\(\int x^4 (a+\frac {b}{c+d x^2})^{3/2} \, dx\) [183]

Optimal result

Integrand size = 21, antiderivative size = 357 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\left (b^2-14 a b c+a^2 c^2\right ) x \sqrt {a+\frac {b}{c+d x^2}}}{5 a d^2}+\frac {(7 b-a c) x \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{5 d^2}+\frac {6 a x^3 \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{5 d}-\frac {x^3 \left (b+a c+a d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{d}-\frac {\sqrt {c} \left (b^2-14 a b c+a^2 c^2\right ) \sqrt {a+\frac {b}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 a d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}-\frac {c^{3/2} (7 b-a c) \sqrt {a+\frac {b}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{5 d^{5/2} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \] Output:

1/5*(a^2*c^2-14*a*b*c+b^2)*x*(a+b/(d*x^2+c))^(1/2)/a/d^2+1/5*(-a*c+7*b)*x* 
(d*x^2+c)*(a+b/(d*x^2+c))^(1/2)/d^2+6/5*a*x^3*(d*x^2+c)*(a+b/(d*x^2+c))^(1 
/2)/d-x^3*(a*d*x^2+a*c+b)*(a+b/(d*x^2+c))^(1/2)/d-1/5*c^(1/2)*(a^2*c^2-14* 
a*b*c+b^2)*(a+b/(d*x^2+c))^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^( 
1/2),(b/(a*c+b))^(1/2))/a/d^(5/2)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1 
/2)-1/5*c^(3/2)*(-a*c+7*b)*(a+b/(d*x^2+c))^(1/2)*InverseJacobiAM(arctan(d^ 
(1/2)*x/c^(1/2)),(b/(a*c+b))^(1/2))/d^(5/2)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d* 
x^2+c))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.93 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.87 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} x \left (-a^2 \left (c-d x^2\right ) \left (c+d x^2\right )^2+b^2 \left (7 c+2 d x^2\right )+3 a b \left (2 c^2+3 c d x^2+d^2 x^4\right )\right )-i \left (b^3-13 a b^2 c-13 a^2 b c^2+a^3 c^3\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+i b \left (b^2-6 a b c-7 a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{5 a c^2 \left (\frac {d}{c}\right )^{5/2} \left (b+a \left (c+d x^2\right )\right )} \] Input:

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

Output:

(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(a*Sqrt[d/c]*x*(-(a^2*(c - d*x^2)*( 
c + d*x^2)^2) + b^2*(7*c + 2*d*x^2) + 3*a*b*(2*c^2 + 3*c*d*x^2 + d^2*x^4)) 
 - I*(b^3 - 13*a*b^2*c - 13*a^2*b*c^2 + a^3*c^3)*Sqrt[(b + a*c + a*d*x^2)/ 
(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b 
+ a*c)] + I*b*(b^2 - 6*a*b*c - 7*a^2*c^2)*Sqrt[(b + a*c + a*d*x^2)/(b + a* 
c)]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)] 
))/(5*a*c^2*(d/c)^(5/2)*(b + a*(c + d*x^2)))
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.22, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2057, 2058, 369, 27, 443, 27, 444, 27, 406, 320, 388, 313}

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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 369

