3.4.93 \(\int \frac {1}{x^6 \sqrt {d+e x^2} (a+b x^2+c x^4)} \, dx\) [393]

Optimal. Leaf size=443 \[ -\frac {\sqrt {d+e x^2}}{5 a d x^5}+\frac {b \sqrt {d+e x^2}}{3 a^2 d x^3}+\frac {4 e \sqrt {d+e x^2}}{15 a d^2 x^3}-\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {2 b e \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^3 x}-\frac {c \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {c \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-1/5*(e*x^2+d)^(1/2)/a/d/x^5+1/3*b*(e*x^2+d)^(1/2)/a^2/d/x^3+4/15*e*(e*x^2+d)^(1/2)/a/d^2/x^3-(-a*c+b^2)*(e*x^
2+d)^(1/2)/a^3/d/x-2/3*b*e*(e*x^2+d)^(1/2)/a^2/d^2/x-8/15*e^2*(e*x^2+d)^(1/2)/a/d^3/x-c*arctan(x*(2*c*d-e*(b-(
-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b^2-a*c+b*(-3*a*c+b^2)/(-4*a*c+b^2)^(
1/2))/a^3/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-c*arctan(x*(2*c*d-e*(b+(-4*a*c+b
^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(b^2-a*c-b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a^
3/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]
time = 1.04, antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1317, 277, 270, 1706, 385, 211} \begin {gather*} -\frac {c \left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \text {ArcTan}\left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {c \left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \text {ArcTan}\left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {2 b e \sqrt {d+e x^2}}{3 a^2 d^2 x}+\frac {b \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^3 x}+\frac {4 e \sqrt {d+e x^2}}{15 a d^2 x^3}-\frac {\sqrt {d+e x^2}}{5 a d x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

-1/5*Sqrt[d + e*x^2]/(a*d*x^5) + (b*Sqrt[d + e*x^2])/(3*a^2*d*x^3) + (4*e*Sqrt[d + e*x^2])/(15*a*d^2*x^3) - ((
b^2 - a*c)*Sqrt[d + e*x^2])/(a^3*d*x) - (2*b*e*Sqrt[d + e*x^2])/(3*a^2*d^2*x) - (8*e^2*Sqrt[d + e*x^2])/(15*a*
d^3*x) - (c*(b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*
x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(a^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2
- 4*a*c])*e]) - (c*(b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a
*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(a^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e])

Rule 211

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 1317

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x^2)^q, (f*x)^m/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2
- 4*a*c, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rule 1706

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac {1}{a x^6 \sqrt {d+e x^2}}-\frac {b}{a^2 x^4 \sqrt {d+e x^2}}+\frac {b^2-a c}{a^3 x^2 \sqrt {d+e x^2}}+\frac {-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x^2}{a^3 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac {\int \frac {-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a^3}+\frac {\int \frac {1}{x^6 \sqrt {d+e x^2}} \, dx}{a}-\frac {b \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx}{a^2}+\frac {\left (b^2-a c\right ) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{a^3}\\ &=-\frac {\sqrt {d+e x^2}}{5 a d x^5}+\frac {b \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{a^3 d x}+\frac {\int \left (\frac {-\frac {b c \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-c \left (b^2-a c\right )}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}+\frac {\frac {b c \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-c \left (b^2-a c\right )}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}\right ) \, dx}{a^3}-\frac {(4 e) \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx}{5 a d}+\frac {(2 b e) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{3 a^2 d}\\ &=-\frac {\sqrt {d+e x^2}}{5 a d x^5}+\frac {b \sqrt {d+e x^2}}{3 a^2 d x^3}+\frac {4 e \sqrt {d+e x^2}}{15 a d^2 x^3}-\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {2 b e \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {\left (c \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{a^3}-\frac {\left (c \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{a^3}+\frac {\left (8 e^2\right ) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{15 a d^2}\\ &=-\frac {\sqrt {d+e x^2}}{5 a d x^5}+\frac {b \sqrt {d+e x^2}}{3 a^2 d x^3}+\frac {4 e \sqrt {d+e x^2}}{15 a d^2 x^3}-\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {2 b e \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^3 x}-\frac {\left (c \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{a^3}-\frac {\left (c \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{a^3}\\ &=-\frac {\sqrt {d+e x^2}}{5 a d x^5}+\frac {b \sqrt {d+e x^2}}{3 a^2 d x^3}+\frac {4 e \sqrt {d+e x^2}}{15 a d^2 x^3}-\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {2 b e \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^3 x}-\frac {c \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {c \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A]
time = 11.25, size = 383, normalized size = 0.86 \begin {gather*} -\frac {\frac {15 \left (b^2-a c\right ) \sqrt {d+e x^2}}{d x}-\frac {5 a b \left (d-2 e x^2\right ) \sqrt {d+e x^2}}{d^2 x^3}+\frac {a^2 \sqrt {d+e x^2} \left (3 d^2-4 d e x^2+8 e^2 x^4\right )}{d^3 x^5}+\frac {15 c \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {15 c \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}{15 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

