\(\int \frac {x^2 (a+b x^2)^{5/2}}{\sqrt {c+d x^2}} \, dx\) [1196]

Optimal result

Integrand size = 26, antiderivative size = 436 \[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {c+d x^2}}{105 d^4 \sqrt {a+b x^2}}+\frac {\left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 d^3}-\frac {2 b (3 b c-5 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}+\frac {\sqrt {a} \left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 \sqrt {b} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 \sqrt {b} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/105*(-15*a^3*d^3+103*a^2*b*c*d^2-128*a*b^2*c^2*d+48*b^3*c^3)*x*(d*x^2+c 
)^(1/2)/d^4/(b*x^2+a)^(1/2)+1/105*(45*a^2*d^2-61*a*b*c*d+24*b^2*c^2)*x*(b* 
x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3-2/35*b*(-5*a*d+3*b*c)*x^3*(b*x^2+a)^(1/2) 
*(d*x^2+c)^(1/2)/d^2+1/7*b*x^3*(b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/d+1/105*a^( 
1/2)*(-15*a^3*d^3+103*a^2*b*c*d^2-128*a*b^2*c^2*d+48*b^3*c^3)*(d*x^2+c)^(1 
/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(1/ 
2)/d^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/105*a^(3/2)*(45*a 
^2*d^2-61*a*b*c*d+24*b^2*c^2)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/ 
2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/d^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/ 
c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.93 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (45 a^2 d^2+a b d \left (-61 c+45 d x^2\right )+3 b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )-i c \left (-48 b^3 c^3+128 a b^2 c^2 d-103 a^2 b c d^2+15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+4 i c \left (-12 b^3 c^3+38 a b^2 c^2 d-41 a^2 b c d^2+15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(45*a^2*d^2 + a*b*d*(-61*c + 45*d*x 
^2) + 3*b^2*(8*c^2 - 6*c*d*x^2 + 5*d^2*x^4)) - I*c*(-48*b^3*c^3 + 128*a*b^ 
2*c^2*d - 103*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^ 
2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*c*(-12*b^3*c^ 
3 + 38*a*b^2*c^2*d - 41*a^2*b*c*d^2 + 15*a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt 
[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*Sqrt[ 
b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 414, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {379, 25, 443, 25, 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.

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

\(\Big \downarrow \) 379

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 443

\(\displaystyle \frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {\int -\frac {x^2 \left (b \left (24 b^2 c^2-61 a b d c+45 a^2 d^2\right ) x^2+a \left (18 b^2 c^2-45 a b d c+35 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}+\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{5 d}}{7 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{5 d}-\frac {\int \frac {x^2 \left (b \left (24 b^2 c^2-61 a b d c+45 a^2 d^2\right ) x^2+a \left (18 b^2 c^2-45 a b d c+35 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}}{7 d}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\int \frac {b \left (\left (48 b^3 c^3-128 a b^2 d c^2+103 a^2 b d^2 c-15 a^3 d^3\right ) x^2+a c \left (24 b^2 c^2-61 a b d c+45 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{5 d}}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\int \frac {\left (48 b^3 c^3-128 a b^2 d c^2+103 a^2 b d^2 c-15 a^3 d^3\right ) x^2+a c \left (24 b^2 c^2-61 a b d c+45 a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}}{7 d}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{3 d}-\frac {a c \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}}{7 d}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{5 d}}{7 d}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{5 d}}{7 d}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {b x^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{5 d}}{7 d}\)

Input:

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

Output:

(b*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(7*d) - ((2*b*(3*b*c - 5*a*d)*x^ 
3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - (((24*b^2*c^2 - 61*a*b*c*d + 45 
*a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - ((48*b^3*c^3 - 128*a* 
b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + 
 d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 
 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c 
 + d*x^2])) + (c^(3/2)*(24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*Sqrt[a + b*x 
^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt 
[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*d))/(5*d))/(7*d)
 

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 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 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 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]
 

Maple [A] (verified)

Time = 16.22 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.29

method	result	size
risch	\(\frac {x \left (15 b^{2} d^{2} x^{4}+45 x^{2} a b \,d^{2}-18 x^{2} b^{2} c d +45 a^{2} d^{2}-61 a b c d +24 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 d^{3}}-\frac {\left (\frac {\left (15 a^{3} d^{3}-103 a^{2} b c \,d^{2}+128 a \,b^{2} c^{2} d -48 b^{3} c^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {45 a^{3} c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {24 b^{2} c^{3} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {61 a^{2} b \,c^{2} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{105 d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(563\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {b^{2} x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 d}+\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (3 a^{2} b -\frac {5 b^{2} a c}{7 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}-\frac {\left (3 a^{2} b -\frac {5 b^{2} a c}{7 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (a^{3}-\frac {3 \left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) a c}{5 b d}-\frac {\left (3 a^{2} b -\frac {5 b^{2} a c}{7 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(612\)
default	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-15 \sqrt {-\frac {b}{a}}\, b^{3} d^{4} x^{9}-60 \sqrt {-\frac {b}{a}}\, a \,b^{2} d^{4} x^{7}+3 \sqrt {-\frac {b}{a}}\, b^{3} c \,d^{3} x^{7}-90 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{4} x^{5}+19 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{3} x^{5}-6 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d^{2} x^{5}-45 \sqrt {-\frac {b}{a}}\, a^{3} d^{4} x^{3}-29 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{3} x^{3}+55 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d^{2} x^{3}-24 \sqrt {-\frac {b}{a}}\, b^{3} c^{3} d \,x^{3}+60 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} c \,d^{3}-164 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b \,c^{2} d^{2}+152 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{3} d -48 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{4}-15 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} c \,d^{3}+103 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b \,c^{2} d^{2}-128 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{3} d +48 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{4}-45 \sqrt {-\frac {b}{a}}\, a^{3} c \,d^{3} x +61 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d^{2} x -24 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} d x \right )}{105 d^{4} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}}\)	\(782\)

