3.4.86 \(\int \frac {x^8}{\sqrt {d+e x^2} (a+b x^2+c x^4)} \, dx\) [386]

Optimal. Leaf size=479 \[ -\frac {3 d x \sqrt {d+e x^2}}{8 c e^2}-\frac {b x \sqrt {d+e x^2}}{2 c^2 e}+\frac {x^3 \sqrt {d+e x^2}}{4 c e}-\frac {\left (b^3-2 a b c-\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b^3-2 a b c+\frac {b^4-4 a b^2 c+2 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^3 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 c e^{5/2}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 e^{3/2}}+\frac {\left (b^2-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3 \sqrt {e}} \]

[Out]

3/8*d^2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/c/e^(5/2)+1/2*b*d*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/c^2/e^(3/2)+(-
a*c+b^2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/c^3/e^(1/2)-3/8*d*x*(e*x^2+d)^(1/2)/c/e^2-1/2*b*x*(e*x^2+d)^(1/2)/
c^2/e+1/4*x^3*(e*x^2+d)^(1/2)/c/e-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^3-2*a*b*c+(-2*a^2*c^2+4*a*b^2*c-b^4)/(-4*a*c+b^2)^(1/2))/c^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)-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^3-2*a*b*c+(2*a^2*c^2-4*a*b^2*c+b^4)/(-4*a*c+b^2)^(1/2))/c^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 = 1.25, antiderivative size = 479, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1317, 223, 212, 327, 1706, 385, 211} \begin {gather*} -\frac {\left (-\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\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 )}{c^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (\frac {2 a^2 c^2-4 a b^2 c+b^4}{\sqrt {b^2-4 a c}}-2 a b c+b^3\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 )}{c^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 ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3 \sqrt {e}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 e^{3/2}}-\frac {b x \sqrt {d+e x^2}}{2 c^2 e}+\frac {3 d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 c e^{5/2}}-\frac {3 d x \sqrt {d+e x^2}}{8 c e^2}+\frac {x^3 \sqrt {d+e x^2}}{4 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

(-3*d*x*Sqrt[d + e*x^2])/(8*c*e^2) - (b*x*Sqrt[d + e*x^2])/(2*c^2*e) + (x^3*Sqrt[d + e*x^2])/(4*c*e) - ((b^3 -
2*a*b*c - (b^4 - 4*a*b^2*c + 2*a^2*c^2)/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])])/(c^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 -
4*a*c])*e]) - ((b^3 - 2*a*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(c^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) + (3*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8*c*e^(5/2)) + (b*d*ArcTanh
[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^2*e^(3/2)) + ((b^2 - a*c)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c^3*Sqrt[
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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 327

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

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

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Mathematica [A]
time = 11.29, size = 461, normalized size = 0.96 \begin {gather*} \frac {-\frac {4 b c x \sqrt {d+e x^2}}{e}+\frac {2 c^2 x^3 \sqrt {d+e x^2}}{e}-\frac {8 \left (b^3-2 a b c-\frac {b^4-4 a b^2 c+2 a^2 c^2}{\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 {8 \left (b^3-2 a b c+\frac {b^4-4 a b^2 c+2 a^2 c^2}{\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}}+\frac {4 b c d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}+\frac {8 \left (b^2-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}+\frac {3 c^2 d \left (-\sqrt {e} x \sqrt {d+e x^2}+d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )}{e^{5/2}}}{8 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/(Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 375, normalized size = 0.78

