Integrand size = 26, antiderivative size = 429 \[ \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=-\frac {2 (2 b c-a d) \left (4 b^2 c^2-4 a b c d-a^2 d^2\right ) x \sqrt {c+d x^2}}{35 b d^4 \sqrt {a+b x^2}}+\frac {\left (8 b^2 c^2-11 a b c d+a^2 d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 b d^3}-\frac {2 (3 b c-4 a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}+\frac {2 \sqrt {a} (2 b c-a d) \left (4 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{35 b^{3/2} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (8 b^2 c^2-11 a b c d+a^2 d^2\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{35 b^{3/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-2/35*(-a*d+2*b*c)*(-a^2*d^2-4*a*b*c*d+4*b^2*c^2)*x*(d*x^2+c)^(1/2)/b/d^4/ (b*x^2+a)^(1/2)+1/35*(a^2*d^2-11*a*b*c*d+8*b^2*c^2)*x*(b*x^2+a)^(1/2)*(d*x ^2+c)^(1/2)/b/d^3-2/35*(-4*a*d+3*b*c)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/ d^2+1/7*b*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d+2/35*a^(1/2)*(-a*d+2*b*c)* (-a^2*d^2-4*a*b*c*d+4*b^2*c^2)*(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^(3/2)/d^4/(b*x^2+a)^(1/2)/(a*(d*x^ 2+c)/c/(b*x^2+a))^(1/2)-1/35*a^(3/2)*(a^2*d^2-11*a*b*c*d+8*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^( 3/2)/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 = 2.87 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.71 \[ \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a^2 d^2+a b d \left (-11 c+8 d x^2\right )+b^2 \left (8 c^2-6 c d x^2+5 d^2 x^4\right )\right )+2 i c \left (8 b^3 c^3-12 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (16 b^3 c^3-32 a b^2 c^2 d+15 a^2 b c d^2+a^3 d^3\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{35 b \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(x^4*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(a^2*d^2 + a*b*d*(-11*c + 8*d*x^2) + b^2*(8*c^2 - 6*c*d*x^2 + 5*d^2*x^4)) + (2*I)*c*(8*b^3*c^3 - 12*a*b^2*c^2 *d + 2*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elli pticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(16*b^3*c^3 - 32*a*b^2*c^ 2*d + 15*a^2*b*c*d^2 + a^3*d^3)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El lipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(35*b*Sqrt[b/a]*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.63 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {379, 25, 444, 27, 444, 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.
| \(\displaystyle \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 379 |
| \(\displaystyle \frac {\int -\frac {x^4 \left (2 b (3 b c-4 a d) x^2+a (5 b c-7 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 d}+\frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\int \frac {x^4 \left (2 b (3 b c-4 a d) x^2+a (5 b c-7 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 d}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{5 d}-\frac {\int \frac {3 b x^2 \left (\left (8 b^2 c^2-11 a b d c+a^2 d^2\right ) x^2+2 a c (3 b c-4 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{5 d}-\frac {3 \int \frac {x^2 \left (\left (8 b^2 c^2-11 a b d c+a^2 d^2\right ) x^2+2 a c (3 b c-4 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{5 d}-\frac {3 \left (-\frac {\int \frac {2 (2 b c-a d) \left (4 b^2 c^2-4 a b d c-a^2 d^2\right ) x^2+a c \left (8 b^2 c^2-11 a b d c+a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {a^2 d}{b}+11 a c-\frac {8 b c^2}{d}\right )\right )}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{5 d}-\frac {3 \left (-\frac {a c \left (a^2 d^2-11 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+2 (2 b c-a d) \left (-a^2 d^2-4 a b c d+4 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {a^2 d}{b}+11 a c-\frac {8 b c^2}{d}\right )\right )}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{5 d}-\frac {3 \left (-\frac {2 (2 b c-a d) \left (-a^2 d^2-4 a b c d+4 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-11 a b c d+8 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {a^2 d}{b}+11 a c-\frac {8 b c^2}{d}\right )\right )}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{5 d}-\frac {3 \left (-\frac {2 (2 b c-a d) \left (-a^2 d^2-4 a b c d+4 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-11 a b c d+8 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {a^2 d}{b}+11 a c-\frac {8 b c^2}{d}\right )\right )}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}-\frac {\frac {2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{5 d}-\frac {3 \left (-\frac {2 (2 b c-a d) \left (-a^2 d^2-4 a b c d+4 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (a^2 d^2-11 a b c d+8 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-\frac {a^2 d}{b}+11 a c-\frac {8 b c^2}{d}\right )\right )}{5 d}}{7 d}\) |
Input:
Int[(x^4*(a + b*x^2)^(3/2))/Sqrt[c + d*x^2],x]
Output:
(b*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*d) - ((2*(3*b*c - 4*a*d)*x^3*Sq rt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - (3*(-1/3*((11*a*c - (8*b*c^2)/d - ( a^2*d)/b)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) - (2*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*((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)*(8*b^2*c^2 - 11*a*b*c*d + a^2*d^2)*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*b*d)))/(5*d))/(7*d)
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[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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*e*(m + 2*(p + q) + 1))), x] + Simp[1/(b*(m + 2*(p + q) + 1)) Int[(e *x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*((b*c - a*d)*(m + 1) + b*c*2 *(p + q)) + (d*(b*c - a*d)*(m + 1) + d*2*(q - 1)*(b*c - a*d) + b*c*d*2*(p + q))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0 ] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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[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]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Time = 14.79 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(\frac {x \left (5 b^{2} d^{2} x^{4}+8 x^{2} a b \,d^{2}-6 x^{2} b^{2} c d +a^{2} d^{2}-11 a b c d +8 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{35 b \,d^{3}}-\frac {\left (-\frac {\left (2 a^{3} d^{3}+4 a^{2} b c \,d^{2}-24 a \,b^{2} c^{2} d +16 b^{3} c^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a^{3} c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {8 