Integrand size = 26, antiderivative size = 312 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2}}{3 a c x^3 \sqrt {a+b x^2}}+\frac {2 (2 b c+a d) \sqrt {c+d x^2}}{3 a^2 c^2 x \sqrt {a+b x^2}}+\frac {\sqrt {b} \left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{5/2} c^2 (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {b} d (4 b c-a d) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/3*(d*x^2+c)^(1/2)/a/c/x^3/(b*x^2+a)^(1/2)+2/3*(a*d+2*b*c)*(d*x^2+c)^(1/ 2)/a^2/c^2/x/(b*x^2+a)^(1/2)+1/3*b^(1/2)*(-2*a^2*d^2-3*a*b*c*d+8*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))/a^(5/2)/c^2/(-a*d+b*c)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1 /2)-1/3*b^(1/2)*d*(-a*d+4*b*c)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1 /2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(3/2)/c^2/(-a*d+b*c)/(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 = 3.41 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (c+d x^2\right ) \left (8 b^3 c^2 x^4+a b^2 c x^2 \left (4 c-3 d x^2\right )+a^3 d \left (c-2 d x^2\right )-a^2 b \left (c^2+2 c d x^2+2 d^2 x^4\right )\right )+i b c \left (-8 b^2 c^2+3 a b c d+2 a^2 d^2\right ) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c \left (-8 b^2 c^2+7 a b c d+a^2 d^2\right ) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^3 \sqrt {\frac {b}{a}} c^2 (-b c+a d) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[1/(x^4*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(-(Sqrt[b/a]*(c + d*x^2)*(8*b^3*c^2*x^4 + a*b^2*c*x^2*(4*c - 3*d*x^2) + a^ 3*d*(c - 2*d*x^2) - a^2*b*(c^2 + 2*c*d*x^2 + 2*d^2*x^4))) + I*b*c*(-8*b^2* c^2 + 3*a*b*c*d + 2*a^2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*E llipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(-8*b^2*c^2 + 7*a*b* c*d + a^2*d^2)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*Arc Sinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^3*Sqrt[b/a]*c^2*(-(b*c) + a*d)*x^3*S qrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.59 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.29, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {374, 25, 445, 445, 25, 27, 406, 320, 388, 313}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}-\frac {\int -\frac {3 b d x^2+4 b c-a d}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 b d x^2+4 b c-a d}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {\int \frac {8 b^2 c^2-3 a b d c-2 a^2 d^2+b d (4 b c-a d) x^2}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-a d)}{3 a c x^3}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {b d \left (\left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) x^2+a c (4 b c-a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c}{a}-\frac {2 a d^2}{c}-3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-a d)}{3 a c x^3}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {b d \left (\left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) x^2+a c (4 b c-a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c}{a}-\frac {2 a d^2}{c}-3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-a d)}{3 a c x^3}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {b d \int \frac {\left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) x^2+a c (4 b c-a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c}{a}-\frac {2 a d^2}{c}-3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-a d)}{3 a c x^3}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {-\frac {\frac {b d \left (\left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (4 b c-a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c}{a}-\frac {2 a d^2}{c}-3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-a d)}{3 a c x^3}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {-\frac {\frac {b d \left (\left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (4 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c}{a}-\frac {2 a d^2}{c}-3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-a d)}{3 a c x^3}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {-\frac {\frac {b d \left (\left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (4 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c}{a}-\frac {2 a d^2}{c}-3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-a d)}{3 a c x^3}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {-\frac {\frac {b d \left (\left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (4 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {8 b^2 c}{a}-\frac {2 a d^2}{c}-3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 b c-a d)}{3 a c x^3}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^3 \sqrt {a+b x^2} (b c-a d)}\) |
Input:
Int[1/(x^4*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(b*Sqrt[c + d*x^2])/(a*(b*c - a*d)*x^3*Sqrt[a + b*x^2]) + (-1/3*((4*b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^3) - (-((((8*b^2*c)/a - 3*b*d - (2*a*d^2)/c)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/x) + (b*d*((8*b^2*c^2 - 3 *a*b*c*d - 2*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)*(4*b*c - a*d)*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])))/(a*c))/(3*a*c))/(a*(b*c - a*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[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -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[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Time = 17.97 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.45
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a^{2} c \,x^{3}}+\frac {\left (2 a d +5 b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a^{3} c^{2} x}-\frac {\left (b d \,x^{2}+b c \right ) b^{2} x}{a^{3} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (-\frac {b d}{3 a^{2} c}+\frac {b^{2}}{a^{3}}+\frac {b^{3} c}{a^{3} \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {d \left (2 a d +5 b c \right ) b}{3 a^{3} c^{2}}+\frac {b^{3} d}{\left (a d -b c \right ) a^{3}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(451\) |
