Integrand size = 23, antiderivative size = 425 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx=\frac {\left (15 b^3 c^3+58 a b^2 c^2 d-33 a^2 b c d^2+8 a^3 d^3\right ) x \sqrt {c+d x^2}}{105 b^2 d \sqrt {a+b x^2}}+\frac {\left (15 b^2 c^2+13 a b c d-4 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b^2}+\frac {(5 b c+a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{35 b}+\frac {1}{7} x \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}-\frac {\sqrt {a} \left (15 b^3 c^3+58 a b^2 c^2 d-33 a^2 b c d^2+8 a^3 d^3\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{5/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {4 a^{3/2} \left (15 b^2 c^2-4 a b c d+a^2 d^2\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/105*(8*a^3*d^3-33*a^2*b*c*d^2+58*a*b^2*c^2*d+15*b^3*c^3)*x*(d*x^2+c)^(1/ 2)/b^2/d/(b*x^2+a)^(1/2)+1/105*(-4*a^2*d^2+13*a*b*c*d+15*b^2*c^2)*x*(b*x^2 +a)^(1/2)*(d*x^2+c)^(1/2)/b^2+1/35*(a*d+5*b*c)*x*(b*x^2+a)^(1/2)*(d*x^2+c) ^(3/2)/b+1/7*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(5/2)-1/105*a^(1/2)*(8*a^3*d^3-33 *a^2*b*c*d^2+58*a*b^2*c^2*d+15*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^(5/2)/d/(b*x^2+a)^(1/2)/( a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+4/105*a^(3/2)*(a^2*d^2-4*a*b*c*d+15*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^(5/2)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 2.84 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.73 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a^2 d^2-a b d \left (16 c+3 d x^2\right )-15 b^2 \left (3 c^2+3 c d x^2+d^2 x^4\right )\right )-i c \left (15 b^3 c^3+58 a b^2 c^2 d-33 a^2 b c d^2+8 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (15 b^3 c^3-2 a b^2 c^2 d-17 a^2 b c d^2+4 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 a^2 \left (\frac {b}{a}\right )^{5/2} d \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[Sqrt[a + b*x^2]*(c + d*x^2)^(5/2),x]
Output:
(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a^2*d^2 - a*b*d*(16*c + 3*d*x^ 2) - 15*b^2*(3*c^2 + 3*c*d*x^2 + d^2*x^4))) - I*c*(15*b^3*c^3 + 58*a*b^2*c ^2*d - 33*a^2*b*c*d^2 + 8*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)] + I*c*(15*b^3*c^3 - 2*a*b^ 2*c^2*d - 17*a^2*b*c*d^2 + 4*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*a^2*(b/a)^(5/2)*d *Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.57 (sec) , antiderivative size = 404, 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, 403, 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.
| \(\displaystyle \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\int \sqrt {b x^2+a} \sqrt {d x^2+c} \left (2 d (5 b c-2 a d) x^2+c (7 b c-a d)\right )dx}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {b x^2+a} \left (d \left (45 b^2 c^2-29 a b d c+8 a^2 d^2\right ) x^2+c \left (35 b^2 c^2-15 a b d c+4 a^2 d^2\right )\right )}{\sqrt {d x^2+c}}dx}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {d \left (\left (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\right ) x^2+4 a c \left (15 b^2 c^2-4 a b d c+a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \int \frac {\left (15 b^3 c^3+58 a b^2 d c^2-33 a^2 b d^2 c+8 a^3 d^3\right ) x^2+4 a c \left (15 b^2 c^2-4 a b d c+a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \left (4 a c \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \left (\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {\frac {1}{3} \left (\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2-29 a b c d+45 b^2 c^2\right )+\frac {1}{3} \left (\frac {4 c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-4 a b c d+15 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3-33 a^2 b c d^2+58 a b^2 c^2 d+15 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{5 b}+\frac {2 d x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 b c-2 a d)}{5 b}}{7 b}+\frac {d x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
Input:
Int[Sqrt[a + b*x^2]*(c + d*x^2)^(5/2),x]
Output:
(d*x*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2))/(7*b) + ((2*d*(5*b*c - 2*a*d)*x* (a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*b) + (((45*b^2*c^2 - 29*a*b*c*d + 8* a^2*d^2)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/3 + ((15*b^3*c^3 + 58*a*b^2*c^ 2*d - 33*a^2*b*c*d^2 + 8*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 ])) + (4*c^(3/2)*(15*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*Ellipt icF[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)/(5*b))/(7*b)
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 = 5.25 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.33
| method | result | size |
| risch | \(-\frac {x \left (-15 b^{2} d^{2} x^{4}-3 x^{2} a b \,d^{2}-45 x^{2} b^{2} c d +4 a^{2} d^{2}-16 a b c d -45 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 b^{2}}+\frac {\left (-\frac {\left (8 a^{3} d^{3}-33 a^{2} b c \,d^{2}+58 a \,b^{2} c^{2} d +15 b^{3} c^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {4 a^{3} c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {60 b^{2} c^{3} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {16 a^{2} b \,c^{2} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{105 b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(564\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {d^{2} x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7}+\frac {\left (a \,d^{3}+3 b c \,d^{2}-\frac {d^{2} \left (6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (\frac {16 a c \,d^{2}}{7}+3 b \,c^{2} d -\frac {\left (a \,d^{3}+3 b c \,d^{2}-\frac {d^{2} \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (c^{3} a -\frac {\left (\frac {16 a c \,d^{2}}{7}+3 b \,c^{2} d -\frac {\left (a \,d^{3}+3 b c \,d^{2}-\frac {d^{2} \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (3 a \,c^{2} d +c^{3} b -\frac {3 \left (a \,d^{3}+3 b c \,d^{2}-\frac {d^{2} \left (6 a d +6 b c \right )}{7}\right ) a c}{5 b d}-\frac {\left (\frac {16 a c \,d^{2}}{7}+3 b \,c^{2} d -\frac {\left (a \,d^{3}+3 b c \,d^{2}-\frac {d^{2} \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(634\) |