-1/15*((15*(b^2 - a*c)*Sqrt[d + e*x^2])/(d*x) - (5*a*b*(d - 2*e*x^2)*Sqrt[d + e*x^2])/(d^2*x^3) + (a^2*Sqrt[d
+ e*x^2]*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^4))/(d^3*x^5) + (15*c*(b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*
ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b - S
qrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (15*c*(b^2 - a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 -
4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sq
rt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a^3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 349, normalized size = 0.79

method	result	size
risch	\(-\frac {\sqrt {e \,x^{2}+d}\, \left (8 a^{2} e^{2} x^{4}+10 a b d e \,x^{4}-15 a c \,d^{2} x^{4}+15 b^{2} d^{2} x^{4}-4 a^{2} d e \,x^{2}-5 a b \,d^{2} x^{2}+3 a^{2} d^{2}\right )}{15 d^{3} a^{3} x^{5}}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \left (a c -b^{2}\right ) \textit {\_R}^{2}+2 \left (4 a b c e -a \,c^{2} d -2 b^{3} e +b^{2} c d \right ) \textit {\_R} +a \,c^{2} d^{2}-b^{2} c \,d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 a^{3}}\)	\(305\)
default	\(-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (c \left (a c -b^{2}\right ) \textit {\_R}^{2}+2 \left (4 a b c e -a \,c^{2} d -2 b^{3} e +b^{2} c d \right ) \textit {\_R} +a \,c^{2} d^{2}-b^{2} c \,d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 a^{3}}+\frac {-\frac {\sqrt {e \,x^{2}+d}}{5 d \,x^{5}}-\frac {4 e \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{5 d}}{a}-\frac {\left (-a c +b^{2}\right ) \sqrt {e \,x^{2}+d}}{a^{3} d x}-\frac {b \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{a^{2}}\)	\(349\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a^3*e^(1/2)*sum((c*(a*c-b^2)*_R^2+2*(4*a*b*c*e-a*c^2*d-2*b^3*e+b^2*c*d)*_R+a*c^2*d^2-b^2*c*d^2)/(_R^3*c+3
*_R^2*b*e-3*_R^2*c*d+8*_R*a*e^2-4*_R*b*d*e+3*_R*c*d^2+b*d^2*e-c*d^3)*ln(((e*x^2+d)^(1/2)-e^(1/2)*x)^2-_R),_R=R
ootOf(c*_Z^4+(4*b*e-4*c*d)*_Z^3+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z+d^4*c))+1/a*(-1/5/d/x^5
*(e*x^2+d)^(1/2)-4/5*e/d*(-1/3/d/x^3*(e*x^2+d)^(1/2)+2/3*e/d^2/x*(e*x^2+d)^(1/2)))-(-a*c+b^2)*(e*x^2+d)^(1/2)/
a^3/d/x-b/a^2*(-1/3/d/x^3*(e*x^2+d)^(1/2)+2/3*e/d^2/x*(e*x^2+d)^(1/2))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10039 vs. \(2 (394) = 788\).
time = 157.67, size = 10039, normalized size = 22.66 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(15*sqrt(1/2)*a^3*d^3*x^5*sqrt(-((b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*d - (b^8 - 8*a*b^6*
c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*a^4*c^4)*e + ((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e
+ (a^8*b^2 - 4*a^9*c)*e^2)*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6
- 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5
- 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c
^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^
3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)
*e^4)))/((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2))*log(-((b^7*c^5 -
5*a*b^5*c^6 + 6*a^2*b^3*c^7 - a^3*b*c^8)*d^2*x^2 + 4*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^3*c^6 - 4*a^4*b*c^7
)*x^2*e^2 - 2*(a*b^6*c^5 - 5*a^2*b^4*c^6 + 6*a^3*b^2*c^7 - a^4*c^8)*d^2 + 2*sqrt(1/2)*((a*b^10*c^2 - 10*a^2*b^
