Integrand size = 23, antiderivative size = 445 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) x \sqrt {a+b x^2}}{15 c d^3 \sqrt {c+d x^2}}-\frac {(b c-a d) x \left (a+b x^2\right )^{5/2}}{c d \sqrt {c+d x^2}}-\frac {b \left (24 b^2 c^2-43 a b c d+15 a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 c d^3}+\frac {b (6 b c-5 a d) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 c d^2}-\frac {\left (48 b^3 c^3-128 a b^2 c^2 d+103 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b \sqrt {c} \left (24 b^2 c^2-61 a b c d+45 a^2 d^2\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
-(-a*d+b*c)*x*(b*x^2+a)^(5/2)/c/d/(d*x^2+c)^(1/2)+1/15*(-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*(b*x^2+a)^(1/2)/c/d^3/(d*x^2+c)^(1 /2)-1/15*(-15*a^3*d^3+103*a^2*b*c*d^2-128*a*b^2*c^2*d+48*b^3*c^3)*(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))*(b*x^2+a)^(1/2)/d^(7/2)/c^(1/2)/(c*(b*x^2+a)/a/(d*x ^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/15*b*(45*a^2*d^2-61*a*b*c*d+24*b^2*c^2)*(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))*c^(1/2)*(b*x^2+a)^(1/2)/d^(7/2)/(c*(b*x^2+a)/a /(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/5*b*(-5*a*d+6*b*c)*x*(b*x^2+a)^(3/2)*( d*x^2+c)^(1/2)/c/d^2-1/15*b*(15*a^2*d^2-43*a*b*c*d+24*b^2*c^2)*x*(b*x^2+a) ^(1/2)*(d*x^2+c)^(1/2)/c/d^3
Result contains complex when optimal does not.
Time = 5.17 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (-45 a^2 b c d^2+15 a^3 d^3+a b^2 c d \left (61 c+16 d x^2\right )-3 b^3 c \left (8 c^2+2 c d x^2-d^2 x^4\right )\right )+i b c \left (-48 b^3 c^3+128 a b^2 c^2 d-103 a^2 b c d^2+15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+4 i b c \left (12 b^3 c^3-38 a b^2 c^2 d+41 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} c d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
(Sqrt[b/a]*d*x*(a + b*x^2)*(-45*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*c*d*(61*c + 16*d*x^2) - 3*b^3*c*(8*c^2 + 2*c*d*x^2 - d^2*x^4)) + I*b*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]*Sqr t[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*b* 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)] )/(15*Sqrt[b/a]*c*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.58 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {315, 27, 403, 25, 403, 25, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\int \frac {b \left (b x^2+a\right )^{3/2} \left ((6 b c-5 a d) x^2+a c\right )}{\sqrt {d x^2+c}}dx}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {\left (b x^2+a\right )^{3/2} \left ((6 b c-5 a d) x^2+a c\right )}{\sqrt {d x^2+c}}dx}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {\int -\frac {\sqrt {b x^2+a} \left (\left (24 b^2 c^2-43 a b d c+15 a^2 d^2\right ) x^2+2 a c (3 b c-5 a d)\right )}{\sqrt {d x^2+c}}dx}{5 d}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 d}\right )}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 d}-\frac {\int \frac {\sqrt {b x^2+a} \left (\left (24 b^2 c^2-43 a b d c+15 a^2 d^2\right ) x^2+2 a c (3 b c-5 a d)\right )}{\sqrt {d x^2+c}}dx}{5 d}\right )}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b \left (\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 d}-\frac {\frac {\int -\frac {\left (48 b^3 c^3-128 a b^2 d c^2+103 a^2 b d^2 c-15 a^3 d^3\right ) x^2+a c \left (24 b^2 c^2-61 a b d c+45 a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{3 d}}{5 d}\right )}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\int \frac {\left (48 b^3 c^3-128 a b^2 d c^2+103 a^2 b d^2 c-15 a^3 d^3\right ) x^2+a c \left (24 b^2 c^2-61 a b d c+45 a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}\right )}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{3 d}-\frac {a c \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}\right )}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{5 d}\right )}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{5 d}\right )}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (6 b c-5 a d)}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2-43 a b c d+24 b^2 c^2\right )}{3 d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (45 a^2 d^2-61 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (-15 a^3 d^3+103 a^2 b c d^2-128 a b^2 c^2 d+48 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{5 d}\right )}{c d}-\frac {x \left (a+b x^2\right )^{5/2} (b c-a d)}{c d \sqrt {c+d x^2}}\) |
-(((b*c - a*d)*x*(a + b*x^2)^(5/2))/(c*d*Sqrt[c + d*x^2])) + (b*(((6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*d) - (((24*b^2*c^2 - 43*a* b*c*d + 15*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)*Sq rt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sq rt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*d))/(5*d) ))/(c*d)
3.3.7.3.1 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[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 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 = 9.53 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.58
| method | result | size |