| default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-15 \sqrt {-\frac {b}{a}}\, b^{3} d^{4} x^{9}-18 \sqrt {-\frac {b}{a}}\, a \,b^{2} d^{4} x^{7}-60 \sqrt {-\frac {b}{a}}\, b^{3} c \,d^{3} x^{7}+\sqrt {-\frac {b}{a}}\, a^{2} b \,d^{4} x^{5}-79 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{3} x^{5}-90 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d^{2} x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{3} d^{4} x^{3}-15 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{3} x^{3}-106 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d^{2} x^{3}-45 \sqrt {-\frac {b}{a}}\, b^{3} c^{3} d \,x^{3}+4 \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}-17 \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}-2 \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 +15 \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}-8 \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}+33 \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}-58 \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 -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 ) b^{3} c^{4}+4 \sqrt {-\frac {b}{a}}\, a^{3} c \,d^{3} x -16 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d^{2} x -45 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} d x \right )}{105 d \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) b^{2} \sqrt {-\frac {b}{a}}}\) | \(784\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/105*x*(-15*b^2*d^2*x^4-3*a*b*d^2*x^2-45*b^2*c*d*x^2+4*a^2*d^2-16*a*b*c* d-45*b^2*c^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2+1/105/b^2*(-(8*a^3*d^3-3 3*a^2*b*c*d^2+58*a*b^2*c^2*d+15*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)))+4*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))+60*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))-16*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)
Time = 0.10 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.76 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx=-\frac {{\left (15 \, b^{3} c^{4} + 58 \, a b^{2} c^{3} d - 33 \, a^{2} b c^{2} d^{2} + 8 \, a^{3} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, b^{3} c^{4} + 58 \, a b^{2} c^{3} d + 4 \, a^{3} d^{4} - 3 \, {\left (11 \, a^{2} b - 20 \, a b^{2}\right )} c^{2} d^{2} + 8 \, {\left (a^{3} - 2 \, a^{2} b\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, b^{3} d^{4} x^{6} + 15 \, b^{3} c^{3} d + 58 \, a b^{2} c^{2} d^{2} - 33 \, a^{2} b c d^{3} + 8 \, a^{3} d^{4} + 3 \, {\left (15 \, b^{3} c d^{3} + a b^{2} d^{4}\right )} x^{4} + {\left (45 \, b^{3} c^{2} d^{2} + 16 \, a b^{2} c d^{3} - 4 \, a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{3} d^{2} x} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2),x, algorithm="fricas")
Output:
-1/105*((15*b^3*c^4 + 58*a*b^2*c^3*d - 33*a^2*b*c^2*d^2 + 8*a^3*c*d^3)*sqr t(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (15*b^3* c^4 + 58*a*b^2*c^3*d + 4*a^3*d^4 - 3*(11*a^2*b - 20*a*b^2)*c^2*d^2 + 8*(a^ 3 - 2*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 + 15*b^3*c^3*d + 58*a*b^2*c^2*d^2 - 33*a^2* b*c*d^3 + 8*a^3*d^4 + 3*(15*b^3*c*d^3 + a*b^2*d^4)*x^4 + (45*b^3*c^2*d^2 + 16*a*b^2*c*d^3 - 4*a^2*b*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3* d^2*x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx=\int \sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {5}{2}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(5/2),x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**(5/2), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(5/2), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(5/2), x)
Timed out. \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx=\int \sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{5/2} \,d x \] Input:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(5/2),x)
Output:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(5/2), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \, dx=\frac {-4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x +16 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b c d x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{3}+45 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x +45 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{3}+15 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{5}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} d^{3}-33 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b c \,d^{2}+58 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{2} d +15 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{3} c^{3}+4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} c \,d^{2}-16 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b \,c^{2} d +60 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{3}}{105 b^{2}} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(5/2),x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*x + 16*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a*b*c*d*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2* x**3 + 45*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*x + 45*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 + 8*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 - 33*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 + 58*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 + 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)*b**3*c**3 + 4*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* *3*c*d**2 - 16*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 + 60*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*b**2)