8*c^3 + 35*a^3*b^6*c^4 - 51*a^4*b^4*c^5 + 29*a^5*b^2*c^6 - 4*a^6*c^7)*d^2*x - (2*a*b^11*c - 22*a^2*b^9*c^2 + 8
8*a^3*b^7*c^3 - 155*a^4*b^5*c^4 + 114*a^5*b^3*c^5 - 24*a^6*b*c^6)*d*x*e + (a*b^12 - 12*a^2*b^10*c + 54*a^3*b^8
*c^2 - 112*a^4*b^6*c^3 + 104*a^5*b^4*c^4 - 32*a^6*b^2*c^5)*x*e^2 - ((a^8*b^5*c^2 - 7*a^9*b^3*c^3 + 12*a^10*b*c
^4)*d^3*x - (2*a^8*b^6*c - 15*a^9*b^4*c^2 + 30*a^10*b^2*c^3 - 8*a^11*c^4)*d^2*x*e + (a^8*b^7 - 7*a^9*b^5*c + 1
1*a^10*b^3*c^2 + 4*a^11*b*c^3)*d*x*e^2 - (a^9*b^6 - 8*a^10*b^4*c + 18*a^11*b^2*c^2 - 8*a^12*c^3)*x*e^3)*sqrt((
(b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 -
2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*
d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*
c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c -
8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)))*sqrt(x^2*e + d)*sqrt(-((b
^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*
a^4*c^4)*e + ((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)*sqrt(((b^12*c
^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^1
3*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (
b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^
2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*
c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)))/((a^7*b^2*c - 4*a^8*c^2)*d^2 - (
a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)) - ((b^8*c^4 - 2*a*b^6*c^5 - 10*a^2*b^4*c^6 + 20*a^3*b^2*c
^7 - 4*a^4*c^8)*d*x^2 - 2*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^3*c^6 - 4*a^4*b*c^7)*d)*e + ((a^7*b^2*c^5 - 4*
a^8*c^6)*d^3*x^2 - (a^7*b^3*c^4 - 4*a^8*b*c^5)*d^2*x^2*e + (a^8*b^2*c^4 - 4*a^9*c^5)*d*x^2*e^2)*sqrt(((b^12*c^
2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13
*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b
^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2
)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c
^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)))/x^2) - 15*sqrt(1/2)*a^3*d^3*x^5*s
qrt(-((b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*
c^3 + 2*a^4*c^4)*e + ((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)*sqrt(
((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2
- 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)
*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2
*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c -
8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)))/((a^7*b^2*c - 4*a^8*c^2)
*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2))*log(-((b^7*c^5 - 5*a*b^5*c^6 + 6*a^2*b^3*c^7 - a^
3*b*c^8)*d^2*x^2 + 4*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^3*c^6 - 4*a^4*b*c^7)*x^2*e^2 - 2*(a*b^6*c^5 - 5*a^2
*b^4*c^6 + 6*a^3*b^2*c^7 - a^4*c^8)*d^2 - 2*sqrt(1/2)*((a*b^10*c^2 - 10*a^2*b^8*c^3 + 35*a^3*b^6*c^4 - 51*a^4*
b^4*c^5 + 29*a^5*b^2*c^6 - 4*a^6*c^7)*d^2*x - (...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{6} \sqrt {d + e x^{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**6*sqrt(d + e*x**2)*(a + b*x**2 + c*x**4)), x)

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^6\,\sqrt {e\,x^2+d}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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