\(\int \frac {x^2 (c+d x^2)^{3/2}}{(a+b x^2)^2} \, dx\) [998]

Optimal result

Integrand size = 24, antiderivative size = 149 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {d x \sqrt {c+d x^2}}{b^2}-\frac {x \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {(b c-4 a d) \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^3}+\frac {\sqrt {d} (3 b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1365\) vs. \(2(149)=298\).

Time = 6.15 (sec) , antiderivative size = 1365, normalized size of antiderivative = 9.16 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((b*x*(-(b*c) + 2*a*d + b*d*x^2)*(64*c^4 + 144*c^3*d*x^2 + 104*c^2*d^2*x^4 
 + 25*c*d^3*x^6 + d^4*x^8 - 64*c^(7/2)*Sqrt[c + d*x^2] - 112*c^(5/2)*d*x^2 
*Sqrt[c + d*x^2] - 56*c^(3/2)*d^2*x^4*Sqrt[c + d*x^2] - 7*Sqrt[c]*d^3*x^6* 
Sqrt[c + d*x^2]))/((a + b*x^2)*(-64*c^(7/2) - 112*c^(5/2)*d*x^2 - 56*c^(3/ 
2)*d^2*x^4 - 7*Sqrt[c]*d^3*x^6 + 64*c^3*Sqrt[c + d*x^2] + 80*c^2*d*x^2*Sqr 
t[c + d*x^2] + 24*c*d^2*x^4*Sqrt[c + d*x^2] + d^3*x^6*Sqrt[c + d*x^2])) + 
(b^(3/2)*c^(3/2)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt 
[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/(Sqrt[a]*S 
qrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (4*Sqrt[a]*Sqrt[b] 
*Sqrt[c]*d*Sqrt[b*c - a*d]*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sq 
rt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/Sqrt[2*b*c - a*d 
 + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]] + (4*Sqrt[a]*Sqrt[b]*Sqrt[c]*d*Sqrt[ 
b*c - a*d]*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x 
)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sq 
rt[c]*Sqrt[b*c - a*d]] + (b*c*(b*c - a*d)*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqr 
t[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/ 
(Sqrt[a]*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (4*Sqrt[ 
a]*d*(-(b*c) + a*d)*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c 
- a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/Sqrt[2*b*c - a*d - 2*S 
qrt[b]*Sqrt[c]*Sqrt[b*c - a*d]] + (b^(3/2)*c^(3/2)*Sqrt[b*c - a*d]*ArcT...
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {369, 403, 27, 398, 224, 219, 291, 218}

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.