b^{2} c^{3} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {11 a^{2} b \,c^{2} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{35 b \,d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(568\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {b \,x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 d}+\frac {\left (2 a b -\frac {b \left (6 a d +6 b c \right )}{7 d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (a^{2}-\frac {5 b a c}{7 d}-\frac {\left (2 a b -\frac {b \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}-\frac {\left (a^{2}-\frac {5 b a c}{7 d}-\frac {\left (2 a b -\frac {b \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {3 \left (2 a b -\frac {b \left (6 a d +6 b c \right )}{7 d}\right ) a c}{5 b d}-\frac {\left (a^{2}-\frac {5 b a c}{7 d}-\frac {\left (2 a b -\frac {b \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(572\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (5 \sqrt {-\frac {b}{a}}\, b^{3} d^{4} x^{9}+13 \sqrt {-\frac {b}{a}}\, a \,b^{2} d^{4} x^{7}-\sqrt {-\frac {b}{a}}\, b^{3} c \,d^{3} x^{7}+9 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{4} x^{5}-4 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{3} x^{5}+2 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a^{3} d^{4} x^{3}-2 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{3} x^{3}-9 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d^{2} x^{3}+8 \sqrt {-\frac {b}{a}}\, b^{3} c^{3} d \,x^{3}+\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}+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 ) a^{2} b \,c^{2} d^{2}-32 \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 +16 \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}-2 \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}-4 \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}+24 \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 -16 \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}+\sqrt {-\frac {b}{a}}\, a^{3} c \,d^{3} x -11 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d^{2} x +8 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} d x \right )}{35 b \,d^{4} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}}\) | \(782\) |
Input:
int(x^4*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/35/b*x*(5*b^2*d^2*x^4+8*a*b*d^2*x^2-6*b^2*c*d*x^2+a^2*d^2-11*a*b*c*d+8*b ^2*c^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^3-1/35/b/d^3*(-(2*a^3*d^3+4*a^2* b*c*d^2-24*a*b^2*c^2*d+16*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)))+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))+8*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))-11*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 = 0.12 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.75 \[ \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {2 \, {\left (8 \, b^{3} c^{4} - 12 \, a b^{2} c^{3} d + 2 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (16 \, b^{3} c^{4} - 24 \, a b^{2} c^{3} d + a^{3} d^{4} + 4 \, {\left (a^{2} b + 2 \, a b^{2}\right )} c^{2} d^{2} + {\left (2 \, a^{3} - 11 \, a^{2} b\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (5 \, b^{3} d^{4} x^{6} - 16 \, b^{3} c^{3} d + 24 \, a b^{2} c^{2} d^{2} - 4 \, a^{2} b c d^{3} - 2 \, a^{3} d^{4} - 2 \, {\left (3 \, b^{3} c d^{3} - 4 \, a b^{2} d^{4}\right )} x^{4} + {\left (8 \, b^{3} c^{2} d^{2} - 11 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{35 \, b^{2} d^{5} x} \] Input:
integrate(x^4*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
1/35*(2*(8*b^3*c^4 - 12*a*b^2*c^3*d + 2*a^2*b*c^2*d^2 + a^3*c*d^3)*sqrt(b* d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (16*b^3*c^4 - 24*a*b^2*c^3*d + a^3*d^4 + 4*(a^2*b + 2*a*b^2)*c^2*d^2 + (2*a^3 - 11*a^2 *b)*c*d^3)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b* c)) + (5*b^3*d^4*x^6 - 16*b^3*c^3*d + 24*a*b^2*c^2*d^2 - 4*a^2*b*c*d^3 - 2 *a^3*d^4 - 2*(3*b^3*c*d^3 - 4*a*b^2*d^4)*x^4 + (8*b^3*c^2*d^2 - 11*a*b^2*c *d^3 + a^2*b*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^5*x)
\[ \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{4} \left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x**4*(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(x**4*(a + b*x**2)**(3/2)/sqrt(c + d*x**2), x)
\[ \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{4}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*x^4/sqrt(d*x^2 + c), x)
\[ \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{4}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*x^4/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^4\,{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {d\,x^2+c}} \,d x \] Input:
int((x^4*(a + b*x^2)^(3/2))/(c + d*x^2)^(1/2),x)
Output:
int((x^4*(a + b*x^2)^(3/2))/(c + d*x^2)^(1/2), x)
\[ \int \frac {x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} d^{2} x -11 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b c d x +8 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{3}+8 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x -6 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{3}+5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{5}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} d^{3}-4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b c \,d^{2}+24 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{2} d -16 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{3} c^{3}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} c \,d^{2}+11 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b \,c^{2} d -8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{2} c^{3}}{35 b \,d^{3}} \] Input:
int(x^4*(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*x - 11*sqrt(c + d*x**2)*sqrt( a + b*x**2)*a*b*c*d*x + 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*x**3 + 8*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*x - 6*sqrt(c + d*x**2)*sqr t(a + b*x**2)*b**2*c*d*x**3 + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d** 2*x**5 - 2*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 - 4*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 + 24*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 - 16*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 - 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 + 11*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 - 8*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)/(35*b*d**3)