Integrand size = 30, antiderivative size = 530 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\left (15 a^3 d^3 f-a b^2 c d (161 d e-128 c f)+a^2 b d^2 (161 d e-103 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) x \sqrt {c+d x^2}}{105 d^4 \sqrt {a+b x^2}}+\frac {\left (15 a^2 d^2 f+a b d (56 d e-43 c f)-4 b^2 c (7 d e-6 c f)\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 d^3}+\frac {(7 b d e-6 b c f+5 a d f) x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{35 d^2}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}-\frac {\sqrt {a} \left (15 a^3 d^3 f-a b^2 c d (161 d e-128 c f)+a^2 b d^2 (161 d e-103 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 \sqrt {b} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (a b c d (77 d e-61 c f)-4 b^2 c^2 (7 d e-6 c f)-15 a^2 d^2 (7 d e-3 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 \sqrt {b} c d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/105*(15*a^3*d^3*f-a*b^2*c*d*(-128*c*f+161*d*e)+a^2*b*d^2*(-103*c*f+161*d *e)+8*b^3*c^2*(-6*c*f+7*d*e))*x*(d*x^2+c)^(1/2)/d^4/(b*x^2+a)^(1/2)+1/105* (15*a^2*d^2*f+a*b*d*(-43*c*f+56*d*e)-4*b^2*c*(-6*c*f+7*d*e))*x*(b*x^2+a)^( 1/2)*(d*x^2+c)^(1/2)/d^3+1/35*(5*a*d*f-6*b*c*f+7*b*d*e)*x*(b*x^2+a)^(3/2)* (d*x^2+c)^(1/2)/d^2+1/7*f*x*(b*x^2+a)^(5/2)*(d*x^2+c)^(1/2)/d-1/105*a^(1/2 )*(15*a^3*d^3*f-a*b^2*c*d*(-128*c*f+161*d*e)+a^2*b*d^2*(-103*c*f+161*d*e)+ 8*b^3*c^2*(-6*c*f+7*d*e))*(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)*(a*b*c*d*(-61*c*f+77*d*e)-4*b^2*c^2*(-6*c *f+7*d*e)-15*a^2*d^2*(-3*c*f+7*d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arcta n(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/d^3/(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 = 4.91 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (45 a^2 d^2 f+a b d \left (77 d e-61 c f+45 d f x^2\right )+b^2 \left (24 c^2 f+3 d^2 x^2 \left (7 e+5 f x^2\right )-2 c d \left (14 e+9 f x^2\right )\right )\right )-i c \left (15 a^3 d^3 f+a^2 b d^2 (161 d e-103 c f)-8 b^3 c^2 (-7 d e+6 c f)+a b^2 c d (-161 d e+128 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) \left (a b c d (133 d e-104 c f)+15 a^2 d^2 (-7 d e+4 c f)+8 b^2 c^2 (-7 d e+6 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((a + b*x^2)^(5/2)*(e + f*x^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*f + a*b*d*(77*d*e - 61* c*f + 45*d*f*x^2) + b^2*(24*c^2*f + 3*d^2*x^2*(7*e + 5*f*x^2) - 2*c*d*(14* e + 9*f*x^2))) - I*c*(15*a^3*d^3*f + a^2*b*d^2*(161*d*e - 103*c*f) - 8*b^3 *c^2*(-7*d*e + 6*c*f) + a*b^2*c*d*(-161*d*e + 128*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*( -(b*c) + a*d)*(a*b*c*d*(133*d*e - 104*c*f) + 15*a^2*d^2*(-7*d*e + 4*c*f) + 8*b^2*c^2*(-7*d*e + 6*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellip ticF[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])
Time = 0.73 (sec) , antiderivative size = 482, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {403, 403, 25, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left ((7 b d e-6 b c f+5 a d f) x^2+a (7 d e-c f)\right )}{\sqrt {d x^2+c}}dx}{7 d}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int -\frac {\sqrt {b x^2+a} \left (a (b c (7 d e-6 c f)-5 a d (7 d e-2 c f))-\left (-4 c (7 d e-6 c f) b^2+a d (56 d e-43 c f) b+15 a^2 d^2 f\right ) x^2\right )}{\sqrt {d x^2+c}}dx}{5 d}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 a d f-6 b c f+7 b d e)}{5 d}}{7 