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

\(\Big \downarrow \) 369

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 369

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* 
b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1))   Int[(e*x)^(m - 2)*(a + b*x^2)^(p 
 + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] 
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 
] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(-\frac {\left (a d -b c \right ) \left (-\frac {\sqrt {x^{2} d +c}\, b x}{b \,x^{2}+a}-\frac {\left (4 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )-d \left (\sqrt {x^{2} d +c}\, b x -\frac {\left (4 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )}{\sqrt {d}}\right )}{2 b^{3}}\)	\(136\)
risch	\(\frac {d x \sqrt {x^{2} d +c}}{2 b^{2}}-\frac {\frac {\sqrt {d}\, \left (4 a d -3 b c \right ) \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{b}+\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} b^{2} c^{2}\right ) \left (\frac {b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}+\frac {\left (-\frac {1}{2} a^{2} d^{2}+a b c d -\frac {1}{2} b^{2} c^{2}\right ) \left (\frac {b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{2}}+\frac {\left (5 a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (5 a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{2 b^{2}}\)	\(956\)
default	\(\text {Expression too large to display}\)	\(3381\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 996, normalized size of antiderivative = 6.68 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/8*(2*(3*a*b*c - 4*a^2*d + (3*b^2*c - 4*a*b*d)*x^2)*sqrt(d)*log(-2*d*x^ 
2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + (a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d 
)*x^2)*sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a 
^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*(a^2*c*x - (a*b*c - 2*a^2*d)*x^ 
3)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4* 
(b^2*d*x^3 - (b^2*c - 2*a*b*d)*x)*sqrt(d*x^2 + c))/(b^4*x^2 + a*b^3), -1/8 
*(4*(3*a*b*c - 4*a^2*d + (3*b^2*c - 4*a*b*d)*x^2)*sqrt(-d)*arctan(sqrt(-d) 
*x/sqrt(d*x^2 + c)) + (a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)*sqrt(-(b*c 
 - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b 
*c^2 - 4*a^2*c*d)*x^2 + 4*(a^2*c*x - (a*b*c - 2*a^2*d)*x^3)*sqrt(d*x^2 + c 
)*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(b^2*d*x^3 - (b^2 
*c - 2*a*b*d)*x)*sqrt(d*x^2 + c))/(b^4*x^2 + a*b^3), 1/4*((a*b*c - 4*a^2*d 
 + (b^2*c - 4*a*b*d)*x^2)*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^ 
2 - a*c)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b*c^2 
 - a*c*d)*x)) - (3*a*b*c - 4*a^2*d + (3*b^2*c - 4*a*b*d)*x^2)*sqrt(d)*log( 
-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(b^2*d*x^3 - (b^2*c - 2*a* 
b*d)*x)*sqrt(d*x^2 + c))/(b^4*x^2 + a*b^3), -1/4*(2*(3*a*b*c - 4*a^2*d + ( 
3*b^2*c - 4*a*b*d)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (a*b 
*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c 
 - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^...
 

Sympy [F]

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

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

Output:

Integral(x**2*(c + d*x**2)**(3/2)/(a + b*x**2)**2, x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (123) = 246\).

Time = 0.15 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.26 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {d x^{2} + c} d x}{2 \, b^{2}} - \frac {{\left (3 \, b c \sqrt {d} - 4 \, a d^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, b^{3}} - \frac {{\left (b^{2} c^{2} \sqrt {d} - 5 \, a b c d^{\frac {3}{2}} + 4 \, a^{2} d^{\frac {5}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{3}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{2} \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c d^{\frac {3}{2}} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {5}{2}} - b^{2} c^{3} \sqrt {d} + a b c^{2} d^{\frac {3}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{3}} \] Input:

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

Output:

1/2*sqrt(d*x^2 + c)*d*x/b^2 - 1/4*(3*b*c*sqrt(d) - 4*a*d^(3/2))*log((sqrt( 
d)*x - sqrt(d*x^2 + c))^2)/b^3 - 1/2*(b^2*c^2*sqrt(d) - 5*a*b*c*d^(3/2) + 
4*a^2*d^(5/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d 
)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*b^3) + ((sqrt(d)*x - s 
qrt(d*x^2 + c))^2*b^2*c^2*sqrt(d) - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b* 
c*d^(3/2) + 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*d^(5/2) - b^2*c^3*sqrt(d 
) + a*b*c^2*d^(3/2))/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - 
sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*b^ 
3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 919, normalized size of antiderivative = 6.17 \[ \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 
 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**2*d - sqr 
t(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d 
 + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*c + 4*sqrt(a)* 
sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b* 
c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*d*x**2 - sqrt(a)*sq 
rt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) 
 + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b**2*c*x**2 + 4*sqrt(a)*s 
qrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + 
 sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a**2*d - sqrt(a)*sqrt(a*d - 
 b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)* 
sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*c + 4*sqrt(a)*sqrt(a*d - b*c)*lo 
g(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + 
 d*x**2) + sqrt(d)*sqrt(b)*x)*a*b*d*x**2 - sqrt(a)*sqrt(a*d - b*c)*log(sqr 
t(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x* 
*2) + sqrt(d)*sqrt(b)*x)*b**2*c*x**2 - 4*sqrt(a)*sqrt(a*d - b*c)*log(2*sqr 
t(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2* 
b*d*x**2)*a**2*d + sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d 
- b*c) + 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*a*b*c - 4*sq 
rt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)...