3.8.95 \(\int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx\) [795]

Optimal. Leaf size=857 \[ \frac {x^3 \sqrt {c+d x^4}}{5 b}+\frac {(2 b c-5 a d) x \sqrt {c+d x^4}}{5 b^2 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x^2\right )}-\frac {a \sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \tan ^{-1}\left (\frac {\sqrt {-\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 b^2}-\frac {a \sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} \tan ^{-1}\left (\frac {\sqrt {\frac {b c-a d}{\sqrt {-a} \sqrt {b}}} x}{\sqrt {c+d x^4}}\right )}{4 b^2}-\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{5 b^2 d^{3/4} \sqrt {c+d x^4}}+\frac {\sqrt [4]{c} \left (b^2 c^2+a b c d-5 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{5 b^2 d^{3/4} (b c+a d) \sqrt {c+d x^4}}+\frac {a \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^{5/2} \sqrt [4]{c} \left (\sqrt {-a} \sqrt {b} \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {a \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^{5/2} \sqrt [4]{c} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}} \]

[Out]

1/5*x^3*(d*x^4+c)^(1/2)/b+1/5*(-5*a*d+2*b*c)*x*(d*x^4+c)^(1/2)/b^2/d^(1/2)/(c^(1/2)+x^2*d^(1/2))-1/4*a*arctan(
x*((a*d-b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/(d*x^4+c)^(1/2))*((a*d-b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/b^2-1/4*a*arctan(
x*((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/(d*x^4+c)^(1/2))*((-a*d+b*c)/(-a)^(1/2)/b^(1/2))^(1/2)/b^2-1/5*c^(1/4)
*(-5*a*d+2*b*c)*(cos(2*arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticE(sin(2*ar
ctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b^2/d^(3
/4)/(d*x^4+c)^(1/2)+1/5*c^(1/4)*(-5*a^2*d^2+a*b*c*d+b^2*c^2)*(cos(2*arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*
arctan(d^(1/4)*x/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*((d*x
^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b^2/d^(3/4)/(a*d+b*c)/(d*x^4+c)^(1/2)+1/8*a*(-a*d+b*c)*(cos(2*arctan(d^(1
/4)*x/c^(1/4)))^2)^(1/2)/cos(2*arctan(d^(1/4)*x/c^(1/4)))*EllipticPi(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/4*(b^(
1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))^2/(-a)^(1/2)/b^(1/2)/c^(1/2)/d^(1/2),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*(b^(1
/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b^(5/2)/c^(1/4)/d^(1/4)/((-a)^(1/2)*
b^(1/2)*c^(1/2)-a*d^(1/2))/(d*x^4+c)^(1/2)-1/8*a*(-a*d+b*c)*(cos(2*arctan(d^(1/4)*x/c^(1/4)))^2)^(1/2)/cos(2*a
rctan(d^(1/4)*x/c^(1/4)))*EllipticPi(sin(2*arctan(d^(1/4)*x/c^(1/4))),-1/4*c^(1/2)*(b^(1/2)-(-a)^(1/2)*d^(1/2)
/c^(1/2))^2/(-a)^(1/2)/b^(1/2)/d^(1/2),1/2*2^(1/2))*(c^(1/2)+x^2*d^(1/2))*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))
*((d*x^4+c)/(c^(1/2)+x^2*d^(1/2))^2)^(1/2)/b^(5/2)/c^(1/4)/d^(1/4)/((-a)^(1/2)*b^(1/2)*c^(1/2)+a*d^(1/2))/(d*x
^4+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.33, antiderivative size = 1067, normalized size of antiderivative = 1.25, number of steps used = 13, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {489, 598, 311, 226, 1210, 504, 1231, 1721} \begin {gather*} \frac {\sqrt {d x^4+c} x^3}{5 b}+\frac {(2 b c-5 a d) \sqrt {d x^4+c} x}{5 b^2 \sqrt {d} \left (\sqrt {d} x^2+\sqrt {c}\right )}+\frac {(-a)^{3/4} \sqrt {b c-a d} \text {ArcTan}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 b^{9/4}}+\frac {(-a)^{3/4} \sqrt {a d-b c} \text {ArcTan}\left (\frac {\sqrt {a d-b c} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 b^{9/4}}-\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{5 b^2 d^{3/4} \sqrt {d x^4+c}}+\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{10 b^2 d^{3/4} \sqrt {d x^4+c}}+\frac {a \left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 b^2 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}+\frac {a \left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 b^2 \sqrt [4]{c} (b c+a d) \sqrt {d x^4+c}}+\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^{5/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \text {ArcTan}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^{5/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(x^6*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

