\(\int \frac {x^4 (c+d x^2)^{3/2}}{(a+b x^2)^2} \, dx\) [996]

Optimal result

Integrand size = 24, antiderivative size = 197 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {3 (3 b c-4 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x^3 \sqrt {c+d x^2}}{4 b^2}-\frac {x^3 \left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {a} (b c-2 a d) \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^4}+\frac {3 \left (b^2 c^2-8 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4 \sqrt {d}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(3517\) vs. \(2(197)=394\).

Time = 13.64 (sec) , antiderivative size = 3517, normalized size of antiderivative = 17.85 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Result too large to show} \] Input:

Integrate[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]
 

Output:

(-32*a*b*c^(9/2)*x + 6*a^2*Sqrt[c]*d^4*x^7 + 32*a*b*c^4*x*Sqrt[c + d*x^2] 
- a^2*d^4*x^7*Sqrt[c + d*x^2] - a*c^3*x*Sqrt[c + d*x^2]*(32*a*d - 48*b*d*x 
^2) - a*c^(7/2)*x*(-32*a*d + 64*b*d*x^2) - a*c^2*x*Sqrt[c + d*x^2]*(48*a*d 
^2*x^2 - 18*b*d^2*x^4) - a*c^(5/2)*x*(-64*a*d^2*x^2 + 38*b*d^2*x^4) - a*c* 
x*Sqrt[c + d*x^2]*(18*a*d^3*x^4 - b*d^3*x^6) - a*c^(3/2)*x*(-38*a*d^3*x^4 
+ 6*b*d^3*x^6))/(64*b^3*c^3*(a + b*x^2) + 96*b^3*c^2*d*x^2*(a + b*x^2) + 3 
6*b^3*c*d^2*x^4*(a + b*x^2) + 2*b^3*d^3*x^6*(a + b*x^2) - 64*b^3*c^(5/2)*( 
a + b*x^2)*Sqrt[c + d*x^2] - 64*b^3*c^(3/2)*d*x^2*(a + b*x^2)*Sqrt[c + d*x 
^2] - 12*b^3*Sqrt[c]*d^2*x^4*(a + b*x^2)*Sqrt[c + d*x^2]) + (3*Sqrt[a]*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]))])/(2*b^(5/2)*Sqrt[2*b*c 
 - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) - (3*a^(3/2)*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]))])/(b^(7/2)*Sqrt[2*b*c - a*d - 2*Sq 
rt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + ((-3*Sqrt[a]*c^2)/(2*b^2*Sqrt[2*b*c - a* 
d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (3*a^(3/2)*c*d)/(2*b^3*Sqrt[2*b* 
c - a*d - 2*Sqrt[b]*Sqrt[c]*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] 
))] + ((3*a^(3/2)*c*d)/(b^3*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c 
- a*d]]) - (3*a^(5/2)*d^2)/(b^4*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sq...
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {369, 27, 443, 27, 444, 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.

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

\(\Big \downarrow \) 369

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 443

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 444

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[(x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x]
 

Output:

-1/2*(x^3*(c + d*x^2)^(3/2))/(b*(a + b*x^2)) + (3*((d*x^3*Sqrt[c + d*x^2]) 
/(2*b) + (((3*b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(2*b) - ((4*Sqrt[a]*(b*c - 2 
*a*d)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2]) 
])/b - ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x 
^2]])/(b*Sqrt[d]))/(2*b))/(2*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 443

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

rule 444

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

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(-\frac {-\left (a d -b c \right ) a \left (-\frac {\sqrt {x^{2} d +c}\, b x}{b \,x^{2}+a}-\frac {3 \left (2 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )+\frac {b \sqrt {x^{2} d +c}\, \left (-2 b d \,x^{2}+8 a d -5 b c \right ) x}{4}-\frac {3 \left (8 a^{2} d^{2}-8 a b c d +b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )}{4 \sqrt {d}}}{2 b^{4}}\)	\(164\)
risch	\(-\frac {x \left (-2 b d \,x^{2}+8 a d -5 b c \right ) \sqrt {x^{2} d +c}}{8 b^{3}}+\frac {\frac {3 \left (8 a^{2} d^{2}-8 a b c d +b^{2} c^{2}\right ) \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{b \sqrt {d}}-\frac {2 a \left (a^{2} d^{2}-2 a b c d +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 {2 a \left (a^{2} d^{2}-2 a b c d +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 {2 a \left (7 a^{2} d^{2}-10 a b c d +3 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 )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {2 a \left (7 a^{2} d^{2}-10 a b c d +3 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 )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{8 b^{3}}\)	\(991\)
default	\(\text {Expression too large to display}\)	\(3440\)

Input:

int(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1249, normalized size of antiderivative = 6.34 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