Input:

int(x^2*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/105*x*(15*b^2*d^2*x^4+45*a*b*d^2*x^2-18*b^2*c*d*x^2+45*a^2*d^2-61*a*b*c* 
d+24*b^2*c^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3-1/105/d^3*((15*a^3*d^3-1 
03*a^2*b*c*d^2+128*a*b^2*c^2*d-48*b^3*c^3)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2 
)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b 
/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c) 
/c/b)^(1/2)))+45*a^3*c*d^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2 
)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b* 
c)/c/b)^(1/2))+24*b^2*c^3*a/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/ 
2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b 
*c)/c/b)^(1/2))-61*a^2*b*c^2*d/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^ 
(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a* 
d+b*c)/c/b)^(1/2)))*((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^ 
(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (48 \, b^{3} c^{4} - 128 \, a b^{2} c^{3} d + 103 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (48 \, b^{3} c^{4} - 128 \, a b^{2} c^{3} d + 45 \, a^{3} d^{4} + {\left (103 \, a^{2} b + 24 \, a b^{2}\right )} c^{2} d^{2} - {\left (15 \, a^{3} + 61 \, a^{2} b\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (15 \, b^{3} d^{4} x^{6} - 48 \, b^{3} c^{3} d + 128 \, a b^{2} c^{2} d^{2} - 103 \, a^{2} b c d^{3} + 15 \, a^{3} d^{4} - 9 \, {\left (2 \, b^{3} c d^{3} - 5 \, a b^{2} d^{4}\right )} x^{4} + {\left (24 \, b^{3} c^{2} d^{2} - 61 \, a b^{2} c d^{3} + 45 \, a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b d^{5} x} \] Input:

integrate(x^2*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

1/105*((48*b^3*c^4 - 128*a*b^2*c^3*d + 103*a^2*b*c^2*d^2 - 15*a^3*c*d^3)*s 
qrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (48*b^ 
3*c^4 - 128*a*b^2*c^3*d + 45*a^3*d^4 + (103*a^2*b + 24*a*b^2)*c^2*d^2 - (1 
5*a^3 + 61*a^2*b)*c*d^3)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/ 
d)/x), a*d/(b*c)) + (15*b^3*d^4*x^6 - 48*b^3*c^3*d + 128*a*b^2*c^2*d^2 - 1 
03*a^2*b*c*d^3 + 15*a^3*d^4 - 9*(2*b^3*c*d^3 - 5*a*b^2*d^4)*x^4 + (24*b^3* 
c^2*d^2 - 61*a*b^2*c*d^3 + 45*a^2*b*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + 
 c))/(b*d^5*x)
 

Sympy [F]

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

integrate(x**2*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(x^2*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(x^2*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((x^2*(a + b*x^2)^(5/2))/(c + d*x^2)^(1/2),x)
 

Output:

int((x^2*(a + b*x^2)^(5/2))/(c + d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {45 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x -61 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b c d x +45 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{3}+24 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x -18 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{3}+15 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{5}+15 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} d^{3}-103 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b c \,d^{2}+128 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{2} d -48 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{3} c^{3}-45 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} c \,d^{2}+61 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b \,c^{2} d -24 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{3}}{105 d^{3}} \] Input:

int(x^2*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)
 

Output:

(45*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*x - 61*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a*b*c*d*x + 45*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*x 
**3 + 24*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*x - 18*sqrt(c + d*x** 
2)*sqrt(a + b*x**2)*b**2*c*d*x**3 + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b 
**2*d**2*x**5 + 15*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d 
*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*d**3 - 103*int((sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b*c*d**2 
 + 128*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c* 
x**2 + b*d*x**4),x)*a*b**2*c**2*d - 48*int((sqrt(c + d*x**2)*sqrt(a + b*x* 
*2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**3*c**3 - 45*int((sq 
rt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x) 
*a**3*c*d**2 + 61*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 
+ b*c*x**2 + b*d*x**4),x)*a**2*b*c**2*d - 24*int((sqrt(c + d*x**2)*sqrt(a 
+ b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b**2*c**3)/(105*d** 
3)