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 443

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 444

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {3 \left (\frac {1}{5} \left (\frac {x (7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 d}-\frac {\int \frac {a d \left (c (7 b-a c) (b+a c)-\left (b^2-14 a c b+a^2 c^2\right ) d x^2\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 a d^2}\right )+\frac {2}{5} a x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}\right )}{d}-\frac {x^3 \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {3 \left (\frac {1}{5} \left (\frac {x (7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 d}-\frac {\int \frac {c (7 b-a c) (b+a c)-\left (b^2-14 a c b+a^2 c^2\right ) d x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 d}\right )+\frac {2}{5} a x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}\right )}{d}-\frac {x^3 \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {3 \left (\frac {1}{5} \left (\frac {x (7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 d}-\frac {c (7 b-a c) (a c+b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx-d \left (a^2 c^2-14 a b c+b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 d}\right )+\frac {2}{5} a x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}\right )}{d}-\frac {x^3 \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {3 \left (\frac {1}{5} \left (\frac {x (7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 d}-\frac {\frac {c^{3/2} (7 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-d \left (a^2 c^2-14 a b c+b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 d}\right )+\frac {2}{5} a x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}\right )}{d}-\frac {x^3 \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {3 \left (\frac {1}{5} \left (\frac {x (7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 d}-\frac {\frac {c^{3/2} (7 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-d \left (a^2 c^2-14 a b c+b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )}{3 d}\right )+\frac {2}{5} a x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}\right )}{d}-\frac {x^3 \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {3 \left (\frac {1}{5} \left (\frac {x (7 b-a c) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 d}-\frac {\frac {c^{3/2} (7 b-a c) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-d \left (a^2 c^2-14 a b c+b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{3 d}\right )+\frac {2}{5} a x^3 \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}\right )}{d}-\frac {x^3 \left (a c+a d x^2+b\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\)

Input:

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

Output:

(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-((x^3*(b + a*c + 
a*d*x^2)^(3/2))/(d*Sqrt[c + d*x^2])) + (3*((2*a*x^3*Sqrt[c + d*x^2]*Sqrt[b 
 + a*c + a*d*x^2])/5 + (((7*b - a*c)*x*Sqrt[c + d*x^2]*Sqrt[b + a*c + a*d* 
x^2])/(3*d) - (-((b^2 - 14*a*b*c + a^2*c^2)*d*((x*Sqrt[b + a*c + a*d*x^2]) 
/(a*d*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[b + a*c + a*d*x^2]*EllipticE[ArcTan 
[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(a*d^(3/2)*Sqrt[c + d*x^2]*Sqrt[(c*(b 
 + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]))) + (c^(3/2)*(7*b - a*c)*Sqrt 
[b + a*c + a*d*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/( 
Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2 
))]))/(3*d))/5))/d))/Sqrt[b + a*c + a*d*x^2]
 

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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

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 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

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

rule 2057

Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* 
((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
 

rule 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1061\) vs. \(2(330)=660\).

Time = 17.50 (sec) , antiderivative size = 1062, normalized size of antiderivative = 2.97

method	result	size
risch	\(\text {Expression too large to display}\)	\(1062\)
default	\(\text {Expression too large to display}\)	\(1101\)

Input:

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

Output:

-1/5/d^2*x*(-a*d*x^2+a*c-2*b)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+ 
1/5/d^2*(a^2*c^3/(-a*d/(a*c+b))^(1/2)*(1+a*d*x^2/(a*c+b))^(1/2)*(1+1/c*x^2 
*d)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a* 
d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-2*d*(a^2*c^2-9*a*b*c+b^2) 
*(a*c^2+b*c)/(-a*d/(a*c+b))^(1/2)*(1+a*d*x^2/(a*c+b))^(1/2)*(1+1/c*x^2*d)^ 
(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(2*a*c*d+2*b*d)*(Ell 
ipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-EllipticE(x* 
(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2)))-7*b^2*c/(-a*d/(a*c+b 
))^(1/2)*(1+a*d*x^2/(a*c+b))^(1/2)*(1+1/c*x^2*d)^(1/2)/(a*d^2*x^4+2*a*c*d* 
x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d 
+b*d)/d/c/a)^(1/2))-a*b*c^2/(-a*d/(a*c+b))^(1/2)*(1+a*d*x^2/(a*c+b))^(1/2) 
*(1+1/c*x^2*d)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*Ellip 
ticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))+5*b^2*c^2*((a* 
d^2*x^2+a*c*d+b*d)/c/b*x/d/((x^2+c/d)*(a*d^2*x^2+a*c*d+b*d))^(1/2)+(1/c-(a 
*c*d+b*d)/c/b/d)/(-a*d/(a*c+b))^(1/2)*(1+a*d*x^2/(a*c+b))^(1/2)*(1+1/c*x^2 
*d)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a* 
d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))+2*a*d/b/c*(a*c^2+b*c)/(-a 
*d/(a*c+b))^(1/2)*(1+a*d*x^2/(a*c+b))^(1/2)*(1+1/c*x^2*d)^(1/2)/(a*d^2*x^4 
+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(2*a*c*d+2*b*d)*(EllipticF(x*(-a*d/( 
a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-EllipticE(x*(-a*d/(a*c+b)...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.65 \[ \int x^4 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {{\left (a^{2} c^{3} - 14 \, a b c^{2} + b^{2} c\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} c^{3} - 14 \, a b c^{2} + b^{2} c + {\left (a^{2} c^{2} - 6 \, a b c - 7 \, b^{2}\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{2} d^{3} x^{6} + 2 \, a b d^{2} x^{4} + a^{2} c^{3} - 14 \, a b c^{2} - {\left (7 \, a b c - b^{2}\right )} d x^{2} + b^{2} c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{5 \, a d^{3} x} \] Input:

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

Output:

-1/5*((a^2*c^3 - 14*a*b*c^2 + b^2*c)*sqrt(a)*x*sqrt(-c/d)*elliptic_e(arcsi 
n(sqrt(-c/d)/x), (a*c + b)/(a*c)) - (a^2*c^3 - 14*a*b*c^2 + b^2*c + (a^2*c 
^2 - 6*a*b*c - 7*b^2)*d)*sqrt(a)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d) 
/x), (a*c + b)/(a*c)) - (a^2*d^3*x^6 + 2*a*b*d^2*x^4 + a^2*c^3 - 14*a*b*c^ 
2 - (7*a*b*c - b^2)*d*x^2 + b^2*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/ 
(a*d^3*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a**2*d**2*x**5 - 3*sqrt(c + d*x 
**2)*sqrt(a*c + a*d*x**2 + b)*a*b*c*x + 2*sqrt(c + d*x**2)*sqrt(a*c + a*d* 
x**2 + b)*a*b*d*x**3 - 3*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*b**2*x 
- 4*int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*x**4)/(a*c**3 + 3*a*c** 
2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c**2 + 2*b*c*d*x**2 + b*d**2* 
x**4),x)*a**2*b*c**2*d**2 - 4*int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + 
b)*x**4)/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c** 
2 + 2*b*c*d*x**2 + b*d**2*x**4),x)*a**2*b*c*d**3*x**2 + 4*int((sqrt(c + d* 
x**2)*sqrt(a*c + a*d*x**2 + b)*x**4)/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d** 
2*x**4 + a*d**3*x**6 + b*c**2 + 2*b*c*d*x**2 + b*d**2*x**4),x)*a*b**2*c*d* 
*2 + 4*int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*x**4)/(a*c**3 + 3*a* 
c**2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c**2 + 2*b*c*d*x**2 + b*d* 
*2*x**4),x)*a*b**2*d**3*x**2 + 3*int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 
 + b))/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c**2 
+ 2*b*c*d*x**2 + b*d**2*x**4),x)*a**2*b*c**4 + 3*int((sqrt(c + d*x**2)*sqr 
t(a*c + a*d*x**2 + b))/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d**2*x**4 + a*d** 
3*x**6 + b*c**2 + 2*b*c*d*x**2 + b*d**2*x**4),x)*a**2*b*c**3*d*x**2 + 6*in 
t((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b))/(a*c**3 + 3*a*c**2*d*x**2 + 
3*a*c*d**2*x**4 + a*d**3*x**6 + b*c**2 + 2*b*c*d*x**2 + b*d**2*x**4),x)*a* 
b**2*c**3 + 6*int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b))/(a*c**3 +...