(x^3*Sqrt[c + d*x^4])/(5*b) + ((2*b*c - 5*a*d)*x*Sqrt[c + d*x^4])/(5*b^2*Sqrt[d]*(Sqrt[c] + Sqrt[d]*x^2)) + ((
-a)^(3/4)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*b^(9/4)) + ((-a
)^(3/4)*Sqrt[-(b*c) + a*d]*ArcTan[(Sqrt[-(b*c) + a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*b^(9/4)) -
(c^(1/4)*(2*b*c - 5*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticE[2*ArcTa
n[(d^(1/4)*x)/c^(1/4)], 1/2])/(5*b^2*d^(3/4)*Sqrt[c + d*x^4]) + (c^(1/4)*(2*b*c - 5*a*d)*(Sqrt[c] + Sqrt[d]*x^
2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(10*b^2*d^(3/4)*
Sqrt[c + d*x^4]) + (a*(Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[
(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*b^2*c^(1/4)*(b*c + a*
d)*Sqrt[c + d*x^4]) + (a*(Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b])*d^(1/4)*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sq
rt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*b^2*c^(1/4)*(b*c +
a*d)*Sqrt[c + d*x^4]) + (Sqrt[-a]*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^2)*
Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[-1/4*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2/(Sqrt[-a]*S
qrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*b^(5/2)*c^(1/4)*d^(1/4)*(b*c + a*d)*Sqrt[c +
d*x^4]) - (Sqrt[-a]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])^2*(b*c - a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4
)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sqrt[-a]*Sqrt[b]*Sqrt[c]*Sqr
t[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(8*b^(5/2)*c^(1/4)*d^(1/4)*(b*c + a*d)*Sqrt[c + d*x^4])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 489

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Dist[e^n/(b*(m + n*(p +
q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b
*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] &&
GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1231

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(c*d + a*e*q
)/(c*d^2 - a*e^2), Int[1/Sqrt[a + c*x^4], x], x] - Dist[(a*e*(e + d*q))/(c*d^2 - a*e^2), Int[(1 + q*x^2)/((d +
e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0]
&& PosQ[c/a]

Rule 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rubi steps

\begin {align*} \int \frac {x^6 \sqrt {c+d x^4}}{a+b x^4} \, dx &=\frac {x^3 \sqrt {c+d x^4}}{5 b}-\frac {\int \frac {x^2 \left (3 a c+(-2 b c+5 a d) x^4\right )}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx}{5 b}\\ &=\frac {x^3 \sqrt {c+d x^4}}{5 b}-\frac {\int \left (-\frac {(2 b c-5 a d) x^2}{b \sqrt {c+d x^4}}-\frac {5 \left (-a b c+a^2 d\right ) x^2}{b \left (a+b x^4\right ) \sqrt {c+d x^4}}\right ) \, dx}{5 b}\\ &=\frac {x^3 \sqrt {c+d x^4}}{5 b}+\frac {(2 b c-5 a d) \int \frac {x^2}{\sqrt {c+d x^4}} \, dx}{5 b^2}-\frac {(a (b c-a d)) \int \frac {x^2}{\left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx}{b^2}\\ &=\frac {x^3 \sqrt {c+d x^4}}{5 b}+\frac {\left (\sqrt {c} (2 b c-5 a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{5 b^2 \sqrt {d}}-\frac {\left (\sqrt {c} (2 b c-5 a d)\right ) \int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c}}}{\sqrt {c+d x^4}} \, dx}{5 b^2 \sqrt {d}}+\frac {(a (b c-a d)) \int \frac {1}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {c+d x^4}} \, dx}{2 b^{5/2}}-\frac {(a (b c-a d)) \int \frac {1}{\left (\sqrt {-a}+\sqrt {b} x^2\right ) \sqrt {c+d x^4}} \, dx}{2 b^{5/2}}\\ &=\frac {x^3 \sqrt {c+d x^4}}{5 b}+\frac {(2 b c-5 a d) x \sqrt {c+d x^4}}{5 b^2 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x^2\right )}-\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{5 b^2 d^{3/4} \sqrt {c+d x^4}}+\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{10 b^2 d^{3/4} \sqrt {c+d x^4}}+\frac {\left (a \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (b c-a d)\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (\sqrt {-a}-\sqrt {b} x^2\right ) \sqrt {c+d x^4}} \, dx}{2 b^2 (b c+a d)}-\frac {\left (a \sqrt {c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (b c-a d)\right ) \int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c}}}{\left (\sqrt {-a}+\sqrt {b} x^2\right ) \sqrt {c+d x^4}} \, dx}{2 b^2 (b c+a d)}+\frac {\left (a \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt {d} (b c-a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{2 b^{5/2} (b c+a d)}+\frac {\left (a \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt {d} (b c-a d)\right ) \int \frac {1}{\sqrt {c+d x^4}} \, dx}{2 b^{5/2} (b c+a d)}\\ &=\frac {x^3 \sqrt {c+d x^4}}{5 b}+\frac {(2 b c-5 a d) x \sqrt {c+d x^4}}{5 b^2 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x^2\right )}+\frac {(-a)^{3/4} \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 b^{9/4}}+\frac {(-a)^{3/4} \sqrt {-b c+a d} \tan ^{-1}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 b^{9/4}}-\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{5 b^2 d^{3/4} \sqrt {c+d x^4}}+\frac {\sqrt [4]{c} (2 b c-5 a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{10 b^2 d^{3/4} \sqrt {c+d x^4}}+\frac {a \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 b^{5/2} \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}+\frac {a \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{4 b^{5/2} \sqrt [4]{c} (b c+a d) \sqrt {c+d x^4}}+\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^{5/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}-\frac {\sqrt {-a} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2 (b c-a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \Pi \left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}};2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{8 b^{5/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt {c+d x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 10.10, size = 141, normalized size = 0.16 \begin {gather*} \frac {7 a x^3 \left (c+d x^4\right )-7 a c x^3 \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+(2 b c-5 a d) x^7 \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{35 a b \sqrt {c+d x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^6*Sqrt[c + d*x^4])/(a + b*x^4),x]

