\(\int \frac {x^4 (c+d x^2)^{3/2}}{a+b x^2} \, dx\) [934]

Optimal result

Integrand size = 24, antiderivative size = 210 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {\left (b^2 c^2-10 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}}{16 b^3 d}+\frac {(7 b c-6 a d) x^3 \sqrt {c+d x^2}}{24 b^2}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}+\frac {a^{3/2} (b c-a d)^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^4}-\frac {(b c-2 a d) \left (b^2 c^2+8 a b c d-8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 b^4 d^{3/2}} \] Output:

1/16*(8*a^2*d^2-10*a*b*c*d+b^2*c^2)*x*(d*x^2+c)^(1/2)/b^3/d+1/24*(-6*a*d+7 
*b*c)*x^3*(d*x^2+c)^(1/2)/b^2+1/6*d*x^5*(d*x^2+c)^(1/2)/b+a^(3/2)*(-a*d+b* 
c)^(3/2)*arctan((-a*d+b*c)^(1/2)*x/a^(1/2)/(d*x^2+c)^(1/2))/b^4-1/16*(-2*a 
*d+b*c)*(-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^(3/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(457\) vs. \(2(210)=420\).

Time = 1.94 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.18 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {b \sqrt {d} x \sqrt {c+d x^2} \left (24 a^2 d^2-6 a b d \left (5 c+2 d x^2\right )+b^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )\right )+48 \sqrt {a} \sqrt {d} (-b c+a d) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \left (-b c+a d-\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )+48 \sqrt {a} \sqrt {d} (-b c+a d) \left (-b c+a d+\sqrt {b} \sqrt {c} \sqrt {b c-a d}\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )+6 \left (b^3 c^3+6 a b^2 c^2 d-24 a^2 b c d^2+16 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )}{48 b^4 d^{3/2}} \] Input:

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

Output:

(b*Sqrt[d]*x*Sqrt[c + d*x^2]*(24*a^2*d^2 - 6*a*b*d*(5*c + 2*d*x^2) + b^2*( 
3*c^2 + 14*c*d*x^2 + 8*d^2*x^4)) + 48*Sqrt[a]*Sqrt[d]*(-(b*c) + a*d)*Sqrt[ 
2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*(-(b*c) + a*d - Sqrt[b]*S 
qrt[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]))] + 48*Sqrt[a]*Sqrt[d] 
*(-(b*c) + a*d)*(-(b*c) + a*d + Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d])*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]))] 
 + 6*(b^3*c^3 + 6*a*b^2*c^2*d - 24*a^2*b*c*d^2 + 16*a^3*d^3)*ArcTanh[(Sqrt 
[d]*x)/(Sqrt[c] - Sqrt[c + d*x^2])])/(48*b^4*d^(3/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {379, 444, 27, 444, 25, 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.