| risch | \(\frac {b^{2} x \left (3 b d \,x^{2}+16 a d -9 b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{15 d^{3}}+\frac {\left (-\frac {b^{2} \left (58 a^{2} d^{2}-83 a b c d +33 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}+\frac {b \left (60 a^{3} d^{3}-106 a^{2} b c \,d^{2}+69 a \,b^{2} c^{2} d -15 b^{3} c^{3}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {\left (15 a^{4} d^{4}-60 a^{3} b c \,d^{3}+90 a^{2} b^{2} c^{2} d^{2}-60 a \,b^{3} c^{3} d +15 b^{4} c^{4}\right ) \left (\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{c \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{15 d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(701\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {\left (b d \,x^{2}+a d \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{c \,d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {b^{3} x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5 d^{2}}+\frac {\left (\frac {b^{3} \left (4 a d -b c \right )}{d^{2}}-\frac {b^{3} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}+\frac {\left (\frac {b \left (4 a^{3} d^{3}-6 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (a d -b c \right )}{d^{4} c}-\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{3} c}-\frac {\left (\frac {b^{3} \left (4 a d -b c \right )}{d^{2}}-\frac {b^{3} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (\frac {b^{2} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right )}{d^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{d^{3} c}-\frac {3 b^{3} a c}{5 d^{2}}-\frac {\left (\frac {b^{3} \left (4 a d -b c \right )}{d^{2}}-\frac {b^{3} \left (4 a d +4 b c \right )}{5 d^{2}}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(744\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (3 \sqrt {-\frac {b}{a}}\, b^{4} c \,d^{3} x^{7}+19 \sqrt {-\frac {b}{a}}\, a \,b^{3} c \,d^{3} x^{5}-6 \sqrt {-\frac {b}{a}}\, b^{4} c^{2} d^{2} x^{5}+15 \sqrt {-\frac {b}{a}}\, a^{3} b \,d^{4} x^{3}-29 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c \,d^{3} x^{3}+55 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{2} d^{2} x^{3}-24 \sqrt {-\frac {b}{a}}\, b^{4} c^{3} d \,x^{3}+60 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} b c \,d^{3}-164 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c^{2} d^{2}+152 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{3} d -48 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{4} c^{4}-15 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} b c \,d^{3}+103 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c^{2} d^{2}-128 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{3} d +48 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{4} c^{4}+15 \sqrt {-\frac {b}{a}}\, a^{4} d^{4} x -45 \sqrt {-\frac {b}{a}}\, a^{3} b c \,d^{3} x +61 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c^{2} d^{2} x -24 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{3} d x \right )}{15 d^{4} \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}\, c}\) | \(755\) |
1/15*b^2*x*(3*b*d*x^2+16*a*d-9*b*c)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3+1/ 15/d^3*(-b^2*(58*a^2*d^2-83*a*b*c*d+33*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*(60*a^3*d^3-106*a^2*b*c*d^2+69*a*b^2*c^2*d-15*b^3*c^3 )/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))+(15*a^4* d^4-60*a^3*b*c*d^3+90*a^2*b^2*c^2*d^2-60*a*b^3*c^3*d+15*b^4*c^4)/d*((b*d*x ^2+a*d)/c/(a*d-b*c)*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(1/c-1/c/(a*d-b*c)*a *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/(a*d-b *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)*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-Ellipti cE(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 = 472, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (48 \, b^{3} c^{4} d - 128 \, a b^{2} c^{3} d^{2} + 103 \, a^{2} b c^{2} d^{3} - 15 \, a^{3} c d^{4}\right )} x^{3} + {\left (48 \, b^{3} c^{5} - 128 \, a b^{2} c^{4} d + 103 \, a^{2} b c^{3} d^{2} - 15 \, a^{3} c^{2} d^{3}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (48 \, b^{3} c^{4} d - 128 \, a b^{2} c^{3} d^{2} + 45 \, a^{3} d^{5} + {\left (103 \, a^{2} b + 24 \, a b^{2}\right )} c^{2} d^{3} - {\left (15 \, a^{3} + 61 \, a^{2} b\right )} c d^{4}\right )} x^{3} + {\left (48 \, b^{3} c^{5} - 128 \, a b^{2} c^{4} d + 45 \, a^{3} c d^{4} + {\left (103 \, a^{2} b + 24 \, a b^{2}\right )} c^{3} d^{2} - {\left (15 \, a^{3} + 61 \, a^{2} b\right )} c^{2} d^{3}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{3} c d^{4} x^{6} + 48 \, b^{3} c^{4} d - 128 \, a b^{2} c^{3} d^{2} + 103 \, a^{2} b c^{2} d^{3} - 15 \, a^{3} c d^{4} - 2 \, {\left (3 \, b^{3} c^{2} d^{3} - 8 \, a b^{2} c d^{4}\right )} x^{4} + {\left (24 \, b^{3} c^{3} d^{2} - 67 \, a b^{2} c^{2} d^{3} + 58 \, a^{2} b c d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left (c d^{6} x^{3} + c^{2} d^{5} x\right )}} \]
-1/15*(((48*b^3*c^4*d - 128*a*b^2*c^3*d^2 + 103*a^2*b*c^2*d^3 - 15*a^3*c*d ^4)*x^3 + (48*b^3*c^5 - 128*a*b^2*c^4*d + 103*a^2*b*c^3*d^2 - 15*a^3*c^2*d ^3)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((48*b^3*c^4*d - 128*a*b^2*c^3*d^2 + 45*a^3*d^5 + (103*a^2*b + 24*a*b^2)*c ^2*d^3 - (15*a^3 + 61*a^2*b)*c*d^4)*x^3 + (48*b^3*c^5 - 128*a*b^2*c^4*d + 45*a^3*c*d^4 + (103*a^2*b + 24*a*b^2)*c^3*d^2 - (15*a^3 + 61*a^2*b)*c^2*d^ 3)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ( 3*b^3*c*d^4*x^6 + 48*b^3*c^4*d - 128*a*b^2*c^3*d^2 + 103*a^2*b*c^2*d^3 - 1 5*a^3*c*d^4 - 2*(3*b^3*c^2*d^3 - 8*a*b^2*c*d^4)*x^4 + (24*b^3*c^3*d^2 - 67 *a*b^2*c^2*d^3 + 58*a^2*b*c*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(c* d^6*x^3 + c^2*d^5*x)
\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {7}{2}}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{7/2}}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]