d}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 a d f-6 b c f+7 b d e)}{5 d}-\frac {\int \frac {\sqrt {b x^2+a} \left (a (b c (7 d e-6 c f)-5 a d (7 d e-2 c f))-\left (-4 c (7 d e-6 c f) b^2+a d (56 d e-43 c f) b+15 a^2 d^2 f\right ) x^2\right )}{\sqrt {d x^2+c}}dx}{5 d}}{7 d}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 a d f-6 b c f+7 b d e)}{5 d}-\frac {\frac {\int \frac {a \left (-4 b^2 (7 d e-6 c f) c^2+a b d (77 d e-61 c f) c-15 a^2 d^2 (7 d e-3 c f)\right )-\left (8 c^2 (7 d e-6 c f) b^3-a c d (161 d e-128 c f) b^2+a^2 d^2 (161 d e-103 c f) b+15 a^3 d^3 f\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2 f+a b d (56 d e-43 c f)-4 b^2 c (7 d e-6 c f)\right )}{3 d}}{5 d}}{7 d}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 a d f-6 b c f+7 b d e)}{5 d}-\frac {\frac {a \left (-15 a^2 d^2 (7 d e-3 c f)+a b c d (77 d e-61 c f)-4 b^2 c^2 (7 d e-6 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\left (15 a^3 d^3 f+a^2 b d^2 (161 d e-103 c f)-a b^2 c d (161 d e-128 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2 f+a b d (56 d e-43 c f)-4 b^2 c (7 d e-6 c f)\right )}{3 d}}{5 d}}{7 d}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 a d f-6 b c f+7 b d e)}{5 d}-\frac {\frac {\frac {\sqrt {c} \sqrt {a+b x^2} \left (-15 a^2 d^2 (7 d e-3 c f)+a b c d (77 d e-61 c f)-4 b^2 c^2 (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (15 a^3 d^3 f+a^2 b d^2 (161 d e-103 c f)-a b^2 c d (161 d e-128 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2 f+a b d (56 d e-43 c f)-4 b^2 c (7 d e-6 c f)\right )}{3 d}}{5 d}}{7 d}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 a d f-6 b c f+7 b d e)}{5 d}-\frac {\frac {\frac {\sqrt {c} \sqrt {a+b x^2} \left (-15 a^2 d^2 (7 d e-3 c f)+a b c d (77 d e-61 c f)-4 b^2 c^2 (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (15 a^3 d^3 f+a^2 b d^2 (161 d e-103 c f)-a b^2 c d (161 d e-128 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2 f+a b d (56 d e-43 c f)-4 b^2 c (7 d e-6 c f)\right )}{3 d}}{5 d}}{7 d}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (5 a d f-6 b c f+7 b d e)}{5 d}-\frac {\frac {\frac {\sqrt {c} \sqrt {a+b x^2} \left (-15 a^2 d^2 (7 d e-3 c f)+a b c d (77 d e-61 c f)-4 b^2 c^2 (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (15 a^3 d^3 f+a^2 b d^2 (161 d e-103 c f)-a b^2 c d (161 d e-128 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (15 a^2 d^2 f+a b d (56 d e-43 c f)-4 b^2 c (7 d e-6 c f)\right )}{3 d}}{5 d}}{7 d}+\frac {f x \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{7 d}\) |
Input:
Int[((a + b*x^2)^(5/2)*(e + f*x^2))/Sqrt[c + d*x^2],x]
Output:
(f*x*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(7*d) + (((7*b*d*e - 6*b*c*f + 5*a *d*f)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*d) - (-1/3*((15*a^2*d^2*f + a*b*d*(56*d*e - 43*c*f) - 4*b^2*c*(7*d*e - 6*c*f))*x*Sqrt[a + b*x^2]*Sqrt[ c + d*x^2])/d + (-((15*a^3*d^3*f - a*b^2*c*d*(161*d*e - 128*c*f) + a^2*b*d ^2*(161*d*e - 103*c*f) + 8*b^3*c^2*(7*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]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2 ))]*Sqrt[c + d*x^2]))) + (Sqrt[c]*(a*b*c*d*(77*d*e - 61*c*f) - 4*b^2*c^2*( 7*d*e - 6*c*f) - 15*a^2*d^2*(7*d*e - 3*c*f))*Sqrt[a + b*x^2]*EllipticF[Arc Tan[(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)
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[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 = 14.56 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.30