[1/16*(3*(a*b^2*c^2 - 8*a^2*b*c*d + 8*a^3*d^2 + (b^3*c^2 - 8*a*b^2*c*d + 8 
*a^2*b*d^2)*x^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 
 6*(a*b*c*d - 2*a^2*d^2 + (b^2*c*d - 2*a*b*d^2)*x^2)*sqrt(-a*b*c + a^2*d)* 
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*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 
 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*(2*b^3*d^2*x^5 + (5*b^3*c*d - 6*a* 
b^2*d^2)*x^3 + 3*(3*a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(d*x^2 + c))/(b^5*d*x^ 
2 + a*b^4*d), -1/8*(3*(a*b^2*c^2 - 8*a^2*b*c*d + 8*a^3*d^2 + (b^3*c^2 - 8* 
a*b^2*c*d + 8*a^2*b*d^2)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) 
+ 3*(a*b*c*d - 2*a^2*d^2 + (b^2*c*d - 2*a*b*d^2)*x^2)*sqrt(-a*b*c + a^2*d) 
*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*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^ 
2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - (2*b^3*d^2*x^5 + (5*b^3*c*d - 6*a*b 
^2*d^2)*x^3 + 3*(3*a*b^2*c*d - 4*a^2*b*d^2)*x)*sqrt(d*x^2 + c))/(b^5*d*x^2 
 + a*b^4*d), -1/16*(12*(a*b*c*d - 2*a^2*d^2 + (b^2*c*d - 2*a*b*d^2)*x^2)*s 
qrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c 
)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 3*( 
a*b^2*c^2 - 8*a^2*b*c*d + 8*a^3*d^2 + (b^3*c^2 - 8*a*b^2*c*d + 8*a^2*b*d^2 
)*x^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 2*(2*b^3* 
d^2*x^5 + (5*b^3*c*d - 6*a*b^2*d^2)*x^3 + 3*(3*a*b^2*c*d - 4*a^2*b*d^2)...
 

Sympy [F]

\[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{4} \left (c + d x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \] Input:

integrate(x**4*(d*x**2+c)**(3/2)/(b*x**2+a)**2,x)
 

Output:

Integral(x**4*(c + d*x**2)**(3/2)/(a + b*x**2)**2, x)
 

Maxima [F]

\[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \] Input:

integrate(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^(3/2)*x^4/(b*x^2 + a)^2, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (165) = 330\).

Time = 0.15 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.97 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {1}{8} \, \sqrt {d x^{2} + c} x {\left (\frac {2 \, d x^{2}}{b^{2}} + \frac {5 \, b^{7} c d^{2} - 8 \, a b^{6} d^{3}}{b^{9} d^{2}}\right )} + \frac {3 \, {\left (a b^{2} c^{2} \sqrt {d} - 3 \, a^{2} b c d^{\frac {3}{2}} + 2 \, a^{3} d^{\frac {5}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{4}} - \frac {3 \, {\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{16 \, b^{4} \sqrt {d}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{2} \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac {3}{2}} + 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} d^{\frac {5}{2}} - a b^{2} c^{3} \sqrt {d} + a^{2} b c^{2} d^{\frac {3}{2}}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{4}} \] Input:

integrate(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x, algorithm="giac")
 

Output:

1/8*sqrt(d*x^2 + c)*x*(2*d*x^2/b^2 + (5*b^7*c*d^2 - 8*a*b^6*d^3)/(b^9*d^2) 
) + 3/2*(a*b^2*c^2*sqrt(d) - 3*a^2*b*c*d^(3/2) + 2*a^3*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^4) - 3/16*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*l 
og((sqrt(d)*x - sqrt(d*x^2 + c))^2)/(b^4*sqrt(d)) - ((sqrt(d)*x - sqrt(d*x 
^2 + c))^2*a*b^2*c^2*sqrt(d) - 3*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*b*c*d 
^(3/2) + 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^3*d^(5/2) - a*b^2*c^3*sqrt(d) 
 + a^2*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 
^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^4\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (b\,x^2+a\right )}^2} \,d x \] Input:

int((x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2,x)
 

Output:

int((x^4*(c + d*x^2)^(3/2))/(a + b*x^2)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 1049, normalized size of antiderivative = 5.32 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(x^4*(d*x^2+c)^(3/2)/(b*x^2+a)^2,x)
 

Output:

( - 12*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** 
2 + 6*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*d - 
 12*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**2*x* 
*2 + 6*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)*b**2*c*d 
*x**2 - 12*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* 
*2 + 6*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*d - 1 
2*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**2*x**2 + 
6*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)*b**2*c*d*x**2 + 
12*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**2*d**2 - 6*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*d + 12*sqrt(a)*sqrt(a*d - b*c)*log(2...