method	result	size
risch	\(-\frac {x \left (-2 c e \,x^{2}+4 e b +3 c d \right ) \sqrt {e \,x^{2}+d}}{8 e^{2} c^{2}}-\frac {\ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) a}{c^{2} \sqrt {e}}+\frac {\ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) b^{2}}{c^{3} \sqrt {e}}+\frac {\ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) b d}{2 c^{2} e^{\frac {3}{2}}}+\frac {3 \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) d^{2}}{8 c \,e^{\frac {5}{2}}}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (b \left (2 a c -b^{2}\right ) \textit {\_R}^{2}+2 \left (2 a^{2} c e -2 a \,b^{2} e -2 a b c d +b^{3} d \right ) \textit {\_R} +2 a b c \,d^{2}-b^{3} d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 c^{3}}\)	\(353\)
default	\(-\frac {-c^{2} \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )+b c \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )+\frac {a c \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{\sqrt {e}}-\frac {b^{2} \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{\sqrt {e}}}{c^{3}}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (b \left (2 a c -b^{2}\right ) \textit {\_R}^{2}+2 \left (2 a^{2} c e -2 a \,b^{2} e -2 a b c d +b^{3} d \right ) \textit {\_R} +2 a b c \,d^{2}-b^{3} d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 c^{3}}\)	\(375\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/c^3*(-c^2*(1/4*x^3/e*(e*x^2+d)^(1/2)-3/4*d/e*(1/2*x/e*(e*x^2+d)^(1/2)-1/2*d/e^(3/2)*ln(e^(1/2)*x+(e*x^2+d)^
(1/2))))+b*c*(1/2*x/e*(e*x^2+d)^(1/2)-1/2*d/e^(3/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2)))+a*c*ln(e^(1/2)*x+(e*x^2+d)^
(1/2))/e^(1/2)-b^2*ln(e^(1/2)*x+(e*x^2+d)^(1/2))/e^(1/2))-1/2/c^3*e^(1/2)*sum((b*(2*a*c-b^2)*_R^2+2*(2*a^2*c*e
-2*a*b^2*e-2*a*b*c*d+b^3*d)*_R+2*a*b*c*d^2-b^3*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=RootOf(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))

