Integrand size = 23, antiderivative size = 328 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\frac {\left (7 a c-\frac {2 b c^2}{d}+\frac {3 a^2 d}{b}\right ) x \sqrt {a+b x^2}}{15 \sqrt {c+d x^2}}-\frac {2 (b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d}+\frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}+\frac {\sqrt {c} \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} (b c-9 a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
1/5*b*x*(d*x^2+c)^(3/2)*(b*x^2+a)^(1/2)/d+1/15*(7*a*c-2*b*c^2/d+3*a^2*d/b) *x*(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)-1/15*c^(3/2)*(-9*a*d+b*c)*(1/(1+d*x^2/c ))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),( 1-b*c/a/d)^(1/2))*(b*x^2+a)^(1/2)/d^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/ (d*x^2+c)^(1/2)+1/15*(-3*a^2*d^2-7*a*b*c*d+2*b^2*c^2)*(1/(1+d*x^2/c))^(1/2 )*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a /d)^(1/2))*c^(1/2)*(b*x^2+a)^(1/2)/b/d^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/ 2)/(d*x^2+c)^(1/2)-2/15*(-3*a*d+b*c)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d
Result contains complex when optimal does not.
Time = 2.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.74 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a d+b \left (c+3 d x^2\right )\right )-i c \left (-2 b^2 c^2+7 a b c d+3 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-2 i c \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(6*a*d + b*(c + 3*d*x^2)) - I*c*(-2 *b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]* EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (2*I)*c*(b^2*c^2 - 4*a*b* c*d + 3*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSi nh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.45 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {318, 25, 403, 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.
| \(\displaystyle \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\int -\frac {\sqrt {d x^2+c} \left (2 b (b c-3 a d) x^2+a (b c-5 a d)\right )}{\sqrt {b x^2+a}}dx}{5 d}+\frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {\int \frac {\sqrt {d x^2+c} \left (2 b (b c-3 a d) x^2+a (b c-5 a d)\right )}{\sqrt {b x^2+a}}dx}{5 d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {\frac {\int \frac {b \left (\left (2 b^2 c^2-7 a b d c-3 a^2 d^2\right ) x^2+a c (b c-9 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b}+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-3 a d)}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {\frac {1}{3} \int \frac {\left (2 b^2 c^2-7 a b d c-3 a^2 d^2\right ) x^2+a c (b c-9 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-3 a d)}{5 d}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {\frac {1}{3} \left (\left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (b c-9 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-3 a d)}{5 d}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {\frac {1}{3} \left (\left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-3 a d)}{5 d}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {\frac {1}{3} \left (\left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-3 a d)}{5 d}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {\frac {1}{3} \left (\left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {2}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-3 a d)}{5 d}\) |
(b*x*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(5*d) - ((2*(b*c - 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/3 + ((2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*((x*Sq rt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[Ar cTan[(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)*(b*c - 9*a*d)*Sqrt[a + b* x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqr t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/3)/(5*d)
3.2.66.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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]
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]
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]
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]
Time = 4.06 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(\frac {x \left (3 b d \,x^{2}+6 a d +b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{15 d}+\frac {\left (\frac {9 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (3 a^{2} d^{2}+7 a b c d -2 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{15 d \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(412\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5}+\frac {\left (2 a b d +b^{2} c -\frac {b \left (4 a d +4 b c \right )}{5}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}+\frac {\left (a^{2} c -\frac {\left (2 a b d +b^{2} c -\frac {b \left (4 a d +4 b c \right )}{5}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (a^{2} d +\frac {7 a b c}{5}-\frac {\left (2 a b d +b^{2} c -\frac {b \left (4 a d +4 b c \right )}{5}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(423\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+9 \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+4 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+6 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}+10 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+6 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x +\sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{15 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) d^{2} \sqrt {-\frac {b}{a}}}\) | \(543\) |
1/15*x*(3*b*d*x^2+6*a*d+b*c)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d+1/15/d*(9*a ^2*c*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*c^2 *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))-(3*a^2*d^ 2+7*a*b*c*d-2*b^2*c^2)*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))))*((b*x ^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.71 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\frac {{\left (2 \, b^{2} c^{3} - 7 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{2} c^{3} - 7 \, a b c^{2} d - 9 \, a^{2} d^{3} - {\left (3 \, a^{2} - a b\right )} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, b^{2} d^{3} x^{4} - 2 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 3 \, a^{2} d^{3} + {\left (b^{2} c d^{2} + 6 \, a b d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b d^{3} x} \]
1/15*((2*b^2*c^3 - 7*a*b*c^2*d - 3*a^2*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*ellip tic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (2*b^2*c^3 - 7*a*b*c^2*d - 9*a^2* d^3 - (3*a^2 - a*b)*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(- c/d)/x), a*d/(b*c)) + (3*b^2*d^3*x^4 - 2*b^2*c^2*d + 7*a*b*c*d^2 + 3*a^2*d ^3 + (b^2*c*d^2 + 6*a*b*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b*d^3* x)
\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\int \left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}\, dx \]
\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \,d x } \]
\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \,d x } \]
Timed out. \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\int {\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c} \,d x \]