Integrand size = 30, antiderivative size = 407 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\left (15 a^2 d^2 f+a b d (30 d e-41 c f)-4 b^2 c (5 d e-6 c f)\right ) x \sqrt {a+b x^2}}{15 d^3 \sqrt {c+d x^2}}+\frac {(5 b d e-6 b c f+5 a d f) x \left (a+b x^2\right )^{3/2}}{15 d^2 \sqrt {c+d x^2}}+\frac {f x \left (a+b x^2\right )^{5/2}}{5 d \sqrt {c+d x^2}}-\frac {\left (a b c d (65 d e-88 c f)-a^2 d^2 (15 d e-38 c f)-8 b^2 c^2 (5 d e-6 c f)\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \left (15 a^2 d^2 f+a b d (30 d e-41 c f)-4 b^2 c (5 d e-6 c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/15*(15*a^2*d^2*f+a*b*d*(-41*c*f+30*d*e)-4*b^2*c*(-6*c*f+5*d*e))*x*(b*x^2 +a)^(1/2)/d^3/(d*x^2+c)^(1/2)+1/15*(5*a*d*f-6*b*c*f+5*b*d*e)*x*(b*x^2+a)^( 3/2)/d^2/(d*x^2+c)^(1/2)+1/5*f*x*(b*x^2+a)^(5/2)/d/(d*x^2+c)^(1/2)-1/15*(a *b*c*d*(-88*c*f+65*d*e)-a^2*d^2*(-38*c*f+15*d*e)-8*b^2*c^2*(-6*c*f+5*d*e)) *(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d) ^(1/2))/c^(1/2)/d^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/ 15*c^(1/2)*(15*a^2*d^2*f+a*b*d*(-41*c*f+30*d*e)-4*b^2*c*(-6*c*f+5*d*e))*(b *x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2)) /d^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 6.58 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (15 a^2 d^2 (d e-c f)+a b c d \left (-30 d e+41 c f+11 d f x^2\right )+b^2 c \left (-24 c^2 f+c d \left (20 e-6 f x^2\right )+d^2 x^2 \left (5 e+3 f x^2\right )\right )\right )-i b c \left (a b c d (65 d e-88 c f)+8 b^2 c^2 (-5 d e+6 c f)+a^2 d^2 (-15 d e+38 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (15 a^2 d^2 f+a b d (45 d e-64 c f)+8 b^2 c (-5 d e+6 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} c d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(15*a^2*d^2*(d*e - c*f) + a*b*c*d*(-30*d*e + 41 *c*f + 11*d*f*x^2) + b^2*c*(-24*c^2*f + c*d*(20*e - 6*f*x^2) + d^2*x^2*(5* e + 3*f*x^2))) - I*b*c*(a*b*c*d*(65*d*e - 88*c*f) + 8*b^2*c^2*(-5*d*e + 6* c*f) + a^2*d^2*(-15*d*e + 38*c*f))*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*(-(b*c) + a*d)*(15*a ^2*d^2*f + a*b*d*(45*d*e - 64*c*f) + 8*b^2*c*(-5*d*e + 6*c*f))*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 = 0.65 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {401, 25, 403, 403, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{c d \sqrt {c+d x^2}}-\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (a c f-b (5 d e-6 c f) x^2\right )}{\sqrt {d x^2+c}}dx}{c d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (a c f-b (5 d e-6 c f) x^2\right )}{\sqrt {d x^2+c}}dx}{c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {b x^2+a} \left (a c (5 b d e-6 b c f+5 a d f)-b (a d (15 d e-23 c f)-4 b c (5 d e-6 c f)) x^2\right )}{\sqrt {d x^2+c}}dx}{5 d}-\frac {b x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}}{c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b \left (-8 b^2 (5 d e-6 c f) c^2+a b d (65 d e-88 c f) c-a^2 d^2 (15 d e-38 c f)\right ) x^2+a c \left (-4 c (5 d e-6 c f) b^2+a d (30 d e-41 c f) b+15 a^2 d^2 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (15 d e-23 c f)-4 b c (5 d e-6 c f))}{3 d}}{5 d}-\frac {b x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}}{c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {\frac {b \left (-a^2 d^2 (15 d e-38 c f)+a b c d (65 d e-88 c f)-8 b^2 c^2 (5 d e-6 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c \left (15 a^2 d^2 f+a b d (30 d e-41 c f)-4 b^2 c (5 d e-6 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (15 d e-23 c f)-4 b c (5 d e-6 c f))}{3 d}}{5 d}-\frac {b x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}}{c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {b \left (-a^2 d^2 (15 d e-38 c f)+a b c d (65 d e-88 c f)-8 b^2 c^2 (5 d e-6 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (15 a^2 d^2 f+a b d (30 d e-41 c f)-4 b^2 c (5 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (15 d e-23 c f)-4 b c (5 d e-6 c f))}{3 d}}{5 d}-\frac {b x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}}{c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {\frac {b \left (-a^2 d^2 (15 d e-38 c f)+a b c d (65 d e-88 c f)-8 b^2 c^2 (5 d e-6 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (15 a^2 d^2 f+a b d (30 d e-41 c f)-4 b^2 c (5 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (15 d e-23 c f)-4 b c (5 d e-6 c f))}{3 d}}{5 d}-\frac {b x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}}{c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{c d \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (15 a^2 d^2 f+a b d (30 d e-41 c f)-4 b^2 c (5 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b \left (-a^2 d^2 (15 d e-38 c f)+a b c d (65 d e-88 c f)-8 b^2 c^2 (5 d e-6 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (15 d e-23 c f)-4 b c (5 d e-6 c f))}{3 d}}{5 d}-\frac {b x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 d e-6 c f)}{5 d}}{c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{c d \sqrt {c+d x^2}}\) |
Input:
Int[((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(3/2),x]
Output:
((d*e - c*f)*x*(a + b*x^2)^(5/2))/(c*d*Sqrt[c + d*x^2]) + (-1/5*(b*(5*d*e - 6*c*f)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/d + (-1/3*(b*(a*d*(15*d*e - 23*c*f) - 4*b*c*(5*d*e - 6*c*f))*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/d + (b *(a*b*c*d*(65*d*e - 88*c*f) - a^2*d^2*(15*d*e - 38*c*f) - 8*b^2*c^2*(5*d*e - 6*c*f))*((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]*S qrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(15*a^2* d^2*f + a*b*d*(30*d*e - 41*c*f) - 4*b^2*c*(5*d*e - 6*c*f))*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))/(c*d)
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[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[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(792\) vs. \(2(372)=744\).
Time = 15.14 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.95
| method | result | size |
| risch | \(\frac {b x \left (3 b d f \,x^{2}+11 a d f -9 b c f +5 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 d^{3}}+\frac {\left (\frac {\left (15 f \,d^{3} a^{3}-56 a^{2} b c \,d^{2} f +45 a^{2} b \,d^{3} e +54 a \,b^{2} c^{2} d f -50 a \,b^{2} c \,d^{2} e -15 b^{3} c^{3} f +15 b^{3} c^{2} d e \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 )}{d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b \left (23 f \,d^{2} a^{2}-58 f d c b a +35 a b \,d^{2} e +33 f \,c^{2} b^{2}-25 d \,b^{2} c e \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 {15 \left (a^{3} c f \,d^{3}-a^{3} e \,d^{4}-3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -b^{3} c^{4} f +b^{3} c^{3} d e \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}{\left (a d -b c \right ) c}\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 {b \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 )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(793\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+a d \right ) \left (a^{2} c f \,d^{2}-a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right ) x}{c \,d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {b^{2} f \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 d^{2}}+\frac {\left (\frac {b^{2} \left (3 a d f -b c f +b d e \right )}{d^{2}}-\frac {b^{2} f \left (4 a d +4 b c \right )}{5 d^{2}}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (\frac {f \,d^{3} a^{3}-3 a^{2} b c \,d^{2} f +3 a^{2} b \,d^{3} e +3 a \,b^{2} c^{2} d f -3 a \,b^{2} c \,d^{2} e -b^{3} c^{3} f +b^{3} c^{2} d e}{d^{4}}-\frac {\left (a^{2} c f \,d^{2}-a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right ) \left (a d -b c \right )}{d^{4} c}+\frac {a \left (a^{2} c f \,d^{2}-a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right )}{d^{3} c}-\frac {\left (\frac {b^{2} \left (3 a d f -b c f +b d e \right )}{d^{2}}-\frac {b^{2} f \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}}\, \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 (\frac {b \left (3 f \,d^{2} a^{2}-3 f d c b a +3 a b \,d^{2} e +f \,c^{2} b^{2}-d \,b^{2} c e \right )}{d^{3}}+\frac {\left (a^{2} c f \,d^{2}-a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right ) b}{d^{3} c}-\frac {3 a \,b^{2} c f}{5 d^{2}}-\frac {\left (\frac {b^{2} \left (3 a d f -b c f +b d e \right )}{d^{2}}-\frac {b^{2} f \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 (\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}}\) | \(892\) |
| default | \(\text {Expression too large to display}\) | \(1169\) |
Input:
int((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/15*b*x*(3*b*d*f*x^2+11*a*d*f-9*b*c*f+5*b*d*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^ (1/2)/d^3+1/15/d^3*((15*a^3*d^3*f-56*a^2*b*c*d^2*f+45*a^2*b*d^3*e+54*a*b^2 *c^2*d*f-50*a*b^2*c*d^2*e-15*b^3*c^3*f+15*b^3*c^2*d*e)/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)*Ellipt icF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-b*(23*a^2*d^2*f-58*a*b*c*d*f+ 35*a*b*d^2*e+33*b^2*c^2*f-25*b^2*c*d*e)*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)))-15*(a^3*c*d^3*f-a^3*d^4*e-3*a^2*b*c^2*d^2*f+3*a^2*b*c*d^3*e+3*a *b^2*c^3*d*f-3*a*b^2*c^2*d^2*e-b^3*c^4*f+b^3*c^3*d*e)/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/(a*d-b*c)/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))-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 = 0.11 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (5 \, {\left (8 \, b^{3} c^{3} d^{2} - 13 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} e - 2 \, {\left (24 \, b^{3} c^{4} d - 44 \, a b^{2} c^{3} d^{2} + 19 \, a^{2} b c^{2} d^{3}\right )} f\right )} x^{3} + {\left (5 \, {\left (8 \, b^{3} c^{4} d - 13 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3}\right )} e - 2 \, {\left (24 \, b^{3} c^{5} - 44 \, a b^{2} c^{4} d + 19 \, a^{2} b c^{3} d^{2}\right )} f\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 (5 \, {\left (8 \, b^{3} c^{3} d^{2} - 13 \, a b^{2} c^{2} d^{3} - 6 \, a^{2} b d^{5} + {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} c d^{4}\right )} e - {\left (48 \, b^{3} c^{4} d - 88 \, a b^{2} c^{3} d^{2} - 41 \, a^{2} b c d^{4} + 15 \, a^{3} d^{5} + 2 \, {\left (19 \, a^{2} b + 12 \, a b^{2}\right )} c^{2} d^{3}\right )} f\right )} x^{3} + {\left (5 \, {\left (8 \, b^{3} c^{4} d - 13 \, a b^{2} c^{3} d^{2} - 6 \, a^{2} b c d^{4} + {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} c^{2} d^{3}\right )} e - {\left (48 \, b^{3} c^{5} - 88 \, a b^{2} c^{4} d - 41 \, a^{2} b c^{2} d^{3} + 15 \, a^{3} c d^{4} + 2 \, {\left (19 \, a^{2} b + 12 \, a b^{2}\right )} c^{3} d^{2}\right )} f\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} f x^{6} + {\left (5 \, b^{3} c d^{4} e - {\left (6 \, b^{3} c^{2} d^{3} - 11 \, a b^{2} c d^{4}\right )} f\right )} x^{4} - {\left (5 \, {\left (4 \, b^{3} c^{2} d^{3} - 7 \, a b^{2} c d^{4}\right )} e - {\left (24 \, b^{3} c^{3} d^{2} - 47 \, a b^{2} c^{2} d^{3} + 23 \, a^{2} b c d^{4}\right )} f\right )} x^{2} - 5 \, {\left (8 \, b^{3} c^{3} d^{2} - 13 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} e + 2 \, {\left (24 \, b^{3} c^{4} d - 44 \, a b^{2} c^{3} d^{2} + 19 \, a^{2} b c^{2} d^{3}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left (b c d^{6} x^{3} + b c^{2} d^{5} x\right )}} \] Input:
integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
1/15*(((5*(8*b^3*c^3*d^2 - 13*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4)*e - 2*(24*b^3 *c^4*d - 44*a*b^2*c^3*d^2 + 19*a^2*b*c^2*d^3)*f)*x^3 + (5*(8*b^3*c^4*d - 1 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3)*e - 2*(24*b^3*c^5 - 44*a*b^2*c^4*d + 19 *a^2*b*c^3*d^2)*f)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x) , a*d/(b*c)) - ((5*(8*b^3*c^3*d^2 - 13*a*b^2*c^2*d^3 - 6*a^2*b*d^5 + (3*a^ 2*b + 4*a*b^2)*c*d^4)*e - (48*b^3*c^4*d - 88*a*b^2*c^3*d^2 - 41*a^2*b*c*d^ 4 + 15*a^3*d^5 + 2*(19*a^2*b + 12*a*b^2)*c^2*d^3)*f)*x^3 + (5*(8*b^3*c^4*d - 13*a*b^2*c^3*d^2 - 6*a^2*b*c*d^4 + (3*a^2*b + 4*a*b^2)*c^2*d^3)*e - (48 *b^3*c^5 - 88*a*b^2*c^4*d - 41*a^2*b*c^2*d^3 + 15*a^3*c*d^4 + 2*(19*a^2*b + 12*a*b^2)*c^3*d^2)*f)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/ d)/x), a*d/(b*c)) + (3*b^3*c*d^4*f*x^6 + (5*b^3*c*d^4*e - (6*b^3*c^2*d^3 - 11*a*b^2*c*d^4)*f)*x^4 - (5*(4*b^3*c^2*d^3 - 7*a*b^2*c*d^4)*e - (24*b^3*c ^3*d^2 - 47*a*b^2*c^2*d^3 + 23*a^2*b*c*d^4)*f)*x^2 - 5*(8*b^3*c^3*d^2 - 13 *a*b^2*c^2*d^3 + 3*a^2*b*c*d^4)*e + 2*(24*b^3*c^4*d - 44*a*b^2*c^3*d^2 + 1 9*a^2*b*c^2*d^3)*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b*c*d^6*x^3 + b*c^2* d^5*x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(5/2)*(f*x**2+e)/(d*x**2+c)**(3/2),x)
Output:
Integral((a + b*x**2)**(5/2)*(e + f*x**2)/(c + d*x**2)**(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)/(d*x^2 + c)^(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)/(d*x^2 + c)^(3/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\left (f\,x^2+e\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int(((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(3/2),x)
Output:
int(((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x)
Output:
(15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**2*f*x - 33*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a**2*b*c*d*f*x + 45*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a** 2*b*d**2*e*x + 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*f*x - 15*s qrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d*e*x + 22*sqrt(c + d*x**2)*sqrt (a + b*x**2)*a*b**2*c*d*f*x**3 - 12*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3 *c**2*f*x**3 + 10*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d*e*x**3 + 6*sq rt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d*f*x**5 - 15*int((sqrt(c + d*x**2) *sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**3*b*c*d**3*f - 15*int((sqrt(c + d*x** 2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x* *2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**3*b*d**4*f*x**2 + 79*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c* *2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b**2*c**2*d**2*f - 45*int((s qrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x** 4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b**2*c*d**3*e + 79*i nt((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d** 2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b**2*c*d**3*f*x **2 - 45*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x* *2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b**2* d**4*e*x**2 - 112*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 ...