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {b^{2} f \,x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 d}+\frac {\left (3 a \,b^{2} f +b^{3} e -\frac {b^{2} f \left (6 a d +6 b c \right )}{7 d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (3 a^{2} b f +3 a \,b^{2} e -\frac {5 a \,b^{2} c f}{7 d}-\frac {\left (3 a \,b^{2} f +b^{3} e -\frac {b^{2} f \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a^{3} e -\frac {\left (3 a^{2} b f +3 a \,b^{2} e -\frac {5 a \,b^{2} c f}{7 d}-\frac {\left (3 a \,b^{2} f +b^{3} e -\frac {b^{2} f \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (a^{3} f +3 a^{2} b e -\frac {3 \left (3 a \,b^{2} f +b^{3} e -\frac {b^{2} f \left (6 a d +6 b c \right )}{7 d}\right ) a c}{5 b d}-\frac {\left (3 a^{2} b f +3 a \,b^{2} e -\frac {5 a \,b^{2} c f}{7 d}-\frac {\left (3 a \,b^{2} f +b^{3} e -\frac {b^{2} f \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(691\) |
| risch | \(\frac {x \left (15 b^{2} d^{2} f \,x^{4}+45 a b \,d^{2} f \,x^{2}-18 b^{2} c f \,x^{2} d +21 b^{2} d^{2} e \,x^{2}+45 f \,d^{2} a^{2}-61 f d c b a +77 a b \,d^{2} e +24 f \,c^{2} b^{2}-28 d \,b^{2} c e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 d^{3}}-\frac {\left (\frac {\left (15 f \,d^{3} a^{3}-103 a^{2} b c \,d^{2} f +161 a^{2} b \,d^{3} e +128 a \,b^{2} c^{2} d f -161 a \,b^{2} c \,d^{2} e -48 b^{3} c^{3} f +56 b^{3} c^{2} d 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 {105 a^{3} d^{3} e \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 a \,b^{2} c^{3} f \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 {45 a^{3} c \,d^{2} f \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 {28 a \,c^{2} e d \,b^{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 {77 a^{2} b c \,d^{2} e \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 f \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}}\) | \(921\) |
| default | \(\text {Expression too large to display}\) | \(1386\) |
Input:
int((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/7*b^2/d*f*x ^5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(3*a*b^2*f+b^3*e-1/7*b^2/d*f*(6 *a*d+6*b*c))/b/d*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(3*a^2*b*f+3* a*b^2*e-5/7*a*b^2*c/d*f-1/5*(3*a*b^2*f+b^3*e-1/7*b^2/d*f*(6*a*d+6*b*c))/b/ d*(4*a*d+4*b*c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(a^3*e-1/3*(3*a ^2*b*f+3*a*b^2*e-5/7*a*b^2*c/d*f-1/5*(3*a*b^2*f+b^3*e-1/7*b^2/d*f*(6*a*d+6 *b*c))/b/d*(4*a*d+4*b*c))/b/d*a*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))-(a^3*f+3*a^2*b*e-3/5*(3*a*b^2*f+b^3*e-1/7*b^2/d*f*( 6*a*d+6*b*c))/b/d*a*c-1/3*(3*a^2*b*f+3*a*b^2*e-5/7*a*b^2*c/d*f-1/5*(3*a*b^ 2*f+b^3*e-1/7*b^2/d*f*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*(2*a*d+2*b*c)) *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))-Ellipt icE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
