880*a^3*b^12*c^17 
*d^6 + 7680*a^4*b^11*c^15*d^8 + 13440*a^5*b^10*c^13*d^10 + 16128*a^6*b^9*c 
^11*d^12 + 13440*a^7*b^8*c^9*d^14 + 7680*a^8*b^7*c^7*d^16 + 2880*a^9*b^6*c 
^5*d^18 + 640*a^10*b^5*c^3*d^20) - 32*a^11*b^3*d^21 + 736*a^3*b^11*c^16*d^ 
5 + 2432*a^4*b^10*c^14*d^7 + 4480*a^5*b^9*c^12*d^9 + 4928*a^6*b^8*c^10*d^1 
1 + 3136*a^7*b^7*c^8*d^13 + 896*a^8*b^6*c^6*d^15 - 128*a^9*b^5*c^4*d^17 - 
160*a^10*b^4*c^2*d^19))*(-(a*b^3*c^5 + a^2*d^5*(-a^3*b)^(1/2) + 5*b^2*c^4* 
d*(-a^3*b)^(1/2) - 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c*d^4 - 10*a*b*c^2*d^3*(-a 
^3*b)^(1/2))/(4*(b^6*c^10 + a^5*b*d^10 + 5*a*b^5*c^8*d^2 + 10*a^2*b^4*c^6* 
d^4 + 10*a^3*b^3*c^4*d^6 + 5*a^4*b^2*c^2*d^8)))^(1/2)*1i + ((c + d*x)^(...
 

Reduce [B] (verification not implemented)

Time = 22.33 (sec) , antiderivative size = 3434, normalized size of antiderivative = 6.01 \[ \int \frac {x^2}{(c+d x)^{5/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 18*sqrt(c + d*x)*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c* 
*2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) 
 - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqr 
t(2)))*a*b*c**2*d**2 - 18*sqrt(c + d*x)*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b) 
*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c 
**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + 
 b*c**2) - b*c)*sqrt(2)))*a*b*c*d**3*x + 6*sqrt(c + d*x)*sqrt(a*d**2 + b*c 
**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)* 
sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt 
(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*b**2*c**4 + 6*sqrt(c + d*x)*sqr 
t(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan( 
(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c + d* 
x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*b**2*c**3*d*x - 6 
*sqrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*a 
tan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c 
+ d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a**2*c*d**4 - 
 6*sqrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2) 
*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt( 
c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a**2*d**5*x 
 + 6*sqrt(b)*sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sq...