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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^8/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^8/((c*x^4 + b*x^2 + a)*sqrt(x^2*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9162 vs. \(2 (408) = 816\).
time = 135.04, size = 9162, normalized size = 19.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/16*(4*sqrt(1/2)*c^3*sqrt(-((b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*d - (a*b^6 - 6*a^2*b^4*c + 9*a^3
*b^2*c^2 - 2*a^4*c^3)*e + ((b^2*c^7 - 4*a*c^8)*d^2 - (b^3*c^6 - 4*a*b*c^7)*d*e + (a*b^2*c^6 - 4*a^2*c^7)*e^2)*
sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 -
2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8
*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/((b^2*c^14 - 4*a*c^15)*d^4 - 2*(b^3*c^13 -
4*a*b*c^14)*d^3*e + (b^4*c^12 - 2*a*b^2*c^13 - 8*a^2*c^14)*d^2*e^2 - 2*(a*b^3*c^12 - 4*a^2*b*c^13)*d*e^3 + (a^
2*b^2*c^12 - 4*a^3*c^13)*e^4)))/((b^2*c^7 - 4*a*c^8)*d^2 - (b^3*c^6 - 4*a*b*c^7)*d*e + (a*b^2*c^6 - 4*a^2*c^7)
*e^2))*e^3*log(((a^3*b^7 - 5*a^4*b^5*c + 6*a^5*b^3*c^2 - a^6*b*c^3)*d^2*x^2 + 4*(a^5*b^5 - 4*a^6*b^3*c + 3*a^7
*b*c^2)*x^2*e^2 - 2*(a^4*b^6 - 5*a^5*b^4*c + 6*a^6*b^2*c^2 - a^7*c^3)*d^2 + 2*sqrt(1/2)*((b^11 - 11*a*b^9*c +
44*a^2*b^7*c^2 - 77*a^3*b^5*c^3 + 54*a^4*b^3*c^4 - 8*a^5*b*c^5)*d^2*x - (2*a*b^10 - 20*a^2*b^8*c + 70*a^3*b^6*
c^2 - 101*a^4*b^4*c^3 + 53*a^5*b^2*c^4 - 4*a^6*c^5)*d*x*e + (a^2*b^9 - 9*a^3*b^7*c + 27*a^4*b^5*c^2 - 31*a^5*b
^3*c^3 + 12*a^6*b*c^4)*x*e^2 - ((b^6*c^7 - 8*a*b^4*c^8 + 18*a^2*b^2*c^9 - 8*a^3*c^10)*d^3*x - (b^7*c^6 - 7*a*b
^5*c^7 + 11*a^2*b^3*c^8 + 4*a^3*b*c^9)*d^2*x*e + (2*a*b^6*c^6 - 15*a^2*b^4*c^7 + 30*a^3*b^2*c^8 - 8*a^4*c^9)*d
*x*e^2 - (a^2*b^5*c^6 - 7*a^3*b^3*c^7 + 12*a^4*b*c^8)*x*e^3)*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a
^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^
4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*
a^6*b^2*c^4)*e^2)/((b^2*c^14 - 4*a*c^15)*d^4 - 2*(b^3*c^13 - 4*a*b*c^14)*d^3*e + (b^4*c^12 - 2*a*b^2*c^13 - 8*
a^2*c^14)*d^2*e^2 - 2*(a*b^3*c^12 - 4*a^2*b*c^13)*d*e^3 + (a^2*b^2*c^12 - 4*a^3*c^13)*e^4)))*sqrt(x^2*e + d)*s
qrt(-((b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*d - (a*b^6 - 6*a^2*b^4*c + 9*a^3*b^2*c^2 - 2*a^4*c^3)*e
+ ((b^2*c^7 - 4*a*c^8)*d^2 - (b^3*c^6 - 4*a*b*c^7)*d*e + (a*b^2*c^6 - 4*a^2*c^7)*e^2)*sqrt(((b^12 - 10*a*b^10
*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*
c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6
*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/((b^2*c^14 - 4*a*c^15)*d^4 - 2*(b^3*c^13 - 4*a*b*c^14)*d^3*e + (b^
4*c^12 - 2*a*b^2*c^13 - 8*a^2*c^14)*d^2*e^2 - 2*(a*b^3*c^12 - 4*a^2*b*c^13)*d*e^3 + (a^2*b^2*c^12 - 4*a^3*c^13
)*e^4)))/((b^2*c^7 - 4*a*c^8)*d^2 - (b^3*c^6 - 4*a*b*c^7)*d*e + (a*b^2*c^6 - 4*a^2*c^7)*e^2)) - ((5*a^4*b^6 -
24*a^5*b^4*c + 27*a^6*b^2*c^2 - 4*a^7*c^3)*d*x^2 - 2*(a^5*b^5 - 4*a^6*b^3*c + 3*a^7*b*c^2)*d)*e - ((a^3*b^2*c^
7 - 4*a^4*c^8)*d^3*x^2 - (a^3*b^3*c^6 - 4*a^4*b*c^7)*d^2*x^2*e + (a^4*b^2*c^6 - 4*a^5*c^7)*d*x^2*e^2)*sqrt(((b
^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^
11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8
*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/((b^2*c^14 - 4*a*c^15)*d^4 - 2*(b^3*c^13 - 4*a*b*c^
14)*d^3*e + (b^4*c^12 - 2*a*b^2*c^13 - 8*a^2*c^14)*d^2*e^2 - 2*(a*b^3*c^12 - 4*a^2*b*c^13)*d*e^3 + (a^2*b^2*c^
12 - 4*a^3*c^13)*e^4)))/x^2) - 4*sqrt(1/2)*c^3*sqrt(-((b^7 - 7*a*b^5*c + 14*a^2*b^3*c^2 - 7*a^3*b*c^3)*d - (a*
b^6 - 6*a^2*b^4*c + 9*a^3*b^2*c^2 - 2*a^4*c^3)*e + ((b^2*c^7 - 4*a*c^8)*d^2 - (b^3*c^6 - 4*a*b*c^7)*d*e + (a*b
^2*c^6 - 4*a^2*c^7)*e^2)*sqrt(((b^12 - 10*a*b^10*c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^4*c^4 - 12*a^5
*b^2*c^5 + a^6*c^6)*d^2 - 2*(a*b^11 - 9*a^2*b^9*c + 29*a^3*b^7*c^2 - 40*a^4*b^5*c^3 + 22*a^5*b^3*c^4 - 3*a^6*b
*c^5)*d*e + (a^2*b^10 - 8*a^3*b^8*c + 22*a^4*b^6*c^2 - 24*a^5*b^4*c^3 + 9*a^6*b^2*c^4)*e^2)/((b^2*c^14 - 4*a*c
^15)*d^4 - 2*(b^3*c^13 - 4*a*b*c^14)*d^3*e + (b^4*c^12 - 2*a*b^2*c^13 - 8*a^2*c^14)*d^2*e^2 - 2*(a*b^3*c^12 -
4*a^2*b*c^13)*d*e^3 + (a^2*b^2*c^12 - 4*a^3*c^13)*e^4)))/((b^2*c^7 - 4*a*c^8)*d^2 - (b^3*c^6 - 4*a*b*c^7)*d*e
+ (a*b^2*c^6 - 4*a^2*c^7)*e^2))*e^3*log(((a^3*b^7 - 5*a^4*b^5*c + 6*a^5*b^3*c^2 - a^6*b*c^3)*d^2*x^2 + 4*(a^5*
b^5 - 4*a^6*b^3*c + 3*a^7*b*c^2)*x^2*e^2 - 2*(a^4*b^6 - 5*a^5*b^4*c + 6*a^6*b^2*c^2 - a^7*c^3)*d^2 - 2*sqrt(1/
2)*((b^11 - 11*a*b^9*c + 44*a^2*b^7*c^2 - 77*a^3*b^5*c^3 + 54*a^4*b^3*c^4 - 8*a^5*b*c^5)*d^2*x - (2*a*b^10 - 2
0*a^2*b^8*c + 70*a^3*b^6*c^2 - 101*a^4*b^4*c^3 + 53*a^5*b^2*c^4 - 4*a^6*c^5)*d*x*e + (a^2*b^9 - 9*a^3*b^7*c +
27*a^4*b^5*c^2 - 31*a^5*b^3*c^3 + 12*a^6*b*c^4)*x*e^2 - ((b^6*c^7 - 8*a*b^4*c^8 + 18*a^2*b^2*c^9 - 8*a^3*c^10)
*d^3*x - (b^7*c^6 - 7*a*b^5*c^7 + 11*a^2*b^3*c^8 + 4*a^3*b*c^9)*d^2*x*e + (2*a*b^6*c^6 - 15*a^2*b^4*c^7 + 30*a
^3*b^2*c^8 - 8*a^4*c^9)*d*x*e^2 - (a^2*b^5*c^6 - 7*a^3*b^3*c^7 + 12*a^4*b*c^8)*x*e^3)*sqrt(((b^12 - 10*a*b^10*
c + 37*a^2*b^8*c^2 - 62*a^3*b^6*c^3 + 46*a^4*b^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\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(x**8/(c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)