[Out]

(7*a*x^3*(c + d*x^4) - 7*a*c*x^3*Sqrt[1 + (d*x^4)/c]*AppellF1[3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] +
(2*b*c - 5*a*d)*x^7*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(35*a*b*Sqrt[
c + d*x^4])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.40, size = 421, normalized size = 0.49

method	result	size
risch	\(\frac {x^{3} \sqrt {d \,x^{4}+c}}{5 b}-\frac {\frac {i \left (5 a d -2 b c \right ) \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, \sqrt {d}}-\frac {5 a \left (a d -b c \right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\arctanh \left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha }\right )}{8 b^{2}}}{5 b}\)	\(332\)
elliptic	\(\frac {x^{3} \sqrt {d \,x^{4}+c}}{5 b}+\frac {i \left (-\frac {a d -b c}{b^{2}}-\frac {3 c}{5 b}\right ) \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, \sqrt {d}}+\frac {a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -b c \right ) \left (-\frac {\arctanh \left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{8 b^{3}}\)	\(334\)
default	\(\frac {\frac {x^{3} \sqrt {d \,x^{4}+c}}{5}+\frac {2 i c^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, \sqrt {d}}}{b}-\frac {a \left (\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (a d -b c \right ) \left (-\frac {\arctanh \left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +b c}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +b c}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{8 b^{2}}\right )}{b}\)	\(421\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

1/b*(1/5*x^3*(d*x^4+c)^(1/2)+2/5*I*c^(3/2)/(I/c^(1/2)*d^(1/2))^(1/2)*(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/c^(1
/2)*d^(1/2)*x^2)^(1/2)/(d*x^4+c)^(1/2)/d^(1/2)*(EllipticF(x*(I/c^(1/2)*d^(1/2))^(1/2),I)-EllipticE(x*(I/c^(1/2
)*d^(1/2))^(1/2),I)))-a/b*(I*d^(1/2)/b*c^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2)*(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/
c^(1/2)*d^(1/2)*x^2)^(1/2)/(d*x^4+c)^(1/2)*(EllipticF(x*(I/c^(1/2)*d^(1/2))^(1/2),I)-EllipticE(x*(I/c^(1/2)*d^
(1/2))^(1/2),I))-1/8/b^2*sum((a*d-b*c)/_alpha*(-1/((-a*d+b*c)/b)^(1/2)*arctanh(1/2*(2*_alpha^2*d*x^2+2*c)/((-a
*d+b*c)/b)^(1/2)/(d*x^4+c)^(1/2))+2/(I/c^(1/2)*d^(1/2))^(1/2)*_alpha^3*b/a*(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+
I/c^(1/2)*d^(1/2)*x^2)^(1/2)/(d*x^4+c)^(1/2)*EllipticPi(x*(I/c^(1/2)*d^(1/2))^(1/2),I*c^(1/2)/d^(1/2)*_alpha^2
/a*b,(-I/c^(1/2)*d^(1/2))^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=RootOf(_Z^4*b+a)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^4 + c)*x^6/(b*x^4 + a), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} \sqrt {c + d x^{4}}}{a + b x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(d*x**4+c)**(1/2)/(b*x**4+a),x)

[Out]

Integral(x**6*sqrt(c + d*x**4)/(a + b*x**4), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(d*x^4+c)^(1/2)/(b*x^4+a),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^4 + c)*x^6/(b*x^4 + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6\,\sqrt {d\,x^4+c}}{b\,x^4+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(c + d*x^4)^(1/2))/(a + b*x^4),x)

[Out]

int((x^6*(c + d*x^4)^(1/2))/(a + b*x^4), x)

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