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

\(\Big \downarrow \) 379

\(\displaystyle \frac {\int \frac {x^4 \left (d (7 b c-6 a d) x^2+c (6 b c-5 a d)\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {\int \frac {3 d x^2 \left (a c (7 b c-6 a d)-\left (b^2 c^2-10 a b d c+8 a^2 d^2\right ) x^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{4 b d}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {3 \int \frac {x^2 \left (a c (7 b c-6 a d)-\left (b^2 c^2-10 a b d c+8 a^2 d^2\right ) x^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {3 \left (\frac {1}{2} x \sqrt {c+d x^2} \left (-\frac {8 a^2 d}{b}+10 a c-\frac {b c^2}{d}\right )-\frac {\int -\frac {(b c-2 a d) \left (b^2 c^2+8 a b d c-8 a^2 d^2\right ) x^2+a c \left (b^2 c^2-10 a b d c+8 a^2 d^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {3 \left (\frac {\int \frac {(b c-2 a d) \left (b^2 c^2+8 a b d c-8 a^2 d^2\right ) x^2+a c \left (b^2 c^2-10 a b d c+8 a^2 d^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b d}+\frac {1}{2} x \sqrt {c+d x^2} \left (-\frac {8 a^2 d}{b}+10 a c-\frac {b c^2}{d}\right )\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {3 \left (\frac {\frac {(b c-2 a d) \left (-8 a^2 d^2+8 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {d x^2+c}}dx}{b}-\frac {16 a^2 d (b c-a d)^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}+\frac {1}{2} x \sqrt {c+d x^2} \left (-\frac {8 a^2 d}{b}+10 a c-\frac {b c^2}{d}\right )\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {3 \left (\frac {\frac {(b c-2 a d) \left (-8 a^2 d^2+8 a b c d+b^2 c^2\right ) \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}-\frac {16 a^2 d (b c-a d)^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}+\frac {1}{2} x \sqrt {c+d x^2} \left (-\frac {8 a^2 d}{b}+10 a c-\frac {b c^2}{d}\right )\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {3 \left (\frac {\frac {(b c-2 a d) \left (-8 a^2 d^2+8 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {16 a^2 d (b c-a d)^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b d}+\frac {1}{2} x \sqrt {c+d x^2} \left (-\frac {8 a^2 d}{b}+10 a c-\frac {b c^2}{d}\right )\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {3 \left (\frac {\frac {(b c-2 a d) \left (-8 a^2 d^2+8 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {16 a^2 d (b c-a d)^2 \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}}{2 b d}+\frac {1}{2} x \sqrt {c+d x^2} \left (-\frac {8 a^2 d}{b}+10 a c-\frac {b c^2}{d}\right )\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {x^3 \sqrt {c+d x^2} (7 b c-6 a d)}{4 b}-\frac {3 \left (\frac {1}{2} x \sqrt {c+d x^2} \left (-\frac {8 a^2 d}{b}+10 a c-\frac {b c^2}{d}\right )+\frac {\frac {(b c-2 a d) \left (-8 a^2 d^2+8 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}-\frac {16 a^{3/2} d (b c-a d)^{3/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b}}{2 b d}\right )}{4 b}}{6 b}+\frac {d x^5 \sqrt {c+d x^2}}{6 b}\)

Input:

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

Output:

(d*x^5*Sqrt[c + d*x^2])/(6*b) + (((7*b*c - 6*a*d)*x^3*Sqrt[c + d*x^2])/(4* 
b) - (3*(((10*a*c - (b*c^2)/d - (8*a^2*d)/b)*x*Sqrt[c + d*x^2])/2 + ((-16* 
a^(3/2)*d*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d 
*x^2])])/b + ((b*c - 2*a*d)*(b^2*c^2 + 8*a*b*c*d - 8*a^2*d^2)*ArcTanh[(Sqr 
t[d]*x)/Sqrt[c + d*x^2]])/(b*Sqrt[d]))/(2*b*d)))/(4*b))/(6*b)
 

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 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 379

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 
1)/(b*e*(m + 2*(p + q) + 1))), x] + Simp[1/(b*(m + 2*(p + q) + 1))   Int[(e 
*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*((b*c - a*d)*(m + 1) + b*c*2 
*(p + q)) + (d*(b*c - a*d)*(m + 1) + d*2*(q - 1)*(b*c - a*d) + b*c*d*2*(p + 
 q))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0 
] && GtQ[q, 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 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 = 0.78 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {-\frac {2 \left (a d -b c \right )^{2} a^{2} \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}-\frac {b \sqrt {x^{2} d +c}\, \left (8 b^{2} d^{2} x^{4}-12 x^{2} a b \,d^{2}+14 x^{2} b^{2} c d +24 a^{2} d^{2}-30 a b c d +3 b^{2} c^{2}\right ) x}{24 d}+\frac {\left (16 a^{3} d^{3}-24 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )}{8 d^{\frac {3}{2}}}}{2 b^{4}}\)	\(189\)
risch	\(\frac {x \left (8 b^{2} d^{2} x^{4}-12 x^{2} a b \,d^{2}+14 x^{2} b^{2} c d +24 a^{2} d^{2}-30 a b c d +3 b^{2} c^{2}\right ) \sqrt {x^{2} d +c}}{48 d \,b^{3}}-\frac {\frac {\left (16 a^{3} d^{3}-24 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{b \sqrt {d}}+\frac {8 a^{2} d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {8 a^{2} d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{16 b^{3} d}\)	\(502\)
default	\(\text {Expression too large to display}\)	\(1389\)

Input:

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

Output:

-1/2/b^4*(-2*(a*d-b*c)^2*a^2/((a*d-b*c)*a)^(1/2)*arctanh((d*x^2+c)^(1/2)/x 
*a/((a*d-b*c)*a)^(1/2))-1/24*b*(d*x^2+c)^(1/2)*(8*b^2*d^2*x^4-12*a*b*d^2*x 
^2+14*b^2*c*d*x^2+24*a^2*d^2-30*a*b*c*d+3*b^2*c^2)/d*x+1/8*(16*a^3*d^3-24* 
a^2*b*c*d^2+6*a*b^2*c^2*d+b^3*c^3)/d^(3/2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/ 
2)))
 