[Out]

Integral(x**8/(sqrt(d + e*x**2)*(a + b*x**2 + c*x**4)), x)

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Giac [A]
time = 6.42, size = 105, normalized size = 0.22 \begin {gather*} \frac {1}{8} \, \sqrt {x^{2} e + d} {\left (\frac {2 \, x^{2} e^{\left (-1\right )}}{c} - \frac {{\left (3 \, c^{5} d e + 4 \, b c^{4} e^{2}\right )} e^{\left (-3\right )}}{c^{6}}\right )} x - \frac {{\left (3 \, c^{2} d^{2} + 4 \, b c d e + 8 \, b^{2} e^{2} - 8 \, a c e^{2}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2}\right )}{16 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

1/8*sqrt(x^2*e + d)*(2*x^2*e^(-1)/c - (3*c^5*d*e + 4*b*c^4*e^2)*e^(-3)/c^6)*x - 1/16*(3*c^2*d^2 + 4*b*c*d*e +
8*b^2*e^2 - 8*a*c*e^2)*e^(-5/2)*log((x*e^(1/2) - sqrt(x^2*e + d))^2)/c^3

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^8}{\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(x^8/((d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)),x)

[Out]

int(x^8/((d + e*x^2)^(1/2)*(a + b*x^2 + c*x^4)), x)

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