| default | \(\frac {\left (2 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{3} x^{6}+3 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{2} x^{6}-8 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d \,x^{6}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x^{3}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d \,x^{3}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x^{3}-3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d \,x^{3}+8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x^{3}+2 \sqrt {-\frac {b}{a}}\, a^{3} d^{3} x^{4}+4 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{2} x^{4}-\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d \,x^{4}-8 \sqrt {-\frac {b}{a}}\, b^{3} c^{3} x^{4}+\sqrt {-\frac {b}{a}}\, a^{3} c \,d^{2} x^{2}+3 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d \,x^{2}-4 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} x^{2}-\sqrt {-\frac {b}{a}}\, a^{3} c^{2} d +\sqrt {-\frac {b}{a}}\, a^{2} b \,c^{3}\right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{3 \left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, x^{3} c^{2} a^{3} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(668\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-2 a d \,x^{2}-5 x^{2} b c +a c \right )}{3 a^{3} c^{2} x^{3}}-\frac {b \left (\frac {a c 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}}-\frac {2 a d 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}}-\frac {5 b \,c^{2} \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}}-3 b \,c^{2} a \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\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 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 )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 a^{3} c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(724\) |
Input:
int(1/x^4/(b*x^2+a)^(3/2)/(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)*(-1/3/a^2/c*(b *d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3+1/3/a^3/c^2*(2*a*d+5*b*c)*(b*d*x^4+a *d*x^2+b*c*x^2+a*c)^(1/2)/x-(b*d*x^2+b*c)*b^2/a^3/(a*d-b*c)*x/((x^2+a/b)*( b*d*x^2+b*c))^(1/2)+(-1/3/a^2/c*b*d+b^2/a^3+b^3*c/a^3/(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))-(-1/3*d*(2*a*d+5*b*c) *b/a^3/c^2+b^3*d/(a*d-b*c)/a^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) )))
Time = 0.10 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {{\left ({\left (8 \, b^{4} c^{2} - 3 \, a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{5} + {\left (8 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d - 2 \, a^{3} b d^{2}\right )} x^{3}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (8 \, b^{4} c^{2} + {\left (4 \, a^{2} b^{2} - 3 \, a b^{3}\right )} c d - {\left (a^{3} b + 2 \, a^{2} b^{2}\right )} d^{2}\right )} x^{5} + {\left (8 \, a b^{3} c^{2} + {\left (4 \, a^{3} b - 3 \, a^{2} b^{2}\right )} c d - {\left (a^{4} + 2 \, a^{3} b\right )} d^{2}\right )} x^{3}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left (a^{3} b c^{2} - a^{4} c d - {\left (8 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d - 2 \, a^{3} b d^{2}\right )} x^{4} - 2 \, {\left (2 \, a^{2} b^{2} c^{2} - a^{3} b c d - a^{4} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left ({\left (a^{4} b^{2} c^{3} - a^{5} b c^{2} d\right )} x^{5} + {\left (a^{5} b c^{3} - a^{6} c^{2} d\right )} x^{3}\right )}} \] Input:
integrate(1/x^4/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
-1/3*(((8*b^4*c^2 - 3*a*b^3*c*d - 2*a^2*b^2*d^2)*x^5 + (8*a*b^3*c^2 - 3*a^ 2*b^2*c*d - 2*a^3*b*d^2)*x^3)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqr t(-b/a)), a*d/(b*c)) - ((8*b^4*c^2 + (4*a^2*b^2 - 3*a*b^3)*c*d - (a^3*b + 2*a^2*b^2)*d^2)*x^5 + (8*a*b^3*c^2 + (4*a^3*b - 3*a^2*b^2)*c*d - (a^4 + 2* a^3*b)*d^2)*x^3)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d /(b*c)) + (a^3*b*c^2 - a^4*c*d - (8*a*b^3*c^2 - 3*a^2*b^2*c*d - 2*a^3*b*d^ 2)*x^4 - 2*(2*a^2*b^2*c^2 - a^3*b*c*d - a^4*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt (d*x^2 + c))/((a^4*b^2*c^3 - a^5*b*c^2*d)*x^5 + (a^5*b*c^3 - a^6*c^2*d)*x^ 3)
\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^{4} \left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(1/x**4/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(1/(x**4*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*x^4), x)
\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*x^4), x)
Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int(1/(x^4*(a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
Output:
int(1/(x^4*(a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a c +2 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d \,x^{2}+4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c \,x^{2}+2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} b \,d^{2} x^{3}+4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a \,b^{2} c d \,x^{3}+2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a \,b^{2} d^{2} x^{5}+4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) b^{3} c d \,x^{5}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} b c d \,x^{3}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a \,b^{2} c^{2} x^{3}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a \,b^{2} c d \,x^{5}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) b^{3} c^{2} x^{5}}{3 a^{2} c^{2} x^{3} \left (b \,x^{2}+a \right )} \] Input:
int(1/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*c + 2*sqrt(c + d*x**2)*sqrt(a + b* x**2)*a*d*x**2 + 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*x**2 + 2*int((sqr t(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*d**2*x**3 + 4*int((s qrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x** 2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d*x**3 + 2*int(( sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x* *2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*d**2*x**5 + 4*int ((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c + a**2*d*x**2 + 2*a*b*c* x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*b**3*c*d*x**5 + int((s qrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2 *a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*c*d*x**3 + 8*int((sqrt( c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b *d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c**2*x**3 + int((sqrt(c + d *x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x* *4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d*x**5 + 8*int((sqrt(c + d*x** 2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*b**3*c**2*x**5)/(3*a**2*c**2*x**3*(a + b*x* *2))