Fricas [A] (verification not implemented)

Time = 1.20 (sec) , antiderivative size = 1119, normalized size of antiderivative = 5.33 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/96*(3*(b^3*c^3 + 6*a*b^2*c^2*d - 24*a^2*b*c*d^2 + 16*a^3*d^3)*sqrt(d)*l 
og(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 24*(a*b*c*d^2 - a^2*d^3)* 
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*(8*b^3*d^3*x^ 
5 + 2*(7*b^3*c*d^2 - 6*a*b^2*d^3)*x^3 + 3*(b^3*c^2*d - 10*a*b^2*c*d^2 + 8* 
a^2*b*d^3)*x)*sqrt(d*x^2 + c))/(b^4*d^2), 1/48*(3*(b^3*c^3 + 6*a*b^2*c^2*d 
 - 24*a^2*b*c*d^2 + 16*a^3*d^3)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c) 
) - 12*(a*b*c*d^2 - a^2*d^3)*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)) + (8*b^3*d^3*x^5 + 2*(7*b^3*c*d^2 - 6*a*b^2*d^3)*x^3 + 3*(b^3* 
c^2*d - 10*a*b^2*c*d^2 + 8*a^2*b*d^3)*x)*sqrt(d*x^2 + c))/(b^4*d^2), 1/96* 
(48*(a*b*c*d^2 - a^2*d^3)*sqrt(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*(b^3*c^3 + 6*a*b^2*c^2*d - 24*a^2*b*c*d^2 + 16*a 
^3*d^3)*sqrt(d)*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(8*b^3 
*d^3*x^5 + 2*(7*b^3*c*d^2 - 6*a*b^2*d^3)*x^3 + 3*(b^3*c^2*d - 10*a*b^2*c*d 
^2 + 8*a^2*b*d^3)*x)*sqrt(d*x^2 + c))/(b^4*d^2), 1/48*(24*(a*b*c*d^2 - a^2 
*d^3)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.94 \[ \int \frac {x^4 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {24 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a^{2} d^{3}-24 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a b c \,d^{2}+24 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a^{2} d^{3}-24 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) a b c \,d^{2}-24 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) a^{2} d^{3}+24 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) a b c \,d^{2}+24 \sqrt {d \,x^{2}+c}\, a^{2} b \,d^{3} x -30 \sqrt {d \,x^{2}+c}\, a \,b^{2} c \,d^{2} x -12 \sqrt {d \,x^{2}+c}\, a \,b^{2} d^{3} x^{3}+3 \sqrt {d \,x^{2}+c}\, b^{3} c^{2} d x +14 \sqrt {d \,x^{2}+c}\, b^{3} c \,d^{2} x^{3}+8 \sqrt {d \,x^{2}+c}\, b^{3} d^{3} x^{5}-48 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{3} d^{3}+72 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} b c \,d^{2}-18 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a \,b^{2} c^{2} d -3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) b^{3} c^{3}}{48 b^{4} d^{2}} \] Input:

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

Output:

(24*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**3 - 
 24*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**2 
+ 24*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**3 - 2 
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*c*d**2 - 24* 
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**3 + 24*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**2 + 24*sqrt(c + d*x**2)*a**2*b*d**3*x 
- 30*sqrt(c + d*x**2)*a*b**2*c*d**2*x - 12*sqrt(c + d*x**2)*a*b**2*d**3*x* 
*3 + 3*sqrt(c + d*x**2)*b**3*c**2*d*x + 14*sqrt(c + d*x**2)*b**3*c*d**2*x* 
*3 + 8*sqrt(c + d*x**2)*b**3*d**3*x**5 - 48*sqrt(d)*log((sqrt(c + d*x**2) 
+ sqrt(d)*x)/sqrt(c))*a**3*d**3 + 72*sqrt(d)*log((sqrt(c + d*x**2) + sqrt( 
d)*x)/sqrt(c))*a**2*b*c*d**2 - 18*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)* 
x)/sqrt(c))*a*b**2*c**2*d - 3*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/s 
qrt(c))*b**3*c**3)/(48*b**4*d**2)