Integrand size = 26, antiderivative size = 399 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2}}{5 a c x^5 \sqrt {a+b x^2}}+\frac {2 (3 b c+2 a d) \sqrt {c+d x^2}}{15 a^2 c^2 x^3 \sqrt {a+b x^2}}-\frac {\left (24 b^2 c^2+13 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2}}{15 a^3 c^3 x \sqrt {a+b x^2}}-\frac {\sqrt {b} \left (48 b^3 c^3-16 a b^2 c^2 d-9 a^2 b c d^2-8 a^3 d^3\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{7/2} c^3 (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {b} d \left (24 b^2 c^2-5 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 a^{5/2} c^3 (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/5*(d*x^2+c)^(1/2)/a/c/x^5/(b*x^2+a)^(1/2)+2/15*(2*a*d+3*b*c)*(d*x^2+c)^ (1/2)/a^2/c^2/x^3/(b*x^2+a)^(1/2)-1/15*(8*a^2*d^2+13*a*b*c*d+24*b^2*c^2)*( d*x^2+c)^(1/2)/a^3/c^3/x/(b*x^2+a)^(1/2)-1/15*b^(1/2)*(-8*a^3*d^3-9*a^2*b* c*d^2-16*a*b^2*c^2*d+48*b^3*c^3)*(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^(7/2)/c^3/(-a*d+b*c)/(b*x^2+a)^( 1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/15*b^(1/2)*d*(-4*a^2*d^2-5*a*b*c*d+ 24*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))/a^(5/2)/c^3/(-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 = 4.34 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (c+d x^2\right ) \left (-48 b^4 c^3 x^6+8 a b^3 c^2 x^4 \left (-3 c+2 d x^2\right )+a^4 d \left (3 c^2-4 c d x^2+8 d^2 x^4\right )+a^2 b^2 c x^2 \left (6 c^2+11 c d x^2+9 d^2 x^4\right )+a^3 b \left (-3 c^3-2 c^2 d x^2+5 c d^2 x^4+8 d^3 x^6\right )\right )-i b c \left (-48 b^3 c^3+16 a b^2 c^2 d+9 a^2 b c d^2+8 a^3 d^3\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+4 i b c \left (-12 b^3 c^3+10 a b^2 c^2 d+a^2 b c d^2+a^3 d^3\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^4 \sqrt {\frac {b}{a}} c^3 (-b c+a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[1/(x^6*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(-(Sqrt[b/a]*(c + d*x^2)*(-48*b^4*c^3*x^6 + 8*a*b^3*c^2*x^4*(-3*c + 2*d*x^ 2) + a^4*d*(3*c^2 - 4*c*d*x^2 + 8*d^2*x^4) + a^2*b^2*c*x^2*(6*c^2 + 11*c*d *x^2 + 9*d^2*x^4) + a^3*b*(-3*c^3 - 2*c^2*d*x^2 + 5*c*d^2*x^4 + 8*d^3*x^6) )) - I*b*c*(-48*b^3*c^3 + 16*a*b^2*c^2*d + 9*a^2*b*c*d^2 + 8*a^3*d^3)*x^5* Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (4*I)*b*c*(-12*b^3*c^3 + 10*a*b^2*c^2*d + a^2*b*c*d^2 + a^3 *d^3)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt [b/a]*x], (a*d)/(b*c)])/(15*a^4*Sqrt[b/a]*c^3*(-(b*c) + a*d)*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.76 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.29, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {374, 25, 445, 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.
| \(\displaystyle \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}-\frac {\int -\frac {5 b d x^2+6 b c-a d}{x^6 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 b d x^2+6 b c-a d}{x^6 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {\int \frac {24 b^2 c^2-5 a b d c-4 a^2 d^2+3 b d (6 b c-a d) x^2}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3+b d \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right ) x^2}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c}{a}-\frac {4 a d^2}{c}-5 b d\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\int -\frac {b d \left (\left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) x^2+a c \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c}{a}-\frac {4 a d^2}{c}-5 b d\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {b d \left (\left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) x^2+a c \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c}{a}-\frac {4 a d^2}{c}-5 b d\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b d \int \frac {\left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\right ) x^2+a c \left (24 b^2 c^2-5 a b d c-4 a^2 d^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c}{a}-\frac {4 a d^2}{c}-5 b d\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b d \left (a c \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c}{a}-\frac {4 a d^2}{c}-5 b d\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b d \left (\left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c}{a}-\frac {4 a d^2}{c}-5 b d\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b d \left (\left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c}{a}-\frac {4 a d^2}{c}-5 b d\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} \left (-4 a^2 d^2-5 a b c d+24 b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (-8 a^3 d^3-9 a^2 b c d^2-16 a b^2 c^2 d+48 b^3 c^3\right )}{a c x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 b^2 c}{a}-\frac {4 a d^2}{c}-5 b d\right )}{3 x^3}}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 b c-a d)}{5 a c x^5}}{a (b c-a d)}+\frac {b \sqrt {c+d x^2}}{a x^5 \sqrt {a+b x^2} (b c-a d)}\) |