Time = 0.11 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=-\frac {\sqrt {b d} {\left (7 \, {\left (8 \, b^{3} c^{4} d - 23 \, a b^{2} c^{3} d^{2} + 23 \, a^{2} b c^{2} d^{3}\right )} e - {\left (48 \, b^{3} c^{5} - 128 \, a b^{2} c^{4} d + 103 \, a^{2} b c^{3} d^{2} - 15 \, a^{3} c^{2} d^{3}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (7 \, {\left (8 \, b^{3} c^{4} d - 23 \, a b^{2} c^{3} d^{2} - 11 \, a^{2} b c d^{4} + 15 \, a^{3} d^{5} + {\left (23 \, a^{2} b + 4 \, a b^{2}\right )} c^{2} d^{3}\right )} e - {\left (48 \, b^{3} c^{5} - 128 \, a b^{2} c^{4} d + 45 \, a^{3} c d^{4} + {\left (103 \, a^{2} b + 24 \, a b^{2}\right )} c^{3} d^{2} - {\left (15 \, a^{3} + 61 \, a^{2} b\right )} c^{2} d^{3}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, b^{3} c d^{4} f x^{6} + 3 \, {\left (7 \, b^{3} c d^{4} e - 3 \, {\left (2 \, b^{3} c^{2} d^{3} - 5 \, a b^{2} c d^{4}\right )} f\right )} x^{4} - {\left (7 \, {\left (4 \, b^{3} c^{2} d^{3} - 11 \, a b^{2} c d^{4}\right )} e - {\left (24 \, b^{3} c^{3} d^{2} - 61 \, a b^{2} c^{2} d^{3} + 45 \, a^{2} b c d^{4}\right )} f\right )} x^{2} + 7 \, {\left (8 \, b^{3} c^{3} d^{2} - 23 \, a b^{2} c^{2} d^{3} + 23 \, a^{2} b c d^{4}\right )} e - {\left (48 \, b^{3} c^{4} d - 128 \, a b^{2} c^{3} d^{2} + 103 \, a^{2} b c^{2} d^{3} - 15 \, a^{3} c d^{4}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b c d^{5} x} \] Input:
integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
-1/105*(sqrt(b*d)*(7*(8*b^3*c^4*d - 23*a*b^2*c^3*d^2 + 23*a^2*b*c^2*d^3)*e - (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)*f)* x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - sqrt(b*d)*(7*(8 *b^3*c^4*d - 23*a*b^2*c^3*d^2 - 11*a^2*b*c*d^4 + 15*a^3*d^5 + (23*a^2*b + 4*a*b^2)*c^2*d^3)*e - (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)*f)*x*sqrt(-c/d)*e lliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (15*b^3*c*d^4*f*x^6 + 3*(7*b^ 3*c*d^4*e - 3*(2*b^3*c^2*d^3 - 5*a*b^2*c*d^4)*f)*x^4 - (7*(4*b^3*c^2*d^3 - 11*a*b^2*c*d^4)*e - (24*b^3*c^3*d^2 - 61*a*b^2*c^2*d^3 + 45*a^2*b*c*d^4)* f)*x^2 + 7*(8*b^3*c^3*d^2 - 23*a*b^2*c^2*d^3 + 23*a^2*b*c*d^4)*e - (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)*f)*sqrt(b*x ^2 + a)*sqrt(d*x^2 + c))/(b*c*d^5*x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(5/2)*(f*x**2+e)/(d*x**2+c)**(1/2),x)
Output:
Integral((a + b*x**2)**(5/2)*(e + f*x**2)/sqrt(c + d*x**2), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)/sqrt(d*x^2 + c), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\left (f\,x^2+e\right )}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(1/2), x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x)
Output:
(45*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*f*x - 61*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a*b*c*d*f*x + 77*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d* *2*e*x + 45*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*f*x**3 + 24*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f*x - 28*sqrt(c + d*x**2)*sqrt(a + b* x**2)*b**2*c*d*e*x - 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*f*x**3 + 21*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*e*x**3 + 15*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*b**2*d**2*f*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*f - 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*f + 161*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*d**3*e + 128*int((s qrt(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*f - 161*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*d**2*e - 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*f + 56*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**2*d*e - 45*int((sqrt(c + d*x**2)*s qrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*c*d**2*f + 105*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*d**3*e + 61*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(...