Input:
Int[1/(x^6*(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^5*Sqrt[a + b*x^2]) + (-1/5*((6*b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^5) - (-1/3*(((24*b^2*c)/a - 5 *b*d - (4*a*d^2)/c)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/x^3 - (-(((48*b^3*c^3 - 16*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 8*a^3*d^3)*Sqrt[a + b*x^2]*Sqrt[c + d* x^2])/(a*c*x)) + (b*d*((48*b^3*c^3 - 16*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 8*a^ 3*d^3)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2] *EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[ (c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(24*b^2*c^2 - 5*a*b*c*d - 4*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))]*Sqr t[c + d*x^2])))/(a*c))/(3*a*c))/(5*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 = 19.58 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.34
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 a^{2} c \,x^{5}}+\frac {\left (4 a d +9 b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 a^{3} c^{2} x^{3}}-\frac {\left (8 a^{2} d^{2}+17 a b c d +33 b^{2} c^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 a^{4} c^{3} x}+\frac {\left (b d \,x^{2}+b c \right ) b^{3} x}{a^{4} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {d \left (4 a d +9 b c \right ) b}{15 a^{3} c^{2}}-\frac {b^{3}}{a^{4}}-\frac {b^{4} c}{a^{4} \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {d \left (8 a^{2} d^{2}+17 a b c d +33 b^{2} c^{2}\right ) b}{15 a^{4} c^{3}}-\frac {b^{4} d}{\left (a d -b c \right ) a^{4}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(534\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (8 a^{2} d^{2} x^{4}+17 a b c d \,x^{4}+33 b^{2} c^{2} x^{4}-4 a^{2} c d \,x^{2}-9 a b \,c^{2} x^{2}+3 a^{2} c^{2}\right )}{15 a^{4} c^{3} x^{5}}+\frac {b \left (-\frac {\left (8 a^{2} d^{2}+17 a b c d +33 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\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 {4 c \,a^{2} 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 {9 a 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}}-15 b^{2} c^{3} 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 )}}{15 c^{3} a^{4} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(764\) |
| default | \(-\frac {\left (-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 ) a \,b^{3} c^{3} d \,x^{5}-8 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{3} d \,x^{6}+9 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c \,d^{3} x^{8}+16 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{2} d^{2} x^{8}+13 \sqrt {-\frac {b}{a}}\, a^{3} b c \,d^{3} x^{6}+20 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c^{2} d^{2} x^{6}-48 \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^{4} c^{4} x^{5}+48 \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^{4} c^{4} x^{5}+17 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c^{3} d \,x^{4}-5 \sqrt {-\frac {b}{a}}\, a^{3} b \,c^{3} d \,x^{2}+4 \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} b c \,d^{3} x^{5}+4 \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^{2} c^{2} d^{2} x^{5}+40 \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^{3} c^{3} d \,x^{5}-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 ) a^{3} b c \,d^{3} x^{5}-9 \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^{2} c^{2} d^{2} x^{5}+3 \sqrt {-\frac {b}{a}}\, a^{3} b \,c^{2} d^{2} x^{4}+8 \sqrt {-\frac {b}{a}}\, a^{4} d^{4} x^{6}-48 \sqrt {-\frac {b}{a}}\, b^{4} c^{4} x^{6}+3 \sqrt {-\frac {b}{a}}\, a^{4} c^{3} d -3 \sqrt {-\frac {b}{a}}\, a^{3} b \,c^{4}+8 \sqrt {-\frac {b}{a}}\, a^{3} b \,d^{4} x^{8}-48 \sqrt {-\frac {b}{a}}\, b^{4} c^{3} d \,x^{8}+4 \sqrt {-\frac {b}{a}}\, a^{4} c \,d^{3} x^{4}-24 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{4} x^{4}-\sqrt {-\frac {b}{a}}\, a^{4} c^{2} d^{2} x^{2}+6 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c^{4} x^{2}\right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{15 \left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, x^{5} c^{3} a^{4} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(941\) |
Input:
int(1/x^6/(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)*(-1/5/a^2/c*(b *d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^5+1/15/a^3/c^2*(4*a*d+9*b*c)*(b*d*x^4+ a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3-1/15/a^4/c^3*(8*a^2*d^2+17*a*b*c*d+33*b^2*c ^2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x+(b*d*x^2+b*c)*b^3/a^4/(a*d-b*c)* x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(1/15/a^3/c^2*d*(4*a*d+9*b*c)*b-b^3/a^4- b^4*c/a^4/(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))-(1/15*d*(8*a^2*d^2+17*a*b*c*d+33*b^2*c^2)*b/a^4/c^3-b^4*d/(a*d-b* c)/a^4)*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 = 0.11 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {{\left ({\left (48 \, b^{5} c^{3} - 16 \, a b^{4} c^{2} d - 9 \, a^{2} b^{3} c d^{2} - 8 \, a^{3} b^{2} d^{3}\right )} x^{7} + {\left (48 \, a b^{4} c^{3} - 16 \, a^{2} b^{3} c^{2} d - 9 \, a^{3} b^{2} c d^{2} - 8 \, a^{4} b d^{3}\right )} x^{5}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (48 \, b^{5} c^{3} + 8 \, {\left (3 \, a^{2} b^{3} - 2 \, a b^{4}\right )} c^{2} d - {\left (5 \, a^{3} b^{2} + 9 \, a^{2} b^{3}\right )} c d^{2} - 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2}\right )} d^{3}\right )} x^{7} + {\left (48 \, a b^{4} c^{3} + 8 \, {\left (3 \, a^{3} b^{2} - 2 \, a^{2} b^{3}\right )} c^{2} d - {\left (5 \, a^{4} b + 9 \, a^{3} b^{2}\right )} c d^{2} - 4 \, {\left (a^{5} + 2 \, a^{4} b\right )} d^{3}\right )} x^{5}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, a^{4} b c^{3} - 3 \, a^{5} c^{2} d + {\left (48 \, a b^{4} c^{3} - 16 \, a^{2} b^{3} c^{2} d - 9 \, a^{3} b^{2} c d^{2} - 8 \, a^{4} b d^{3}\right )} x^{6} + {\left (24 \, a^{2} b^{3} c^{3} - 11 \, a^{3} b^{2} c^{2} d - 5 \, a^{4} b c d^{2} - 8 \, a^{5} d^{3}\right )} x^{4} - 2 \, {\left (3 \, a^{3} b^{2} c^{3} - a^{4} b c^{2} d - 2 \, a^{5} c d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left ({\left (a^{5} b^{2} c^{4} - a^{6} b c^{3} d\right )} x^{7} + {\left (a^{6} b c^{4} - a^{7} c^{3} d\right )} x^{5}\right )}} \] Input:
integrate(1/x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
1/15*(((48*b^5*c^3 - 16*a*b^4*c^2*d - 9*a^2*b^3*c*d^2 - 8*a^3*b^2*d^3)*x^7 + (48*a*b^4*c^3 - 16*a^2*b^3*c^2*d - 9*a^3*b^2*c*d^2 - 8*a^4*b*d^3)*x^5)* sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((48*b^ 5*c^3 + 8*(3*a^2*b^3 - 2*a*b^4)*c^2*d - (5*a^3*b^2 + 9*a^2*b^3)*c*d^2 - 4* (a^4*b + 2*a^3*b^2)*d^3)*x^7 + (48*a*b^4*c^3 + 8*(3*a^3*b^2 - 2*a^2*b^3)*c ^2*d - (5*a^4*b + 9*a^3*b^2)*c*d^2 - 4*(a^5 + 2*a^4*b)*d^3)*x^5)*sqrt(a*c) *sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (3*a^4*b*c^3 - 3 *a^5*c^2*d + (48*a*b^4*c^3 - 16*a^2*b^3*c^2*d - 9*a^3*b^2*c*d^2 - 8*a^4*b* d^3)*x^6 + (24*a^2*b^3*c^3 - 11*a^3*b^2*c^2*d - 5*a^4*b*c*d^2 - 8*a^5*d^3) *x^4 - 2*(3*a^3*b^2*c^3 - a^4*b*c^2*d - 2*a^5*c*d^2)*x^2)*sqrt(b*x^2 + a)* sqrt(d*x^2 + c))/((a^5*b^2*c^4 - a^6*b*c^3*d)*x^7 + (a^6*b*c^4 - a^7*c^3*d )*x^5)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^{6} \left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(1/x**6/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(1/(x**6*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} x^{6}} \,d x } \] Input:
integrate(1/x^6/(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^6), x)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} x^{6}} \,d x } \] Input:
integrate(1/x^6/(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^6), x)
Timed out. \[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^6\,{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int(1/(x^6*(a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
Output:
int(1/(x^6*(a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
int(1/x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c + 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*x**2 + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*x**2 + 15*sqrt( c + d*x**2)*sqrt(a + b*x**2)*b*d*x**4 + 15*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 + 15*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*d**2*x**7 + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*x**2 + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**6 + b** 2*c*x**6 + b**2*d*x**8),x)*a**3*d**2*x**5 + 28*int((sqrt(c + d*x**2)*sqrt( a + b*x**2))/(a**2*c*x**2 + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**6 + b* *2*c*x**6 + b**2*d*x**8),x)*a**2*b*c*d*x**5 + 8*int((sqrt(c + d*x**2)*sqrt (a + b*x**2))/(a**2*c*x**2 + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**6 + b **2*c*x**6 + b**2*d*x**8),x)*a**2*b*d**2*x**7 + 24*int((sqrt(c + d*x**2)*s qrt(a + b*x**2))/(a**2*c*x**2 + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**6 + b**2*c*x**6 + b**2*d*x**8),x)*a*b**2*c**2*x**5 + 28*int((sqrt(c + d*x**2 )*sqrt(a + b*x**2))/(a**2*c*x**2 + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x* *6 + b**2*c*x**6 + b**2*d*x**8),x)*a*b**2*c*d*x**7 + 24*int((sqrt(c + d*x* *2)*sqrt(a + b*x**2))/(a**2*c*x**2 + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d* x**6 + b**2*c*x**6 + b**2*d*x**8),x)*b**3*c**2*x**7 + 12*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...